TEST BANK for Finite Mathematics & Its Applications plus MyLab Math 12th Edition by digitaldownload87

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Plot the points on the Cartesian coordinate system provided. 1) A(5, 5), B(-4, 3) 8

8 x

-8

A) 8

y A B

8 x

-8

B) 8

y A

8 x

-8

C) 8

y A

8 x

-8 B

-8

D) 8

A B

8 x

-8

Answer: B 2) A(2, -6), B(-1, 2) 8

8 x

-8

A) 8

8 x

-8

A -8

B) 8

y A

8 x

-8

C) 8

y A

8 x

-8

D) 8

8 x

-8

A -8

Answer: D 3) A(-6, -4), B(-1, 1) 8

8 x

-8

A) 8

B 8 x

-8

B) 8

B 8 x

-8 A

-8

C) 8

B 8 x

-8

D) 8

B 8 x

-8 A

-8

Answer: D

4) A(1, 3), B(3, -5) y

8 x

-8

A) 8

A 8 x

-8

B) 8

8 x

-8

C) 8

y B A

8 x

-8

D) 8

8 x

-8

Answer: D 5) A(2, 5), B(-4, -3) 8

8 x

-8

A) 8

8 x

-8

B) 8

y A B

8 x

-8

C) 8

y A

8 x

-8 B

-8

D) 8

8 x

-8 B A

-8

Answer: C 6) A(-

4 , -2), B(-4, 6) 5 8

8 x

-8

A) 8

-8

8 x

-8

B) 8

y B

-8

8 x

-8

C) 8

-8

8 x

-8

D) 8

-8

8 x

-8

Answer: C

7) A(-2, 0), B (0, 6) 8

8 x

-8

A) 8

B 8 x

-8 A

-8

B) 8

y B

8 x

-8 A

-8

C) 8

y B

A 8 x

-8

D) 8

B 8 x

-8

Answer: C Give the coordinates for the points labeled on the graph. 8) y B A

A) A(7, 4); B(-3, 5) B) A(7, 4); B(5, -3) C) A(4, 30); B(5, -3) D) A(7, 5); B(4, 5) Answer: A

9) y

x D

A) C(-6, 2); D(-3, 2) B) C(2, 4); D(-3, 2) C) C(-6, 2); D(2, -3) D) C(-6, -3); D(2, -3) Answer: C 10) y

x G

A) E(2, 10); G(-2, -6) B) E(-3, -2); G(2, -2) C) E(-3, 2); G(-2, -6) D) E(-3, 2); G(-6, -2) Answer: D

11) y

A) A(2, -2); B(1, -4) B) A(2, 2); B(0, -4) C) A(-2, 2); B(0, 4) D) A(2, 2); B(0, 4) Answer: B 12) Determine which of the following points lie on the graph of the linear equation 4x + 5y = 20. A) (-5, 8) B) (5, 4) C) (0, -4) D) (-4, 20) E) (0, 0) Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 13) On the line 2y = 6, is there a point whose first coordinate is 4? If so, name the point. Answer: (4, 3) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the given point is on the graph of the equation. 14) 4x + 5y = 8; (0, 2) A) No B) Yes Answer: A 15)

3 x + y = 1; (-4, 4) 4 A) Yes B) No Answer: A

16) -

2 x + y = -1; (-1, 0) 3 A) No B) Yes

Answer: A The equation is in the form y = mx + b. Identify m and b. 5 9 17) y = x 2 2 A) m =

5 9 ;b=2 2

B) m =

5 9 ;b= 2 2

C) m = D) m =

9 5 ;b= 2 2

Answer: A 18) y = x - 6 A) m = -6; b = 1 B) m = 1; b = -6 C) m = -6; b = -1 D) m = 0; b = 6 Answer: B 19) y = -4x - 5 A) m = -4; b = -5 B) m = 5; b = -4 C) m = -5; b = -4 D) m = -4; b = 5 Answer: A 20) y = 15 A) m = 15; b = 15 B) m = 15; b = 0 C) m = 0; b = 15 D) m = 0; b =0 Answer: C 21) Find the standard form of the linear equation x =3y - 4. 1 4 A) y = x + 3 3 B) x = y -4 C) x -3 y =- 4 D) 3y = -x + 4 E) none of these Answer: A

22) Find the standard form of the linear equation 4x - 7y = 14. 4 A) y = x - 2 7 B) 7y = -4x + 14 2 1 C) x - y = 1 7 2 D) 4x = 7y + 14 E) none of these Answer: A 23) Find the standard form of the equation -2x + 3y = 6. 2 A) y = x + 6 3 B) y = -

2 x+6 3

C) y = -

2 x+2 3

D) y =

2 x+2 3

E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 24) Write the equation of the line x +3y = 1 in standard form. Answer: y = -

1 1 x+ 3 3

25) Write the equation of the line -4x + 2y = 5 in standard form. Answer: y = 2x +

5 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 26) Which of the following equations describe the same line as the equation 3x + 4y = 5? 3 A) y = x + 5 4 B) 5 - 3x - 4y = 0 C) 6x + 8y = 5 3 5 D) y = x + 4 4 E) none of these Answer: B

Write the equation in slope-intercept form. 27) 16x + 10y = 5 A) y = 16x - 5 8 1 B) y = x 5 2 C) y = D) y =

8 1 x+ 5 2

Answer: C 28) 4x - 8y = 7 1 7 A) y = x + 2 8 B) y = 2x +

7 4

C) y = 4x - 7 1 7 D) y = x 2 8 Answer: D 29) x - 5y = 9 A) y = 5x - 9 B) y = x -

9 5

C) y =

1 9 x5 5

D) y =

1 x-9 5

Answer: C 30) Find the x-intercept of the line y = 8x - 12. 3 A) , 0 2 B) (8, 0) C) (0, -12) 3 D) 0, 2 E) none of these Answer: A

31) Find the x-intercept of the line y = 2x + 5. 5 A) - , 0 2 B) (0, 5) C) (5, 0) D) 0, -

5 2

E) none of these Answer: A 32) Find the y-intercept of the line y = 0.5x + 4. A) (0, 4) B) (1, 0) C) (4, 0) D) (0, 1) E) none of these Answer: A 33) Find the y-intercept of the line x-y = 3. A) (3, 0) B) (0, 3) C) (-3, 0) D) (0, -3) E) none of these Answer: D 34) Find the y-intercept of the line 2y -x =6. A) (3, 0) B) (-6, 0) C) (0, 3) D) (0,-6) E) none of these Answer: C 35) Find the y-intercept of the line y = 0.5x + 4. A) (1, 0) B) (0, 4) C) (4, 0) D) (0, 1) E) none of these. Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 36) Find the x-intercept and y-intercept for the line y = -2x + 6. Answer: x-intercept: (3, 0) y-intercept: (0, 6) 37) Find the x-intercept and y-intercept for the line y = 2. Answer: x-intercept: none y-intercept: (0, 2) 18

38) Find the x-intercept and y-intercept for the line x = -3. Answer: x-intercept: (-3, 0) y-intercept: none 39) Find the x-intercept and y-intercept for the line y = 5x. Answer: x-intercept: (0, 0) y-intercept: (0, 0) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 40) The coordinates of the x-intercept are the same as the coordinates of the y-intercept on the graph of the line whose equation is A) x - 3y = 0 B) 2x + y = 2 C) 2x + 2y = 8 D) 2x - 3y = 3 E) none of these Answer: A Find the intercepts for the equation. 41) y = -x + 5 A) x-intercept: (2, 0); y-intercept: (0, 3) B) x-intercept: (0, 0); y-intercept: (0, 5) C) x-intercept: (5, 0); y-intercept: (0, 5) D) x-intercept: (5, 0); y-intercept: (0, -5) Answer: C Find the x- and y-intercepts for the equation. 42) y = 5x - 3 3 A) x-intercept: , 0 , y-intercept: (0, -3) 5 B) x-intercept: (-3, 0) , y-intercept: 0, C) x-intercept:

3 5

3 , 0 , y-intercept: (0, 3) 5

D) x-intercept: -

3 , 0 , y-intercept: (0, -3) 5

Answer: A 43) y = 5x A) x-intercept: (0, 0), y-intercept: (0, 0) 1 B) x-intercept: (0, 0), y-intercept: (0, ) 5 C) x-intercept: (5, 0), y-intercept: (0, 0) D) x-intercept: (-5, 0), y-intercept: (0, 0) Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the graph of the linear equation. Label the coordinates of the y-intercept and the x-intercept. 44) 2x - 5y = -10 y 10

-10

-5

-10

Answer: y 10

5 (0, 2) (-5, 0) -10

-5

-10

45) y = -2x + 3 y 10

-10

-5

-10

Answer: y 10

5 (0, 3) (3/2, 0) -10

-5

-10

46) 3y +2x = 6 y 10

-10

-5

-10

Answer: y 10

5 (0, 2) -10

-5

(3, 0) 5

-5

-10

47) 5x - 10y = 20 y 10

-10

-5

-10

Answer: y 10

5 (4, 0) -10

-5

(0, -2) -5

-10

48) y - 3x = 0 y 10

-10

-5

-10

Answer: y 10

5 (0, 0) -10

-5

-10

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the equation. 1 49) y = - x - 4 2 y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

8 10

-4 -6 -8 -10

A) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

B) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

C) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

D) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

8 10

-4 -6 -8 -10

Answer: C 50) 5y + 7x = 5 y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

A) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

B) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

C) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

D) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

Answer: B

51) 4x - 20y = 0 y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

8 10

-4 -6 -8 -10

A) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

B) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

C) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

8 10

-4 -6 -8 -10

D) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

Answer: C 52) x = -2 y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

A) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

B) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

C) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

D) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

8 10

-4 -6 -8 -10

Answer: C 53) y = 4 y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

A) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

B) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

-4 -6 -8 -10

C) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

D) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10

Answer: B Graph the linear equation.

54) y = 4x y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

5 -5

-10

Answer: A Choose the graph that matches the equation. 55) 2x - y = 4 A) y 10

-10

-5 -5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5

-10

D) y 10

-10

-5 -5

-10

Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 56) The value y of a machine (in dollars) is known to depreciate linearly with time x (measured in years from the time it was bought new). Suppose that y is related to x by the equation y = 2000 - 200x. (a) Sketch the graph of this linear equation for 0 ≤ x ≤ 10. (b) What is the value of the machine when it is 5 years old? (c) What is the economic interpretation of the y-intercept of the graph? (d) When will the value of the machine reach the scrap value of $400? y

Answer: (a) y 2000

(b) $1000 (c) the value of the machine when it is new (d) 8 years

57) A towing company charges a fixed amount in addition to a per-mile charge to tow a car. The total charge y in dollars is related to the towing distance x in miles by the equation y = 30 + 2x. (a) Sketch the graph of this equation for 0 ≤ x ≤ 25. (b) What is the charge of towing a car for 15 miles? (c) If the company charged a customer $50 to tow his car, what was the corresponding towing distance? y 100 80 60 40 20

25 x

Answer: (a) y 100 80 60 40 20

25 x

(b) $60 (c) 10 miles

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 58) The value, v, in hundreds of dollars, of Juan's television is approximated by v = -0.50t + 8, where t is the number of years since he first bought the television. Graph the equation and use the graph to estimate the value of the television 4 years after it was purchased. v

1000 900 800 700 600 500 400 300 200 100

9 t

A) $720 B) $600 C) $400 D) $1000 Answer: B 59) During the month of January, the depth, d, of snow in inches at the base of one ski resort could be approximated by d = -2t + 68, where t is the number of days since December 31st. Graph the equation and use the graph to estimate the depth of snow on January 28th. d 65 60 55 50 45 40 35 30 25 20 15 10 5 4

12 16 20 24 28 32 t

A) 20 in. B) 17 in. C) 12 in. D) 40 in. Answer: C

60) The cost, T, in hundreds of dollars, of tuition at one community college is given by T = 3 + 1.25c, where c is the number of credits for which a student registers. Graph the equation and use the graph to estimate the cost of tuition if a student registers for 8 credits. 20 18 16 14 12 10 8 6 4 2

1 2 3 4 5 6 7 8 9 10 11 c

A) $2200 B) $1800 C) $1000 D) $1300 Answer: D Find an equation for the line with the given conditions. 61) y-intercept (0, 7), x-intercept (-6, 0) 6 A) y = x - 6 7 B) y =

7 x+7 6

C) y = -

7 x+7 6

D) y = -

6 x-6 7

Answer: B The general form of the equation of a line is cx + dy = e, where not both c and d are 0. Find the general form of the equation. 62) y = 9x + 5 A) -9x - y = 5 5 B) x - y = 9 C) 9x - y = -5 D) 9x + y = 5 Answer: C 63) y =

2 x-5 3

A) 2x + 3y = 10 B) 3x - 2y = -15 C) 2x - 3y = 15 D) 2x + 3y = -5 Answer: C

64) Find the slope of the line 3x - 2y + 12 = 0. A) -2 3 B) 2 C) -

2 3

D) 3 E) none of these Answer: B 65) Find the slope of the line x= -2. A) undefined 1 B) 2 C) -2 D) 0 E) none of these Answer: A 66) Find the slope of the line y = 3. A) 0 B) 1 C) 3 1 D) 3 E) none of these Answer: A 67) Find the slope of the line y - 2 = 3(x + 5). 3 A) 2 B) -2 C) 3 3 D) 2 E) none of these Answer: C 68) Find the slope of the line y + 3 = -(x - 2). A) 1 B) -1 3 C) 2 D) 3 E) none of these Answer: B

69) Find the slope of the line 5y + 7x - 43 = 0. 7 A) 5 B) 7 43 C) 5 D) -7 E) none of these Answer: A 70) Find the slope of the line passing through the points (2, 0) and (2, 6). A) 4 B) 0 2 C) 3 D) undefined E) none of these Answer: D 71) Find the slope of the line passing through the points (2, -5) and (-1, 3). 8 A) 3 B) -

2 5

C) -

3 8

D) -

5 2

E) none of these Answer: A Find the slope of the line. 72) y = 2x + 1 A) -2 B) 1 C) 2 D) -1 Answer: C 73) 2x + 5y = 26 2 A) 5 B) C)

2 5

5 2

D) -

5 2

Answer: B 41

74) 4x - 5y = 18 4 A) 5 B)

5 4

C) D)

5 4

4 5

Answer: D 75) y = -2 A) 2 B) 0 C) -2 D) undefined Answer: B Graph the line containing the given pair of points and find the slope. 76) (-6, 0) (0, -1) y 10

-10

-5

-10

1 6 y 10

-10

-5

-10

B) -

1 6 y 10

-10

-5

-10

C) 6 y 10

-10

-5 -5

-10

D) -6 y 10

-10

-5 -5

-10

Answer: B

77) (1, 7) (0, 0) y 10

-10

-5

-10

1 7 y 10

-10

-5

-10

B) -

1 7 y 10

-10

-5 -5

-10

C) -7 y 10

-10

-5

-10

D) 7 y 10

-10

-5 -5

-10

Answer: D 78) (2, -8) (-1, 4) y 10

-10

-5

-10

1 4 y 10

-10

-5

-10

B) -

1 4 y 10

-10

-5 -5

-10

C) - 4 y 10

-10

-5 -5

-10

D) 4 y 10

-10

-5

-10

Answer: C 79) (3, 2) (-4, 2) y 10

-10

-5

-10

A) 0 y 10

-10

-5

-10

B) 1 y 10

-10

-5

-10

C) 1 y 10

-10

-5 -5

-10

D) 0 y 10

-10

-5 -5

-10

Answer: A Graph the line on the coordinate plane using the given information.

80) Passes through (0, 4) and m =

1 2

y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: B 81) Passes through (-2, -10) and m = 6 y 10

-10

-5

-10

A) y 10

-10

-5

5 -5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: C

82) Passes through (0, 5) and m = - 2 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: C 83) Passes through (0, 3) and m = -

1 2

y 10

-10

-5

-10

A) y 10

-10

-5

5 -5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: D

84) Passes through (10, 0) and m = -

1 5

y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: A 85) Passes through (0, 2) and m = -

1 6

y 10

-10

-5

-10

A) y 10

-10

-5

5 -5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: C

86) Passes through (3, 9) and m = 0 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: B 87) Passes through (5, -2) and m = 0 y 10

-10

-5

-10

A) y 10

-10

-5

5 -5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

Answer: B

88) Passes through (1, 9) and the slope is undefined y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

D) y 10

-10

-5

-10

Answer: D Graph the equation using the slope, m, and y-intercept. 89) y = 3x - 3 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: D

90) y = -4x + 5 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: B 91) y = 3 10

-10

-5

-10

A) 10

-10

-5

-10

B) 10

-10

-5

-10

C) 10

-10

-5

-10

D) 10

-10

-5

-10

Answer: B

92) Find the equation of the line passing through the point (-2, 5) and parallel to the line y = -2x - 3. 1 A) y = - x + 4 2 B) y = 2x - 1 C) y = -2x + 3 D) y = -2x + 1 E) none of these Answer: D 93) Find the equation of the line passing through the point (1, 3) and parallel to the line y = -5x + 2. A) y = -5x + 3 B) y - 3 = x - 1 1 14 C) y = x + 5 5 D) y = -5x + 8 E) none of these Answer: D 94) Find the equation of the vertical line passing through (-1, 2). A) y = -1 B) x = -1 C) y = 2 D) x = 2 E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the equation for the line described. 95) the line through (1, 2) and (-2, 11) Answer: y = -3x + 5 96) the line through (5, 2) and (-2, 2) Answer: y = 2 97) the line that crosses the x-axis at x = 2 and the y-axis at y = -4 Answer: y = 2x - 4 98) the line having y-intercept (0, 5) and parallel to 2x + y = 10 Answer: y = -2x + 5 99) the line passing through (-2, -1) and having y-intercept (0, 3) Answer: y = 2x + 3 100) the line having y-intercept (0, 2) and perpendicular to y = -4x + 20 Answer: y =

1 x+2 4

101) the line perpendicular to 3x + 2y = 5 and passing through (1, 3) Answer: y =

2 7 x+ 3 3

102) perpendicular to y = 2x + 5 and passing through the point (3, 11). Answer: y = -

1 25 x+ 2 2

103) perpendicular to y = Answer: y =

3 x + 2 and passing through the point (0, 0). 2

2 x 3

104) parallel to the y-axis with x-intercept of (-2, 0) Answer: x=-2 105) parallel to the x-axis with y-intercept of (0, 5) Answer: y = 5 106) the vertical line passing through the point (-3, 1) Answer: x = -3 107) the horizontal line passing through the point (2, -3) Answer: y = -3 108) the line passing through the point (1, -2) and having slope 3 Answer: y = 3x - 5 109) the line passing through the point (2, 3) and having slope -4 Answer: y = -4x + 11 110) the line passing through (2, 5) and having slope Answer: y = -

2 3

2 19 x+ 3 3

111) Give an equation for the y-axis. Answer: x = 0

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the equation of the graph. 112) 10

8 6 4 2 -10 -8

-6

-4

-2

-2 -4 -6 -8 -10

A) y = 3x - 4 B) y = -3x - 4 4 C) y = x - 4 3 D) y =

3 x+3 4

Answer: C 113) 10

8 6 4 2 -10 -8

-6

-4

-2 -2 -4 -6 -8 -10

A) y = -x + 3 B) y = -x - 3 C) y = x - 3 D) y = x + 3 Answer: A

114) 10

8 6 4 2 -10 -8

-6

-4

-2

-2 -4 -6 -8 -10

A) y = -6 B) y = 6 C) x = 6 D) x = -6 Answer: B Find an equation of the line passing through the given point with the given slope, m. 115) (4, 2); m = - 5 A) y = -5x - 22 B) y = -5x + 22 1 C) y = -5x + 22 D) y = -

1 x + 22 5

Answer: B 116) (-4, -10), m = 4 A) y = 4x - 16 B) y = 4x - 6 C) y = 4x + 16 D) y = 4x + 6 Answer: D 117) (-6, -2), m = A) y =

1 2

1 x+1 2

B) y = 2x + 1 1 C) y = - x + 1 2 D) y =

1 x-1 2

Answer: A

118) (3, -4) , m =

2 3

A) y =

2 x-6 3

B) y =

3 x-6 2 2 x+6 3

C) y = D) y =

2 x+6 3

Answer: A 119) (0, 3); m =

3 5

A) y = -

3 x+3 5

B) y = -

5 1 x+ 3 3

C) y =

3 1 x5 3

D) y =

3 x+3 5

Answer: D 120) (0, 5); m =

7 9

A) y =

7 x-5 9

B) y =

7 x+5 9

C) y =

9 x+5 7

D) y =

7 1 x+ 9 5

Answer: B

Find the equation of the line L. 121) y

(2, 5)

y = -2x + 9

L perpendicular to y = -2x + 9 1 A) y = - x + 8 2 B) y = -

1 x-4 2

C) y =

1 x+4 2

D) y =

1 x-9 2

Answer: C

122) y

L (6, 10)

1 x+5 3

L parallel to y =

1 x+5 3

A) y = -3x + 8 1 B) y = x + 2 3 C) y =

1 x+8 3

D) y =

1 x + 10 3

Answer: C 123) Find the y-intercept of the line passing through the point (14, 12) and having slope

2 . 7

A) (0, 12) B) (0, -4) C) (0, 8) 24 D) 0, 7 E) none of these Answer: C 124) Find the value of a such that the three points (1, 3), (6, 5), and (a, -1) lie on the same line. 1 A) 3 B) -10 3 C) 5 D) -9 E) none of these Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 125) Find the x-intercept of the line passing through (1, 2) and having slope -4. Answer:

3 ,0 2

126) Find the coordinates of points A and B if A has an x-coordinate of 3 and B is the x-intercept. y

A (2, 5)

(0, 2) B x

Answer: A: 3,

13 4 , B: - , 0 2 3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 127) Suppose a manufacturer finds that the cost y of producing x units is given by a formula of the form y = mx + b. If it costs $1300 to produce 20 units and $1750 to produce 35 units, what is the fixed cost? A) $700 B) $450 C) $30 D) $2050 E) none of these Answer: A 128) Suppose a manufacturer finds that the cost y of producing x units is given by a formula of the form y = mx + b. If it costs $8200 to produce 20 units and $14,500 to produce 50 units, what is the marginal cost? A) $290 B) $410 C) $4000 D) $210 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 129) Suppose a manufacturer finds that the number of units x she produces and the cost y of producing x units are related by an equation of the form y = mx + b. If it costs $2300 to produce 10 units and $2450 to produce 15 units, what does it cost to produce 20 units? Answer: $2600

130) For a dial-up service between two cities, a telephone company charges 105 cents for the first minute and 55 cents for each additional minute (or fraction thereof). Write an equation such that y, the total cost of the call in cents, is expressed in terms of x, the length of the call in minutes for 1 ≤ x ≤ 8. Answer: y = 55x + 50 131) A game company has fixed costs of $40,000 per year. Each game costs $12.00 to produce and sells for $18.00. How many games must the company produce and sell each year in order to make a profit of $95,000? Answer: 22,500 games 132) A salesman's weekly pay depends on his volume of sales. He earns $80 each week in addition to $10 for each item he sells. (a) Write an equation relating y, the salesman weekly pay, to x, the number of items he sells. (b) How many items must he sell for his pay to be $300. Answer: (a) y = 10x + 80 (b) 22 133) The cost to run a TV factory depends on the number of items produced. The factory's fixed cost is $10,000 each month in addition to $120 for each TV produced. (a) Write an equation relating y, the factory's costs, to x, the number of TV's produced. (b) When will the factory's costs reach $22,000? Answer: (a) y = 120x + 10,000 (b) 100 134) A water tank is being emptied such that the height y (in feet) of the water inside the tank decreases at a linear rate with time t measured in hours past 12:00 noon. If the height was 15 feet at 1:00 pm and 7.5 feet at 6:00 pm, find (a) the equation relating y to t, (b) the height at 12:00 noon, and (c) the total time required to empty the tank. Answer: (a) y = -1.5t + 16.5 (b) 16.5 feet (c) 11 hours MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 135) It costs $700 to start up a business of selling hot dogs. Each hot dog costs $.90 to produce. Let y be the cost in dollars of producing x hot dogs. Write the cost equation. A) y = 700x + .90 B) y = .90x - 700 C) y = 90x + 700 D) y = .90x + 700 Answer: D 136) It costs $500 to start up a business of selling hot dogs. Each hot dog costs $.75 to produce. Let y be the cost in dollars of producing x hot dogs. What would be the cost to produce 400 hot dogs? A) $300 B) $800 C) $30,500 D) $575 Answer: B

137) It costs $500 to start up a business of selling hot dogs. Each hot dog costs $.90 to produce. Let y be the cost in dollars of producing x hot dogs. How many hot dogs will be produced if total cost is $2300? A) 4000 hot dogs B) 2000 hot dogs C) 200 hot dogs D) 20 hot dogs Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 138) Find a system of inequalities that has the following feasible set. y

A FS

A = (0, 1), B = (4, -1), C = (0, -2), D = (-1, 0) y≤Answer:

1 x+1 2

y≤x+ 1 1 y≥ x- 2 4

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write an inequality whose graph has the following properties. 139) Contains the points on or above the line with slope 5 and y-intercept (0, -3) A) y ≥ -3x + 5 B) y ≤ 5x + 3 C) y ≥ 5x - 3 D) y ≤ 5x - 3 Answer: C 140) Contains the points on or above the straight line passing through the points (3, 1) and (5, 15) A) y ≥ 7x + 22 B) y ≥ 7x - 20 C) y ≤ 7x - 20 D) y ≤ 7x + 20 Answer: B

141) Find the point of intersection of the two lines x + 3y = 1 and x - 3y = 5. 2 A) ,3 3 B)

1 1 , 2 2

C) 3, D) 3,

2 3

1 2

E) none of these Answer: C 142) Find the point of intersection of the two lines x - y = 1 and x + 2y = 2. A) (2, 1) 4 B) ,0 3 C)

4 1 , 3 3

D) 1,

1 2

E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 143) Find the point of intersection of the two lines x + 4y = 6 and x -4y = 2. Answer: 4,

1 2

144) Find the point of intersection of the two lines 2x + y = 5 and y = x - 1. Answer: (2, 1) 145) Find the point of intersection of the two lines x + 3y = 6 and x - y = 2. Answer: (3, 1) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the point of intersection of the given pair of straight lines. 146) x + y = 15 x - y = -1 A) (6, 9) B) No solution C) (-7, 9) D) (7, 8) Answer: D

147) 4 x - y = 17 5x + y = 28 A) (5, 3) B) (3, 5) C) No solution D) (5, 4) Answer: A 148) x - y = -2 x = 3y A) (-1, 3) B) (-3, -1) C) (-1, -3) D) (1, 3) Answer: B 149) x + 2y = -4 y=3 A) (2, 3) B) (-10, 3) C) (3, -10) D) (2, -3) Answer: B Solve the problem. 150) Does (0,-4) satisfy the following system? y = 2x -4 2y = 7x - 8 A) Yes B) No Answer: A 151) Does (3, 4) satisfy the following system? y=x+ 1 y = 7x - 6 A) Yes B) No Answer: B Decide if the given point satisfies the system of linear equations. x+y=1 152) x - y = 3 (2, -1) A) Yes B) No Answer: A

x + y = 11 153) x - y = -1 (-5, 6) A) Yes B) No Answer: B 2x + y = -3 154) 4x + 2y = -6 (-3, -3) A) Yes B) No Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the system of linear equations. 155) y = 5x -3 y = -3x - 11 Answer: (-1, -8) 156) 5x - 3y = 12 x=3 Answer: (3, 1) 157) 4x + 3y = 380 2x + 5y = 330 Answer: (65, 40) 158) 2x + 2y = 1 3x - y = 6 Answer:

13 9 ,8 8

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 159) If two lines intersect in more than one point, then they are A) unique. B) parallel. C) inconsistent. D) the same. E) none of these. Answer: D

Solve the system of linear equations. x + 4y = -1 160) -5x + 3y = -18 A) (-3, 0) B) (2, 0) C) (3, -1) D) No solution Answer: C

161)

x + 4y = 28 -6x + 5y = 35 A) No solution B) (-7, 0) C) (1, 6) D) (0, 7) Answer: D

162)

7x + 9y = 109 4x + 3y = 43 A) (4, 9) B) (4, 10) C) (3, 10) D) No solution Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 163) Find the coordinates of the vertices of the feasible set shown below.

Answer: A =

3 , 2 , B = (5, 2), C = (3, 0), D = (0, 1) 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 164) Consider the feasible set, FS, shown below. Find the coordinates of vertex A.

A) (2, 2) 4 B) 2, 3 C)

8 4 , 3 3

8 ,1 3

E) none of these Answer: C 165) Consider the feasible set, FS, shown below. Find the coordinates of the vertex B.

A) (0, 2) B) (0, 4) C) (2, 0) D) (4, 0) E) none of these Answer: D

166) Consider the feasible set, FS, shown below. Find the coordinates of the vertex B.

A) (0, 20) B) (5, 15) C) (10, 6) D) (10, 10) E) none of these Answer: B 167) Consider the feasible set, FS, shown below. Which of the following is a valid coordinate in the feasible set?

A) (10, 7) B) (5, 16) C) (11, 11) D) (3, 10) E) none of these Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 168) Consider the following system of linear inequalities. 5x + y ≤ 100 5x + 9y ≤ 180 x+y≥5 x ≥ 0, y ≥ 0 (a) Graph the feasible set determined by the system. (b) Find the coordinates of all of the vertices of the feasible set. y

Answer: (a) y 20

FS E

C 10

(b) A: (0, 20), B: (18, 10), C: (20, 0), D: (5, 0), E: (0, 5)

169) Consider the following system of linear inequalities: 2x + 3y ≤ 9 x+y≤4 x ≥ 0, y ≥ 0 (a) Graph the feasible set of the system. (b) Find the coordinates of vertices of the feasible set. y

Answer: (a) y

(b) A: (4, 0), B: (0, 3), C: (4, 0), D: (0, 0) Graph the feasible set of the system of linear inequalities. Shade the region which consists of points that do not belong to the feasible set. ≤ 170) y 2x - 3 y≥0 Answer: y 10

5 FS -10

-5

-10

≥ 171) x + 2y 4 2x - y ≥ 6

Answer: y 10

5 FS -10

-5

-10

≤ 172) y 2x + 5 y≥x

Answer: y 10

5 FS -10

-5

5 -5

-10

2x + 3y ≤ 9 173) x + y ≤ 4 x ≥ 1, y ≥ 0 y

Answer: y

FS 5

x + 3y ≤ 6 174) x - y ≤ 2 -5x + y ≤ 2 y 5

-5

Answer: y 5

FS -5

-5

x+y≥6 175) 3x - 2y ≥ 4 y ≥ 0, y ≤ 10 y

Answer: y 10

5 FS

x + 2y ≤ 12 176) Explain why y ≥ 7 x ≥ 0, y ≥ 0

has no solution.

Answer: If x, y satisfy x ≥ 0 , x + 2y ≤ 12, then the largest possible value for y is 6. Thus y does not satisfy y ≥ 7. Solve the problem. 177) Suppose that the supply and demand equations of a certain commodity are given by q = 5p - 15 and q = -2.5p + 30 respectively, where p is the unit price of the commodity in dollars and q is the quantity. (a) What is the supply when the price is $8? (b) What is the demand when the price is $8? (c) Find the equilibrium price and the corresponding number of units supplied and demanded. (d) Draw the graphs of the supply and demand equations on the same set of axes. (e) Find where the two lines cross the horizontal axis and give an economic interpretation of these points.

Answer: (a) 25 (b) 10 (c) (6, 15); The equilibrium price is $6 when 15 units are supplied and demanded. (d)

(e) (3, 0), (12, 0) Economic interpretation: The intersection of the supply curve with the horizontal axis indicates the lowest price at which the manufacturer is willing to sell the product or service. The intersection of the demand curve with the horizontal axis is the highest price a consumer is willing to pay for a product or service. 178) Suppose that the supply and demand equations of a new CD at a store are given by q = 3p - 12 and q = -2p + 23 respectively, where p is the unit price of the CD's in dollars and q is the quantity. (a) What is the supply when the price is $10? (b) What is the demand when the price is $10? (c) Find the equilibrium price and the corresponding number of units supplied and demanded. (d) Find where the two lines cross the horizontal axis and give an economic interpretation of these points. Answer: (a) 18 (b) 3 (c) (7, 9); The equilibrium price is $7 when 9 units are supplied and demanded. 23 (d) (4, 0), ,0 2 Economic interpretation: The intersection of the supply curve with the horizontal axis indicates the lowest price at which the store is willing to sell the CD's. The intersection of the demand curve with the horizontal axis is the highest price a consumer is willing to pay for the CD. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

179) Let the supply and demand equations for a certain model of electric pencil sharpener be given by 2 2 p = q and p = 18 - q , 3 3 where p is the price in dollars and q is the quantity of pencil sharpeners (in hundreds). Graph these equations on the same axes (graph the supply equation as a dashed line and the demand equation as a solid line). Also, find the equilibrium quantity and the equilibrium price. 20

18 16 14 12 10 8 6 4 2 2

8 10 12 14 16 18 20 q

A) 20

18 16 14 12 10 8 6 4 2 6

8 10 12 14 16 18 20 q

Equilibrium quantity: 950 Equilibrium price: $7 B) 20

18 16 14 12 10 8 6 4 2 2

8 10 12 14 16 18 20 q

Equilibrium quantity: 1350 Equilibrium price: $9

C) 20

18 16 14 12 10 8 6 4 2 2

8 10 12 14 16 18 20 q

Equilibrium quantity: 720 Equilibrium price: $10.80 D) 20

18 16 14 12 10 8 6 4 2 2

8 10 12 14 16 18 20 q

Equilibrium quantity: 1080 Equilibrium price: $7.20 Answer: B 180) Let the supply and demand equations for raspberry-flavored licorice be given by 5 3 p = q and p = 60 - q , 4 4 where p is the price in dollars and q is the number of batches. Graph these equations on the same axes (graph the supply equation as a dashed line and the demand equation as a solid line). Also, find the equilibrium quantity and the equilibrium price. p 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 q

A) p 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 q

Equilibrium quantity: 24 Equilibrium price: $30.00 B) p 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 q

Equilibrium quantity: 37.50 Equilibrium price: $30 C) p 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 q

Equilibrium quantity: 30 Equilibrium price: $37.50

D) p 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 q

Equilibrium quantity: 30.00 Equilibrium price: $24 Answer: C 181) Find the area of the shaded triangle. Note it has its base on one of the axes. The area of a triangle is one-half, times its altitude, times the length of its base. y

5x - y = 10

x+y=8

A) 30 sq. units B) 8 sq. units C) 15 sq. units D) 5 sq. units Answer: C

182) Find the area of the shaded triangle. Note it has its base on one of the axes. The area of a triangle is one-half, times its altitude, times the length of its base. y

4x + 5y = 45

4x - 5y = -5

A) 40 sq. units B) 80 sq. units C) 20 sq. units D) 21 sq. units Answer: C 183) A plane flying with a tailwind flew at a speed of 470 mph, relative to the ground. When flying against the tailwind, it flew at a speed of 330 mph. Express these relationships as equations. Let x represent the speed of the plane in calm air and let y represent the speed of the wind. A) x + y = 470 x - y = 330 B) x + y = 330 y = 470 C) x + y = 470 y = 330 D) x + y = 330 x - y = 470 Answer: A 184) A young married couple had a combined annual income of $43,000. If the wife made $3000 more than the husband, write these relationships as a system of equations. Let x represent the husband's income and let y represent the wife's income. A) x + y = 3000 y = x + 43,000 B) x + y = 43,000 y = 3000 C) x - y = 43,000 y = x + 3000 D) x + y = 43,000 y = x + 3000 Answer: D

185) Mark the electrician charges $120 for a house call, and then $25 per hour for labor. Sara the electrician charges $95 for a house call, and then $45 per hour for labor. Write a cost equation for each electrician, where y is the total cost of the electrical work and x is the number of hours of labor. A) Mark: y = 120x + 25 Sara: y = 95x + 45 B) Mark: y = 95x + 45 Sara: y = 120x + 25 C) Mark: y = 25x + 120 Sara: y = 45x + 95 D) Mark: y = 45x + 95 Sara: y = 25x + 120 Answer: C 186) Two plumbers make house calls. One charges $85 for a visit plus $40 per hour of work. The other charges $65 per visit plus $50 per hour of work. For how many hours of work do the two plumbers charge the same? A) 1 hr B) 2 hr C) 2.5 hr D) 1.5 hr Answer: B 187) Chris and Mary are selling tickets at their class play. Chris is selling student tickets for $1.00 each, and Mary selling adult tickets for $6.50 each. If their total income for 26 tickets was $64.50, how many tickets did Chris sell? A) 21 tickets B) 19 tickets C) 12 tickets D) 7 tickets Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 188) Determine the least-square error when the line y =

11 x + 3 is used to approximate the data points (1, 3), 10

(2, 6), (3, 8) and (4, 6). Answer: 6.70 189) Determine the least-square error when the line y = 2x + 2.5 is used to approximate the data points (1, 5), (3, 8) and (6, 15). Answer: 0.75 190) Find the least-squares line for the data points (3, 6), (1, 9), (4, 3) and (2, 8). Answer: y = -2x+

23 2

191) Find the least-squares line for the data points (1, 5), (3, 8) and (6, 15). Answer: y =

77 49 x+ 38 19

192) Find the least-squares line for the data points (a - 1, b), (a, b + c + 1), and (a + 1, b + c - 1). Answer: y = (

c-1 1 2 1 )x + ( a + b + c - ac) 2 2 3 2

Solve the problem. 193) The following table gives the revenue of a company over several years. Years (after 1990) 0 1 2 3 4

Dollars (in millions) 12.6 13.9 14.1 15.4 16.2

(a) Find the least-squares line for this data. (b) Use the least-squares line to predict the revenues for the year 2000. Answer: (a) y = 0.87x + 12.7 (b) $21.4 million 194) The following table gives the amount of carbon dioxide being released into the atmosphere from a certain foreign country. Years (after 2000) 0 1 2 3 4

Carbon Dioxide (tons) 17 19.4 21.3 23 25.5

(a) Find the least-squares line for this data. (b) If the trend continues, when will the number of tons reach 28? Answer: (a) y = 2.06x + 17.12 (b) 2005

195) The following table gives the amount (in millions of tons) of paper and paperboard waste generated in the United States for certain years. Year 1960 1965 1970 1975 1980 1985 1990 1993

Waste 29.9 38.0 44.2 43.0 54.7 61.5 73.3 77.8

(a) Use the method of least squares to obtain the straight line that best fits these data. Let x = the number of years after 1960. Round answer to the nearest hundredth (b) If the trend determined by the straight line continues, then when will the amount of paper waste exceed 100 million tons? Answer: (a) y = 1.41x + 28.45 (b) 2012 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 196) The paired data below consist of the test scores of 6 randomly selected students and the number of hours they studied for the test. Use the equation of the least squares line to predict the score on the test of a student who studies 3 hours. Hours (x) 5 10 4 6 10 9 Score (y) 64 86 69 86 59 87 A) 75.5 B) 70.5 C) 71.8 D) 65.5 Answer: B 197) The paired data below consist of the costs of advertising (in thousands of dollars) and the number of products sold (in thousands). Use the equation of the least squares line to predict the number of products sold if the cost of advertising is $9000. Cost (x) 9 2 3 4 2 5 9 10 Number (y) 85 52 55 68 67 86 83 73 A) 77.91 products sold B) 87.61 products sold C) 80.91 products sold D) 25,165.8 products sold Answer: C

198) The paired data below consist of the temperatures on randomly chosen days and the amount a certain kind of plant grew (in millimeters). Use the equation of the least squares line to predict the growth of a plant if the temperature is 61. Temp (x) 62 76 50 51 71 46 51 44 79 Growth (y) 36 39 50 13 33 33 17 6 16 A) 27.47 mm B) 26.19 mm C) 28.45 mm D) 27.96 mm Answer: A 199) A study was conducted to compare the average time spent in the lab each week versus course grade for computer students. The results are recorded in the table below. Use the equation of the least squares line to predict the grade of a student who spends 14 hours in the lab. Number of hours spent in lab (x) 10 11 16 9 7 15 16 10

Grade (percent) (y) 96 51 62 58 89 81 46 51

A) 58.6% B) 65.4% C) 74.6% D) 62.6% Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Perform the indicated elementary row operation. 1) x - 5y = 4 R + (-2)R 2 1 2x + 2y = 5 Answer: x - 5y = 4 12y = -3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1 0 3 9 2) Find the result of performing the elementary row operation R3 + (5) R2 on the system 0 1 -3 2 . 0 -5 4 1 A) 1 0 3 9 0 0 -3 2 0 -5 4 1 B) 1 0 3 9 0 1 -3 2 0 0 -11 11 C) 1 0 3 9 0 -5 -11 11 0 -5 4 1 D) 1 0 3 9 0 1 -3 2 10 5 14 11 Answer: B

1 0 3 9 3) Find the result of performing the elementary row operation R2 + (-1) R3 on the system 0 1 -3 2 . 0 -5 4 1 A) 1 0 3 9 0 1 -3 2 0 -7 2 0 B) 1 0 3 9 0 6 -7 1 0 -5 4 1 C) 1 03 9 0 -7 2 0 0 -5 4 1 D) 1 0 3 9 0 -1 -3 2 0 -4 -7 1 Answer: B 1 0 0 -2 4) The system 0 1 3 5 is equivalent to the system 1 1 -3 4 A) 1 0 0 -2 0 1 3 5 . 0 1 -3 6 B) 1 0 0 -2 0 1 -3 5 . 0 0 0 4 C) 1 0 0 -2 0 3 -9 5 . 1 1 -3 4 D) 1 0 0 -2 0 1 -3 5 . 0 1 -3 4 Answer: A

1 0 1 -1 5) The system 3 1 3 7 is equivalent to the system 2 14 4 A) 1 0 1 -1 0 10 7 2 14 4 B) 1 0 1 -1 313 7 013 2 C) 1 0 1 -1 3 13 7 0-12 2 D) 1 0 1 -1 313 7 012 2 Answer: C Use the indicated row operation to change the matrix. 6) Replace R2 by R1 + (-1)R2 . 1 -3 4 2 31 A) 1 -3 3 3 04 B) 1 -3 3 2 31 C) 1 -3 3 1 6 -2 D) 1 -3 4 -1 -6 3 Answer: D

7) Replace R2 by

1 1 R1 + R2 . 2 2

202 -2 2 8 A) 2 0 2 005 B) 2 0 2 015 C) 2 0 2 0 2 10 D) 2 0 2 -1 1 4 Answer: B 8) Replace R2 by

1 1 R + R . 3 1 2 2

3 0 12 -2 4 6 A) 3 0 12 1 4 18 B) 3 0 12 00 7 C) 3 0 12 -1 2 3 D) 3 0 12 02 7 Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. State the next elementary row operation that should be performed to put the matrix into diagonal form. 9) 1 2 -4 5 0 0 3 6 0 1 4 1 Answer: R 2 ↔ R3 State and perform the next elementary row operation that should be performed to put the matrix in diagonal form. 10) 1 0 3 9 0 1 -3 2 0 -5 4 1 Answer: R3 + 5 R 2

10 3 9 0 1 -3 2 0 0 -11 11

11) 1 0 0 -3 0 1 -2 -2 0 0 1 1 Answer: R 2 +2R 3

1 0 0 -3 0 1 0 0 0 0 1 1

12) 1 2 -3 3 0 -4 6 7 2 6 5 -1 Answer: R 3 + (-2)R1

1 2 -3 3 0 -4 6 7 0 2 11 -7

13) Is x = 2, y = -1, z = 4 a solution of the system of equations shown below? Explain your answer. x- y+ z = 7 2x - z = 0 2y + 3z = 9 Answer: x = 2, y = -1, z = 4 is not a solution because these values do not satisfy the third equation. 14) Is x = 2, y = -1, z = 4 a solution of the system of equations shown below? Explain your answer. x-y+ z = 7 2x - z = 0 2y + 3z = 10 Answer: x = 2, y = -1, z = 4 is a solution because these values satisfy every equation. 15) When you solve a system of equations, what set of values are you looking for? Answer: The solution to a system of equations is the set of all values such that each value satisfies every equation in the system. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Gauss-Jordan method to solve the system of equations. 16) 2x + 4y = 14 2x + 3y = 9 A) x = 5, y = -3 B) x = -3, y = -5 C) x = -3, y = 5 D) No solution Answer: C

17) 3x + 5y = 16 3x = -9 A) x = 5, y = -3 B) x = -3, y = 5 C) x = -3, y = -5 D) No solution Answer: B 18) 5x + y = 12 6x + 3y = 9 A) x = -3, y = -3 B) x = -3, y = 3 C) x = 3, y = -3 D) No solution Answer: C 19) x+y+ z= 7 x - y + 2z = 6 5x + y + z = 3 A) x = 5, y = -1, z = 3 B) x = -1, y = 3, z = 5 C) x = 5, y = 3, z = -1 D) No solution Answer: B 20) x - y + 4z = -6 5x + z = -1 x + 3y + z = 5 A) x = -1, y = 0, z = 2 B) x = -1, y = 2, z = 0 C) x = 0, y = 2, z = -1 D) No solution Answer: C 21) x-y+z= 3 x + y + z = -1 x+y-z= 5 A) x = 4, y = -2, z = -3 B) x = 4, y = -3, z = -2 C) x = -3, y = 4, z = -2 D) No solution Answer: A

22) 2x - y - 9z = -71 6x + 6z = 66 9y + z = 53 A) x = 3, y = 8, z = 5 B) x = 3, y = 5, z = 8 C) x = -3, y = 5, z = 6 D) No solution Answer: B Fill in the missing numbers in the next step of the Gauss-Jordan elimination method. 23) 1 7 4 -5 1 7 4 -5 → 4 1 -9 9 0 -27 □ □ 0 25 □ □ -4 -3 5 0 A) 1 7 4 -5 0 -27 -25 29 0 25 21 -20 B) 1 7 4 -5 0 -27 -13 14 0 25 1 -21 C) 1 7 4 -5 0 -27 -24 31 0 25 20 -19 D) 1 7 4 -5 0 -27 -27 28 0 25 24 -20 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 24) The sales division of a company purchased type A cell phones for $95 each and type B cell phones for $110 each. It purchased a total of 18 cell phones at a total cost of $1815. How many type A cell phones did the division purchase? Answer: 11 type A cell phones

25) A marketing company wishes to conduct a $15,000,000 advertising campaign in three media-radio, television, and newspaper. The company wants to spend twice as much money in television advertising as in radio and newspaper advertising combined. Also, the company wants to spend a total of $13,000,000 in radio and television combined. Let x, y and z represent the amount in millions of dollars spent in radio, television, and newspaper respectively. How much should the company spend in each medium? (Express the problem in terms of a system of linear equations and solve it using Gaussian elimination.) Answer: x + y + z = 15 -2x + y - 2z = 0 x+y = 13 x = 3, y = 10, z = 2; The company should spend $3,000,000 in radio advertising, $10,000,000 in television advertising, and $2,000,000 in newspaper advertising. 26) At a certain party, 72 people are invited. Only half of those invited come. Of those who come, twice as many are women as are men. How many men and how many women come? Let x be the number of women and y be the number of men. Answer: x = 24 = number of women; y = 12 = number of men MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 27) A waiter made a deposit of $194. If his deposit consisted of 82 bills, some of them one-dollar bills and the rest five-dollar bills, how many one-dollar bills did he deposit? A) 28 one-dollar bills B) 44 one-dollar bills C) 54 one-dollar bills D) 49 one-dollar bills Answer: C 28) At a local zoo, adult visitors must pay $3 and children pay $1. On one day, the zoo collected a total of $770. If the zoo had 410 visitors that day, how many adults and how many children visited? A) 230 adults and 180 children B) 192 adults and 218 children C) 115 adults and 295 children D) 180 adults and 230 children Answer: D 29) The annual salaries of a software engineer and a project supervisor total $192,400. If the project supervisor makes $15,200 more than the software engineer, find each of their salaries. A) software engineer: $103,800 project supervisor: $119,000 B) software engineer: $88,600 project supervisor: $103,800 C) software engineer: $73,400 project supervisor: $119,000 D) software engineer: $73,400 project supervisor: $88,600 Answer: B

30) A particular computer takes 74 nanoseconds (ns) to carry out 5 sums and 9 products, and 64 nanoseconds to perform 4 sums and 8 products. How long does the computer take to carry out one sum and one product? A) one sum: 4 ns one product: 6 ns B) one sum: 6 ns one product: 8 ns C) one sum: 6 ns one product: 4 ns D) one sum: 4 ns one product: 8 ns Answer: A 31) Barges from ports X and Y went to cities A and B. X sent 30 barges and Y sent 8. City A received 21 barges and B received 17. Shipping costs $220 from X to A, $300 from X to B, $400 from Y to A, and $180 from Y to B. $8760 was spent. How many barges went where? A) 15 from X to A, 15 from X to B, 6 from Y to A, and 2 from Y to B B) 17 from X to A, 17 from X to B, 4 from Y to A, and 4 from Y to B C) 19 from X to A, 11 from X to B, 2 from Y to A, and 6 from Y to B D) 21 from X to A, 9 from X to B, 0 from Y to A, and 8 from Y to B Answer: D 32) Factories A and B sent rice to stores 1 and 2. A sent 14 loads and B sent 22. Store 1 received 20 loads and store 2 received 16. It cost $200 to ship from A to 1, $350 from A to 2, $300 from B to 1, and $250 from B to 2. $8600 was spent. How many loads went where? A) 12 from A to 1, 2 from A to 2, 8 from B to 1, and 14 from B to 2 B) 0 from A to 1, 14 from A to 2, 16 from B to 1, and 6 from B to 2 C) 13 from A to 1, 1 from A to 2, 7 from B to 1, and 4 from B to 2 D) 14 from A to 1, 0 from A to 2, 6 from B to 1, and 16 from B to 2 Answer: D

33) Pivot the matrix 1 3 5 about the element 6. 5 6 2 A) 1 3 5 3 0 -8 B) 1 3 5 5 1 1 6 3 C) 8 3

2 -

23 9

3 0 2

5 1 1 6 3 Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 34) Pivot the matrix 1 3 about the element 3. 4 -2 Answer: 1 3

14 3

1 35) Pivot the matrix 0

2 1

0 -5

4 2 about the underlined element 1. 3

Answer: 1 0 0 0 1 2 0 0 13

3 36) Pivot the matrix 1

2 -2 1 3 1 about the element . 2 2 2

0 -1

Answer: -1 0 4 2 1 -3 2 0 6 Solve the system of linear equations using the Gaussian elimination method. If there is no solution, state so; if there are infinitely many solutions, find two of them. 37) x - y + 2z = 2 y - 2z = 1 -3x + 5y - 10z = -4 Answer: z = any value, x = 3, y = 2z + 1; two possible solutions: x = 3, y = 1, z = 0 and x = 3, y = 3, z = 1 38) x - 3y - 5z = 1 -3x + 7y + 9z = 1 x + 4z = -5 Answer: z = any value, x = -4z - 5, y = -3z -2; two possible solutions: x = -5, y = -2, z = 0 and x = -9, y = -5, z = 1 39) x - y - 2z = 2 y - 2z = 1 -3x + 5y - 10z = -4 Answer: unique solution; x = 3, y = 1, z = 0 Solve the linear system by using the Gauss-Jordan elimination method. 40) 3x + 6y = -6 -2x - 7y = 1 Answer: x = -4, y = 1 41) x - 3y - 5z = 1 -3x + 7y + 9z = 1 x + 4z = -5 Answer: z = any value, x = -4z - 5, y = -3z - 2 For the system of equations, state whether there is one, none, or infinitely many solutions. If there are one or more solutions, give all values of x, y, and z that satisfy the system. 42) x - y + 3z = 0 z=7 Answer: infinitely many solutions: y = any value, x = y - 21, z = 7

43) x + 3y + 2z = 4 y - 3z = -1 z = 17 Answer: one solution: x = - 180, y = 50, z = 17 44) x+ y- z=1 y - 2z = 1 x+ y- z=2 Answer: no solution 45) When solving a system of linear equations with the unknowns x, y, and z using the Gauss-Jordan elimination method, the following matrix was obtained. What can be concluded about the solution of the system? 1 0 0 4 0 1 1 3 0 0 0 0 Answer: The system has infinitely many solutions; the general solution is: z = any value, x = 4 , y = 3 - z. 46) When solving a system of linear equations with the unknowns x, y, and z using the Gauss-Jordan elimination method, the following matrix was obtained. What can be concluded about the general solution of the system? 1 0 2 4 0 1 1 3 Answer: The system has infinitely many solutions; the general solution is: z = any value, x = 4 - 2z, y = 3 - z. 47) When solving a system of linear equations with the unknowns x, y, and z using the Gauss-Jordan elimination method, the following matrix was obtained. What can be concluded about the solution of the system? 1 0 0 4 0 1 -1 3 0 0 0 2 Answer: The system has no solution. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. x - y = 7 . Which of the following statements is true? 2x - 2y = k A) If k = 14, the system has no solution. B) If k ≠ 14, the system has exactly one solution. C) If k = 14, the system has infinitely many solutions. D) none of these

48) Consider the system:

Answer: C x - y = 7 . Which of the following statements is true? 2x - 2y = k A) If k = 10, the system has no solution. B) If k ≠ 10, the system has exactly one solution. C) If k = 10, the system has infinitely many solutions. D) none of these

49) Consider the system:

Answer: A 12

50) Which of the following systems have infinitely many solutions? x - y + 4z + 3w = 2 I. x - 2y + 5z = 7 II. z + 2w = 8 y - 2z = 4 w=3

III.

x + y- z= 1 y - 2z = 1 3y - 6z = 4

IV.

x + 2y - z = 2 x - z=3 2x - 2z = 6

A) III and IV only B) I, II, and III only C) I and II only D) I, II, and IV only E) all of these Answer: D 51) Which of the following systems has no solution? A) 2x + 4y = 10 3x + 6y = 20 B) 2x + 4y = 10 3x + 6y = 15 C) 2x + 4y = 10 2x + 6y = 10 D) 2x + 4y = 10 4x + 8y = 20 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 52) Suppose the line L has equation 2x + 3y = 5 and the line M has equation x + 4y = -5. Use Gauss-Jordan elimination method to determine whether L and M intersect in one point, are distinct but parallel, or coincide. Answer: L and M intersect in one point: (7, -3). 53) Suppose line L has equation 3x + 2y = 7 and line H has equation 9x + Ky = 14. Find the value for K so that L and M are parallel. Answer: K = 6 54) If possible, find the general solution of the following system. x + y+ z = 1 x + y + 2z = -1 2x + 2y + 3z = 0 Answer: z = -2, x + y = 3; the general solution is: x = any value, y = 3 - x, z = -2.

55) Find three solutions to

8x + 2y - z = 7 . y = 2

Answer: Answers will vary. Solutions are of the form: x = any value, y = 2, z = 8x - 3. Three possible solutions are: x = 0, y = 2, z = -3 x = 1, y = 2, z = 5 x = 2, y = 2, z = 13 56) Find a value for K so that the following system of equations 6x + 3y = 15 has infinitely many solutions. Kx + y = 5 Answer: K = 2 57) Find a value for K so that the following system of equations 6x + 3y = 15 has exactly one solution. Kx + y = 5 Answer: K ≠ 2 58) Find a specific solution to a system of linear equations whose general solution is y = any value w = any value x = 3 - 6y z=2 Answer: One possible solution is x = 3, y =0, z = 2, w = 1. 59) Find a specific solution to a system of linear equations whose general solution is y = any value w = 2x - z x = 3 - 6y z=2 Answer: One possible solution is x = 3, y =0, z = 2, w = 4. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 60) The two lines -2x + y = 3 and -3x + y = 2 A) are parallel. B) are perpendicular. C) coincide. D) intersect in exactly one point. E) none of these Answer: D 61) The two lines x + 5y = 3 and 5x - y = 6 A) coincide. B) are perpendicular. C) intersect in exactly one point. D) are parallel. E) none of these Answer: B

Use the Gauss-Jordan method to solve the system of equations. 62) -3x - 8y = -9 -6x - 16y = -9 A) (2, 2) B) (-9, -9) 8 C) 3 - y, y 3 D) No solution Answer: D 63) -3x - 7y = 59 -9x - 21y = 177 A) (-8, -5) 59 7 B) + y, y 3 3 C) -

59 7 - y, y 3 3

D) No solution Answer: C 64) -4x - 2y = 6 -16x - 8y = -1 A) (6, -1) B) (4, 4) 1 2 C) - x - y, y 3 3 D) No solution Answer: D A matrix, A, corresponding to a system of linear equations and the matrix rref(A) obtained after the Gauss-Jordan elimination method is applied to A, are given. Write the system of linear equations corresponding to A, and use rref(A) to give all solutions of the system of linear equations. 65) A = 1 5 -5 12 ; rref(A) = 1 0 5 7 2 11 -12 25 0 1 -2 1 A) x + 5y - 5z = 12 2x + 11y - 12z = 25 z = any value, y = 1 + 2z, x = 7 - 5z B) x + 5y - 5z = 7 2x + 11y - 12z = 1 z = any value, y = 1 - 2z, x = 7 + 5z C) x + 5y - 5z = 7 2x + 11y - 12z = 1 z = any value, y = -2, x = 5 D) x+ 5z = 7 y - 2z = 1 z = any value, y = 1 + 2z, x = 7 - 5z Answer: A 15

1 2 6 7 1 0 -2 -3 66) A = 2 5 16 19 ; rref(A) = 0 1 4 5 00 0 0 -2 -5 -16 -19 A) x + 2y + 6z = 7 2x + 5y + 16z = 19 -2x - 5y - 16z = -19 z = 0, y = 5, x = -3 B) x - 2z = -3 y + 4z = 5 z = any value, y = 5 - 4z, x = -3 + 2z C) x + 2y + 6z = -3 2x + 5y + 16z = 5 -2x - 5y - 16z = 0 z = any value, y = 5 + 4z, x = 3 - 2z D) x + 2y + 6z = 7 2x + 5y + 16z = 19 -2x - 5y - 16z = -19 z = any value, y = 5 - 4z, x = -3 + 2z Answer: D Solve the problem. 67) Janet is planning to visit Arizona, New Mexico, and California on a 14-day vacation. If she plans to spend as much time in New Mexico as she does in the other two states combined, how can she allot her time in the three states? (Let x denote the number of days in Arizona, y the number of days in New Mexico, and z the number of days in California. Let z be the parameter.) A) x = 7 - z, y = 7, 0 ≤ z ≤ 7 B) x = 7, y = z - 7, 0 ≤ z ≤ 7 C) x = z - 7, y = 7, 0 ≤ z ≤ 7 D) x = 7, y = 7 - z, 0 ≤ z ≤ 7 Answer: A 68) A company is introducing a new soft drink and is planning to have 48 advertisements distributed among TV ads, radio ads, and newspaper ads. If the cost of TV ads is $500 each, the cost of radio ads is $200 each, and the cost of newspaper ads is $200 each, how can the ads be distributed among the three types if the company has $13,800 to spend for advertising? (Let x denote the number of TV ads, y the number of radio ads, and z the number of newspaper ads. Let z be the parameter.) A) x = 14, y = z - 34, 0 ≤ z ≤ 34 B) x = 14, y = 34 - z, 0 ≤ z ≤ 34 C) x = z - 34, y = 14, 0 ≤ z ≤ 34 D) x = 34 - z, y = 14, 0 ≤ z ≤ 34 Answer: B

69) An investor has $400,000 to invest in stocks, bonds, and commodities. If he plans to put three times as much into stocks as in bonds, how can he distribute his money among the three types of investments? (Let x denote the amount put into stocks, y the amount put into bonds, and z the amount put into commodities. Let all amounts be in dollars, and let z be the parameter.) A) x = 100,000 - z/2, y = 300,000 - z/2, 0 ≤ z ≤ 400,000 B) x = 300,000 - 3z/4, y = 100,000 - z/4, 0 ≤ z ≤ 400,000 C) x = 300,000 - z/2, y = 100,000 - z/2, 0 ≤ z ≤ 400,000 D) x = 100,000 - 3z/4, y = 300,000 - z/4, 0 ≤ z ≤ 400,000 Answer: B 70) A company has 120 sales representatives, each to be assigned to one of four marketing teams. If the first team is to have three times as many members as the second team and the third team is to have twice as many members as the fourth team, how can the members be distributed among the teams? (Let x denote the number of members assigned to the first team, y the number assigned to the second team, z the number assigned to the third team, and w the number assigned to the fourth team. Let w be the parameter.) A) x = 120 - 3w, y = 40 - w, z = 3w - 40, 0 ≤ w ≤ 40 B) x = 90 - 9w/4, y = 30 - 3w/4, z = 2w, 0 ≤ w ≤ 40 C) x = 30 - 3w/4, y = 10 - w/4, z = 2w, 0 ≤ w ≤ 20 D) x = 60 - 3w, y = 20 - w, z = 40 + 3w, 0 ≤ w ≤ 20 Answer: B 71) A school library has $27,000 to spend on new books among the four categories of biology, chemistry, physics, and mathematics. If the amount spent on biology books is to be the same as the amount spent on chemistry books and if the amount spent on mathematics books is to be the same as the total spent on chemistry and physics books, how can the money be distributed among the four types of books? (Let x denote the amount spent on biology books, y the amount spent on chemistry books, z the amount spent on physics books, and w the amount spent on mathematics books. Let all amounts be in dollars, and let w be the parameter.) A) x = 27,000 - 3w, y = 27,000 - 3w, z = 4w - 27,000, 9000 ≤ w ≤ 18,000 B) x = 27,000 - w, y = 27,000 - w, z = 2w - 27,000, 4500 ≤ w ≤ 9000 C) x = 27,000 - 2w, y = 27,000 - 2w, z = 3w - 27,000, 9000 ≤ w ≤ 13,500 D) x = 27,000 + w, y = 27,000 + w, z = w - 27,000, 4500 ≤ w ≤ 13,500 Answer: C 72) A recording company is to release 210 new CDs in the categories of rock, country, jazz, and classical. If twice the number of rock CDs is to equal three times the number of country CDs and if the number of jazz CDs is to equal the number of classical CDs, how can the CDs be distributed among the four types? (Let x be the number of rock CDs, y the number of country CDs, z the number of jazz CDs, and w the number of classical CDs. Let w be the parameter.) A) x = 126 - w, y = 56 - w, z = w, 0 ≤ w ≤ 70 B) x = 105 - 2w, y = 105 - 2w/5, z = w, 0 ≤ w ≤ 105 C) x = 105 - 3w/5, y = 70 - 2w/5, z = w, 0 ≤ w ≤ 91 D) x = 126 - 6w/5, y = 84 - 4w/5, z = w, 0 ≤ w ≤ 105 Answer: D

73) A politician is planning to spend a total of 40 hours on a campaign swing through the southern states of Arkansas, Louisiana, Mississippi, Alabama, and Georgia. Assume that he spends the same amount of time in Mississippi as in Alabama; half the amount of time in Georgia as in Arkansas; and the same amount of time in Mississippi, Alabama, and Georgia (combined) as in Arkansas and Louisiana (combined). How can he distribute his time among the five states? (Let a be the hours spent in Arkansas, b the hours spent in Louisiana, c the hours spent in Mississippi, d the hours spent in Alabama, and e the hours spent in Georgia. Let e be the parameter.) A) a = e, b = 20 - e/2, c = 10 - e/4, d = 10 - e/4, 0 ≤ e ≤ 20 B) a = 2e, b = 20 - 2e, c = 10 - e/2, d = 10 - e/2, 0 ≤ e ≤ 10 C) a = 2e, b = 20 - e, c = 10 - e, d = 10 - e, 0 ≤ e ≤ 20 D) a = 2e, b = 20 - e, c = 10 - e/2, d = 10 - e/2, 0 ≤ e ≤ 10 Answer: B 74) Which of the following calculations can be performed? 1 I. [4 0 2] + 3 II. [6] + 1 0 III. [5] 1 0 IV. 2 1 2 0 1 0 1 4 0 1 5 A) III only B) IV only C) III and IV only D) I and II only E) all of these Answer: B 75) What is the identity matrix of size 3? A) 001 010 . 100 B) 011 101 . 110 C) 100 010 . 001 D) 111 111 . 111 Answer: C

1 1 2

76) If B is a 1 × 6 matrix and A is a 6 × 1 matrix, determine the size of AB. A) 6 × 6 B) 6 × 1 C) 1 × 1 D) 1 × 6 E) none of these Answer: A 77) If B is a 4 × 2 matrix and A is a 3 × 4 matrix, determine the size of AB. A) 3 × 2 B) 4 × 4 C) 3 × 4 D) 2 × 3 E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine the values of x and y, if any, that make A = B. 78) A = 5 3 -2 and B = 5 y -2 x 4 0 8 4 0 Answer: x = 8, y = 3 x 1 79) A = 2 3 and B = 5 2 4 1 3 -2 4 y Answer: There are no values of x and y that make A = B. 80) In each case below, give an example of a matrix that meets the specified condition. (a) A and B so that AB is not defined (b) a 2 × 2 matrix that has no inverse (c) C and D so that CD is defined but DC is not defined Answer: Answers will vary. (a) Possible solutions: any two matrices such that the number of columns in A is not the same as the number of rows in B. 5 One possible example is A = 2 4 and B = 1 . 31 7 (b) Possible solution: any 2 × 2 matrix such that Δ = 0, or having a row of zeros. Possible examples: 6 4 or 0 1 . 3 2 0 0 (c) Possible solutions: any two matrices such that the number of columns in C is the same as the number of rows in D, but the number of rows in C is not equal to the number of columns in D. 4 One possible example is C = 3 2 1 and D = 8 . 1 -2 0 -2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 81) Which choice best describes the following matrix? 24 26 20 17 18 A) Row matrix B) Square matrix C) 1 × 5 matrix D) Column matrix Answer: D Answer the question about the 2 × 3 matrix A. 82) Find a 12 and a 23. A = -1 7 2 -1 0 9 A) 7, 9 B) 7, -1 C) -1, 9 D) -1, 2 Answer: A 83) For what values of i and j does a ij = 9? A = -6 9 2 -2 0 4 A) i = 2, j = 1 B) i = 2, j = 2 C) i = 1, j = 2 D) i = 1, j = 1 Answer: C

84) 9 -2 -5 -8 -7 2 + 2 -7 8 -7 4 -2 A) 14 6 -9 9 4 -6 B) 4 -10 5 2 12 9 C) 4 2 -5 -5 12 -9 D) 4 -10 -5 -5 12 -9

Answer: D 85) -1 0 3 1 A)

- -1 3 3 1

03 00 B) [-3] C) -2 3 62 D) 0 -3 0 0 Answer: D Perform the indicated operation, where possible. 86) -3 1 + 6 2 25 42 A) 3 -1 1 -1 B) 3 3 67 C) 3 4 37 D) Not possible Answer: B

+ 1 8 A) 4 12 B) 4 12 C) 3 1 48 D) Not possible

87) 3 4

Answer: D 88) -4 8 4 - 4 3 A) -8 5 1 B) -8 8 1 C) -8 5 4 D) Not possible Answer: D 3 89) -3 -4

-5 4 9

2 5 6 B) -2 1 5 3 -5 C) -3 4 -4 9 -2 D) 1 5 A)

Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Let A, B, and C be the following matrices: 1 A= 2 B = 2 0 -2 C = -2 3 1 0 2 3 5 6 -3 -1 State whether or not each of the following calculations is possible. If possible, perform the calculation. 90) A + B Answer: The calculation is not possible. 91) B - C Answer: 4 -3 -3 -5 -4 6 92) 2A - B Answer: The calculation is not possible.

93) 2C + B Answer: -2 6 0 10 14 -3 State whether the calculation is possible. If possible, perform the calculation. 94) 1 -1 4 2 + 1 6 6 0 2 1 2 -3 1 Answer: -1 -5 7 -1 2 95) 3 14 + 30 0 17 8 Answer: 17 30 25 96) [6] + 1 0 01 Answer: The calculation is not possible. 97) 1 3 + [4 0 6] -2 Answer: The calculation is not possible.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 98) [4 0 6]

1 3 = -2

A) [4 0 -12] B) 4 12 -8 0 0 0 6 18 -12 C) 4 0 -12 D) 4 12 -8

0 6 0 18 0 -12

E) [-8] Answer: E

99)

1 3 [4 0 6] = -2 A) 4 0 6 12 0 18 -8 0 -12 B) 4 12 -8 0 0 0 6 18 -12 C) [-8] D) 4 0 -12 E) [4 0 -12] Answer: A

0 1 -3 -2 1 3 100) Let A = 1 0 5 and B = 1 3 -4 . Find the second row of AB. 0 42 3 0 1 A) [15 -1 2] B) [18 1 2] C) [18 -1 2] D) [15 1 2] E) none of these Answer: D 3 2 7 0 1 2 101) Let A = 1 0 3 and B = 1 5 2 . Find the third row of AB. 1 4 2 7 1 3 A) [31 11 16] B) [7 4 6] C) [25 18 58] D) [18 23 16] E) none of these Answer: D 102) Let A = 1 4 and B = 3 -2 . Find the entry in the second row, first column of AB. 2 3 4 6 A) 6 B) 34 C) 18 D) 22 E) 8 Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. State whether the calculation is possible. If possible, perform the calculation. 103) 7 -1 2 1 0 1 0 3 0 2 5 2 Answer: 15 -2 31 1 104) 1 2 [4 0 3] -5 Answer: 4 0 3 8 0 6 -20 0 -15

105) 2 7 6 4 1 6

3 1 -1 0

Answer: The calculation is not possible. 106) 1 0 -2 2 4 3

2 1 -1 50 1

Answer: The calculation is not possible. 107) 1 1 1

2 1 3 4 0 1 Answer:

6 5

108) If A =

1 6 -2 and B = 2 1 , find AB and BA. -2 1 1 3

Answer: AB = BA = ℐ2 109) If A = 1 4 and B = 1 2 , does AB = BA? Explain. 23 43 Answer: No. AB = 1 4 23

1 2 = 17 14 43 14 13

BA = 1 2 43

1 4 = 5 10 23 10 25

1 6 -3 110) Find a and b so that 7 7 4 0 0 1

a b 0 -4 28 0 2 6 1 = 2 36 15 4 2 2 4 2 2

Answer: a = -4, b = -2 111) If A4 = -3 4 and A5 = 4 -5 , what is A? 5 -6 -4 5 Answer: A = 0 -1 1 -2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. 112) A is 4 × 4, and B is 4 × 4. A) 4 × 8; 4 × 8 B) 8 × 4; 8 × 4 C) 4 × 4; 4 × 4 D) 1 × 1; 1 × 1 Answer: C 113) A is 2 × 1, and B is 1 × 2. A) 2 × 2; 1 × 1 B) AB does not exist; 1 × 1 C) 1 × 1; 2 × 2 D) 2 × 2; BA does not exist. Answer: A 114) A is 4 × 3, and B is 4 × 3. A) 4 × 3; 3 × 4 B) 3 × 4; 4 × 3 C) 4 × 4; 3 × 3 D) AB does not exist; BA does not exist. Answer: D Find the matrix product, if possible. -2 0 115) -2 3 32 -1 1 3 1 A) 2 -8 B) 4 -6 -4 -1 C) 1 3 -8 2 D) 4 0 -3 2 Answer: C 116) -1 3 36

0 -2 6 1 -3 2 0 -6 A) 18 3 -18 12 B) 3 -7 0 6 -24 30 3 6 -7 C) -24 0 30 D) Does not exist

Answer: B

50 117) 3 -2 1 0 4 -2 -2 2 15 0 A) 08 15 -6 B) -10 12 5 -6 15 -10 5 C) -6 12 -6 D) Does not exist Answer: D 30 -2 1 05 -3 -7 A) 9 25 3 -6 0 B) 0 0 25 C) -7 -3 25 9 D) Does not exist

118) 1 3 -2 3 0 5

Answer: A 119) Find the system of equations that is equivalent to the matrix equation, -2 5 1 4 A) -2x + 5y = 3 x + 4y = 6 B) -2x + 5x = 3 y + 4y = 6 C) -2x + y = 3 5x + 4y = 6 D) -2x + y = 3 5y + 4y = 6 Answer: A

x = 3 . y 6

120) Find the matrix equation that is equivalent to the system of equations,

x - 2y + z = 4 3x + y + z = 1 . x + y + 2z = 0

A) 1 3 1 -2 1 1 1 1 2

x 4 = y 1 . z 0

B) x y z

1 -2 1 4 = 3 1 1 1 . 1 1 2 0

C) 1 -2 1 3 1 1 1 1 2

x 4 y = 1 . z 0

D) x y z

1 3 1 4 -2 1 1 = 1 . 1 1 2 0

Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 121) Write the system of linear equations as a matrix equation. x + 3y = 6 2x - y = 3 Answer: 1 3 2 -1

x = 6 y 3

122) Write the system of linear equations as a matrix equation. x + 2y + 3z = 4 6y + 7z = 8 x = 5 Answer: 1 2 3 0 6 7 1 0 0

∙

x 4 y = 8 z 5

123) Write the system of linear equations as a matrix equation. x + y + 4z = 3 4x + y - 2z = -6 -3x + 2z = 1 Answer: 1 1 4 4 1 -2 -3 0 2

x 3 y = -6 z 1 29

124) Give the system of linear equations that is equivalent to the matrix equation: 3 1 2 x 1 -1 0 2 y = 2 . 0 4 1 z -1 Answer: 3x + y + 2z = 1 -x + 2z = 2 4y + z = -1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the matrix equation as a system of linear equations without matrices. 125) -7 0 x1 -8 = 8 1 1 x2 -8 -5 -7 A) -7x1 + x2 = -8 x1 + 8x2 = 8 -8x1 - 5x2 = -7 B) -7x1 + x2 + x3 = -8 x1 - 5x2 C) -7x1

=8 = -8

x 1 + x2 = 8 -8x1 - 5x2 = -7 Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 126) In a certain state legislature the percentage of legislators voting for or against lowering the legal drinking age by various party affiliations is summarized by this matrix. For Against Democrat 0.6 0.4 Republican 0.3 0.7 = A Independent 1 0 There are 60 Democratic, 30 Republicans, and 10 Independents in the legislature. Use a matrix calculation to determine how many legislators voted for lowering the drinking age and how many voted against lowering the drinking age. Answer: Let B = [ 60 30 10 ] 0.6 0.4 BA = [ 60 30 10 ] 0.3 0.7 = [ 55 45 ] 1 0 The vote is 55 in favor, 45 against.

127) A company has three factories I, II, and III. Each factory employs both skilled and unskilled worker. The break down of the number of workers by type in the three factories is summarized by matrix A below. The hourly wage of the skilled worker is $15 and that of the unskilled worker is $10. Multiply appropriate matrices and compute the total hourly pay at each factory. I II III skilled 25 34 24 A= unskilled 42 53 62 Answer: Let B = [ 15 10 ]. Then BA = [ 795 1040 980 ]. Therefore, the total hourly pay for factory I = $795, for II = $1,040, and for III = $980. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 128) Suppose one hour's output (measured in bottles) in a brewery is described by the following matrix:

Regular Beer Light Beer Malt Beer

Production line 1 300 400 100

Production line 2 200 100 50

Let H = x where x represents the number of hours Production line 1 is in operation in a day and y represents y the number of hours Production line 2 is in operation in a day. Then the entry in the second row, first column of AH, represents A) the total number of hours in a day spent producing light beer. B) the total number of hours in a day spent producing regular beer. C) the total number of bottles of regular beer produced in a day. D) the total number of bottles of light beer produced in a day. E) none of these Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 129) Two hundred students are registered in a certain mathematics course. Experience suggests that if x and y represent the number of students who earn a grade of C or better on Exam I and the number of students who earn a grade of D or F on Exam I, respectively, and if p and n represent the number of students who earn a grade of C or better on Exam II and the number of students who earn a grade of D or F on Exam II, respectively, then 0.9x + 0.3y = p 0.1x + 0.7y = n (a) Write the system of equations in matrix form. (b) Solve the resulting matrix equation for X = x . y (c) Suppose 120 students earn a grade of C or better on Exam II. How many students earned a grade of D or F on Exam I? Answer: (a) 0.9 0.3 0.1 0.7

x = p y n

(b) 7 1 6 2

x = 1 y 6 (c) y = -

3 2

p n

1 3 (120) + ( 80) = 100 6 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 130) Barnes and Able sell life, health, and auto insurance. Their sales, in dollars, for May and June are given in the following matrices. Life

Health Auto

20,000 15,000 May:

June:

8000

Able

30,000

0 17,000

Barnes

70,000

0 30,000

Able

20,000 25,000 32,000

Barnes

Find a matrix that gives total sales, in dollars, of each type of insurance by each salesman for the two-month period. A) 90,000 15,000 38,000 50,000 25,000 49,000 B) 90,000 0 38,000 50,000 0 49,000 C) 90,000 15,000 38,000 50,000 25,000 32,000 D) 140,000 25,000 49,000 Answer: A

131) The matrices give points and rebounds for five starting players in two games. Find the matrix that gives the totals. Points 20 16 F= 8 5 10

Rebounds 3 5 12 11 2

Points 18 11 S = 12 4 10

Rebounds 4 3 9 10 3

A) 62 5 B) 38 7 27 8 20 21 9 21 20 5 C) 5 62 D) 7 38 27 8 21 20 21 9 5 20 Answer: B

132) Find the inverse of the matrix 1 2 . 34 A) 1 3 2 2 -1

B) 1 -2 3 1 2 2 C) not defined D) 1 1 2 1 3

1 4

E) none of these Answer: B 133) Find the inverse of the matrix 1 -1 . -3 4 A) 4 -1 -3 1 B) not defined C) 4 1 -3 1 D) 4 1 3 1 E) none of these Answer: D 134) The matrix 3 x has no inverse if x is 4 36 A) 0. B) 3. C) 48. 1 D) . 3 E) 27. Answer: E

135) The matrix 3 12 has no inverse if x is x 36 A) 48. B) 9. C) 27. D) 12. E) 3. Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the inverse of the matrix, if it exists. 136) 1 0 01 Answer: 1 0 01 137) 3 1 62 Answer: Inverse does not exist. 138)

2 -1 -6 8 Answer: 0.8 0.6

0.1 0.2

139) 0.6 0.1 0.4 0.9 Answer:

1.8 -0.2 -0.8 1.2

140) 3 2 0 1 Answer:

1 2 3 3 0

141) 3 1 0 0 Answer: Inverse does not exist. 142) [6] Answer:

1 6

143) [0.4] Answer: [2.5]

144) Two n × n matrices A and B are called inverses of each other if both products AB and BA equal In . Are the following matrices inverses of each other? A = 7 4 , B = 3 -4 53 -5 7 Answer: Yes 145) Are the following matrices inverses of each other? A = 1 4 , B = -1 8 83 3 -2 Answer: No. 2 -2 -1 8 -3 5 146) Given that -5 3 4 and 4 -2 3 are inverses of each other, 7 -2 4 -6 5 4 8 -3 5 find A so that (A - I)-1 = 4 -2 3 . 7 -2 4 Answer: 3 -2 -1 A = -5 4 4 -6 5 5 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 147) 120 Let A = 3 1 0 and B = 002

x y 0 3 1 0 5 5 0

1 2

In order for A and B to be inverses, x and y must be 5 A) x = -5, y = . 3 B) x = 1, y = 0. 1 C) x = 1, y = . 2 D) x = -

1 2 ,y= . 5 5

E) none of these Answer: D

148) -4 1 2 xy4 7 -1 -4 Let A = 1 1 and B = 3 2 4 0 116 2 2 In order for A and B to be inverses, x and y must be A) x = 1, y = 1. B) x = -1, y = 1. C) x = 1, y = -1. D) x = -1, y =-1. E) none of these Answer: A Find the inverse, if it exists, for the matrix. 149) 2 -6 -3 1 A) 3 1 16 8 -

1 3 16 8

3 1 16 8

C) -

1 3 8 8

3 1 16 16

1 16

D) 3 8

3 1 16 8 Answer: B

150) 0 -2 5 4 A) 2 1 5 5 1 2

1 0 2

2 1 5 5 C) 2 1 5 5 -

1 0 2

D) 0 -

1 5

1 2 2 5

Answer: C 151) -6 -2 4 6 -

1 1 2 6

1 3

6 2 -4 -6 1 1 2 6

1 2

1 1 3 2

D) No inverse Answer: D

152) -1 4 0 6 A) 0

1 6

-1

2 3

B) -1 -

2 3

1 6

C) 1 6

2 3

0-1 D) -1

2 3

1 6

Answer: D 1 2 2 5 -2 -2 x 153) Given that the matrices A = 1 3 2 and B = -1 1 0 are inverses of each other, find the solution y of the 1 2 3 z -1 0 1 -5x - 2y - 2z = 1 system - x + y = 2. -x + z = -3 A) 5 -2 -2 1 2 . -1 1 0 0 1 -1 -3 B) 1 2 2 1 3 2 1 2 3

1 2 . -3

C) 1 2 -3

1 2 2 1 3 2 . 1 2 3

1 2 -3

5 -2 -2 -1 1 0 . -1 0 1

Answer: B

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 154) Solve the system

x + y = 2 by using the inverse of an appropriate matrix. 2x + y = -1

Answer: x = -3, y = 5 5 -2 -2 1 2 2 155) The matrices -1 1 0 and 1 3 2 are inverses of each other. 1 2 3 -1 0 1 x + 2y + 2z = 1 Use this fact to solve the system x + 3y + 2z = -1 . x + 2y + 3 z = -1 Answer: x = 9, y = -2, z = -2 5 2 -2 1 -2 0 2 1 -1 and -1 4 1 are inverses of each other. 1 -1 1 -3 -1 2 x - 2y = 5 Use this fact to solve the system -x + 4y + z = 1 . x - y+z = 2

156) The matrices

Answer: x = 23, y = 9, z = -12 157) Use the fact that

3 0 -6 0 -1 1 1 1 4 = 0 2 1 6 1 0 0 2

1 9

4 2 2 3 3 3

2 9

5 1 7 3 3 3

1 9

2 1 1 3 3 3

1 2 18 3

1 3

5 6

to solve the system 3x - 6z = 0 x + y + z + 4w = 2 . 2y + z + 6w = 1 x + 2w = 0 Answer: x = 2, y = 3, z = 1, w = -1

158) Consider the system 3x + 2y = 7 y=8 (a) Rewrite it in the form AX = B, where A, B, and X are appropriate matrices. (b) Find the inverse of A. (c) Solve the system by computing A-1B. Answer: (a) 3 2 0 1

x = 7 y 8

(b) 1 2 3 3 0

(c) x = -3, y = 8 2x + 3y = 4 -2x - y = 8 (a) Rewrite it in the form AX = B, where A, B, and X are appropriate matrices. (b) Find the inverse of A. (c) Solve the system by computing A-1B.

159) Consider the system

Answer: (a) 2 3 -2 -1

x = 4 y 8

(b) -

1 4

1 2

3 4 1 2

(c) x = -7, y = 6 160) Consider the system 0.7x + 0.2y = 3 0.3x + 0.8y = 2 (a) Rewrite it in the form AX = B where, A, B, and X are appropriate matrices. (b) Find the inverse of A. (c) Solve the system by computing A-1B. Answer: (a) 0.7 0.2 0.3 0.8

x = 3 y 2

(b) 1.6 -0.4 -0.6 1.4 (c) x = 4, y = 1

161) (a) Find the inverse of the matrix 2 1 . 31 (b) Use the result from (a) to solve the system 2x + y = 2 . 3x + y = -1 Answer: (a)

-1 1 3 -2

(b) x = -3, y = 8

162) Find the 2 × 2 matrix C such that AC = B, where A = 3 2 and B = 4 1 . 75 2 -3 Answer: C=

16 11 -22 -16

1 1 0 123 163) Find the 3 × 3 matrix C such that AC = B, where A = 0 -1 0 and B = 4 5 6 . 3 0 1 789 Answer: 5 7 9 C = -4 -5 -6 -8 -13 -18 164) Solve the equation AX = B, where A = 13 -5 and B = 1 . 2 -5 2 Answer: X = 12 31 165) Let A and B be n × n invertible matrices and C be an n × 1 column vector. Solve AX + BY = C for X. Answer: X = A-1 (C - BY) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the inverse doesn't exist. 166) -3x + 9y = 9 3x + 2y = 13 A) x = -2, y = -3 B) x = 2, y = 3 C) x = -3, y = -2 D) x = 3, y = 2 Answer: D 167) 3x + 6y = -3 2x - 2y = -14 A) x = -5, y = 2 B) No inverse, no solution for system C) x = 2, y = -5 D) x = -5, y = -2 Answer: A

168) 2x + y = 2 5x + 3y = 4 A) x = -2, y = -2 B) x = 2, y = -2 C) x = -2, y = 2 D) No inverse, no solution for system Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 169) A charitable organization estimates that 60% of the people in a certain area who make a contribution in one year will also contribute the next year, and that 10% of those who do not contribute one year will contribute the next. Let x and y denote the number of people who contribute in one year respectively, and let u and v be the corresponding numbers for the following year. (a) Write a matrix equation relating x to u . y v x (b) Solve the equation for . y (c) Suppose that out of 10,000 people, 1500 made a contribution this year. How many made a contribution last year? Answer: (a) 0.6 0.1 0.4 0.9

x = u . y v

(b) x = 1.8 -0.2 y -0.8 1.2

u v

(c) 1000 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 170) A bakery sells three types of cakes, each requiring the amount of ingredients shown.

flour (cups) sugar (cups) eggs

Cake I Cake II Cake III 2 4 2 2 1 2 2 1 3

To fill its orders for these cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 62 eggs. How many cakes of each type were made? A) 6 cake I, 8 cake II, 14 cake III B) 8 cake I, 8 cake II, 14 cake III C) 14 cake I, 8 cake II, 6 cake III D) 30 cake I, 8 cake II, 12 cake III Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 171) Use the Gauss-Jordan method to compute the inverse of the matrix 3 1 . 21

Answer:

1 1 1 1 1 1 1 1 3 3 1 0 3 1 1 0 , 1 3 3 0 , , 3 3 , 1 0 1 -1 1 2 2 1 0 1 0 1 -2 3 0 1 2 1 0 1 0 1 -2 3 3 3 3 1 -1 = 1 -1 2 1 -2 3

1 1 1 172) Use the Gauss-Jordan method to find the inverse of the matrix A, where A = 1 1 2 . 4 5 5 Answer: 5 0 -1 A-1 = -3 -1 1 -1 1 0 173) Use the Gauss-Jordan method to compute the inverse of the matrix, if it exists. -1 2 -3 7 Answer: -7 2 -3 1 1 2 1 174) Use the Gauss-Jordan method to explain why the matrix 1 1 0 has no inverse. 0 1 1 Answer: Row reduction of the original matrix results in a matrix with a row of zeros. -1 2 -4 -1 175) Use the Gauss-Jordan method to compute 1 -1 3 . 0 0 1 Answer: 1 2 -2 1 1 1 0 0 1 176) Use the Gauss-Jordan method to compute the inverse of the matrix, if it exists. 1 2 3 4 5 6 7 8 9 Answer: Inverse does not exist. 177) Use the Gauss-Jordan method to compute the inverse of the matrix, if it exists. 2 6 0 -1 -3 0 7 7 1 Answer: Inverse does not exist.

1 0 0 178) If A-1 = -2 0 1 , find A. 3 1 5 Answer: 1 0 0 -13 -5 1 2 1 0 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the inverse, if it exists, of the given matrix. 5 -6 0 5 179) A = 15 -12 0 10 6 -24 0 15 0 0 -6 0 A) 2 1 0 0 5 5 11 10

3 10

1 6

0 -

1 6

48 25

14 25

1 5

1 6

1 5

1 15

1 12

1 10

1 6

1 24

1 15

1 6

2 5

11 10

48 25

3 10

0 -

14 25

1 6

C) -

1 5

1 6

1 5 0

D) No inverse Answer: A

0 180) A = -3 0 A)

3 0 8

3 6 0

2 3

1 3

1 4

1 8

B) Does not exist C) 2 1 1 3 3 4 0

1 3

1 8 -

1 8

D) -

2 3

1 8

1 3

1 4

Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 181) Use matrix inversion to solve the system of linear equations. 5x + 2y - 2z = 5 2x + y - z = 1 -3x - y + 2z = 2 Answer: x = 3, y =1, z = 6 182) Use matrix inversion to solve the system of linear equations. x + y +3z = 10 2x + y - z = 0 -x - y + 2z = 5 Answer: x = 2, y =-1, z = 3 183) Use matrix inversion to solve the system of linear equations. x+y+z=6 x-y+z=3 x-y-z=0 Answer: x = 3, y =

3 3 ,z= 2 2

1 0 2 184) (a) Using the Gauss-Jordan method, find the inverse of the matrix 0 1 -4 . 0 0 2 1 0 2 x 1 (b) Based on your answer to (a), solve the system 0 1 -4 y = 2 . 0 0 2 z 3 Answer: (a) 1 0 -1 0 1 2 1 0 0 2 (b) x = -2, y = 8, z =

3 2

-2 2 -1 185) (a) Using the Gauss-Jordan method, find the inverse of the matrix 3 -5 4 . 5 -6 4 1 -2 2 -1 x (b) Based on your answer to (a), solve the system 3 -5 4 y = 0 . 5 -6 4 z 3 Answer: (a) 4 -2 3 8 -3 5 7 -2 4 (b) x = 13, y =23, z =19 1 0 1 186) (a) Using the Gauss-Jordan method, find the inverse of the matrix 2 1 0 . 0 1 -1 (b) Based on your answer to (a), solve the following system. x +z=2 2x + y =3 y-z=4 Answer: (a) -1 1 -1 2 -1 2 2 -1 1 (b) x = -3, y = 9, z = 5 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the system of equations by using the inverse of the coefficient matrix. 187) x + y + z = 3 x - y + 5z = 17 5x + y + z = 7 A) x = 1, y = -1, z = 3 B) x = 3, y = 1, z = -1 C) x = 3, y = -1, z = 1 D) No inverse, no solution for system Answer: A

188) x - y + 4z = 17 5x + z = 5 x + 2y + z = 11 A) x = 5, y = 0, z = 3 B) x = 0, y = 3, z = 5 C) x = 5, y = 3, z = 0 D) No inverse, no solution for system Answer: B 189) x - y + z = 6 x+y+z= 8 x+y-z= 4 A) x = 5, y = 1, z = 2 B) x = 5, y = 2, z = 1 C) x = 2, y = 5, z = 1 D) No inverse, no solution for system Answer: A 190) x - y + 4z = 1 5x + z = 0 x + 2y + z = -2 A) x = 0, y = 0, z = -1 B) x = 0, y = -1, z = 0 C) x = 0, y = -1, z = 1 D) No inverse, no solution for system Answer: B Solve the problem. 191) A company makes 3 types of cable. Cable A requires 3 black wires, 3 white wires, and 2 red wires. Cable B requires 1 black, 2 white, and 1 red. Cable C requires 2 black, 1 white, and 2 red. If 100 black wires, 110 white wires, and 90 red wires were used, then how many of each cable were made? A) 10 cable A, 103 cable B, 20 cable C B) 10 cable A, 30 cable B, 20 cable C C) 20 cable A, 30 cable B, 10 cable C D) 10 cable A, 30 cable B, 93 cable C Answer: B 192) A company makes 3 types of cable. Cable A requires 3 black wires, 3 white wires, and 2 red wires. Cable B requires 1 black, 2 white, and 1 red. Cable C requires 2 black, 1 white, and 2 red. If 95 black wires, 100 white wires, and 90 red wires were used, then how many of each cable were made? A) 5 cable A, 18 cable B, 25 cable C B) 30 cable A, 5 cable B, 25 cable C C) 5 cable A, 30 cable B, 25 cable C D) 58 cable A, 30 cable B, 22 cable C Answer: C

193) A basketball fieldhouse seats 15,000. Courtside seats cost $8, endzone seats cost $6, and balcony seats cost $5. The total revenue for a sellout is $86,000. If half the courtside seats, half the balcony seats, and all the endzone seats are sold; then the total revenue is $49,000. How many of each type of seat are there? A) 4000 courtside, 3000 endzone, 8000 balcony B) 3200 courtside, 1800 endzone, 10,000 balcony C) 3000 courtside, 2000 endzone, 10,000 balcony D) 3000 courtside, 4000 endzone, 8000 balcony Answer: C 194) The economy of a small country can be regarded as consisting of three industries, I, II, and III, whose input-output matrix is I II III I 0.20 0.01 0.30 A = II 0.30 0.10 0.02 III 0.05 0.40 0.10 The entry in the third row, second column of matrix A means that A) industry II uses 40% of industry III's output. B) industry III uses 40% of industry II's output. C) to make $1 worth of output, industry III needs $0.40 worth of input from industry II. D) to make $1 worth of output, industry II needs $0.40 worth of input from industry III. E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 195) Solve the following matrix equation for X: AX = BX + C. Answer: X = (A - B)-1 C 196) Solve the matrix equation for AX = BX + C for X =

x1 x2

6 0 , B = -1 -2 , and C = 1 . 15 1 4 -2 2

where A =

Answer: X =

1 -3

197) If A is given by A = 0.6 0.4 , find I - A. 0.2 0.3 Answer: 0.4 -0.4 -0.2 0.7 198) Let A = 3 6 . Find (I - A) -1 . 24 Answer: 1 -1 2 -

1 3

199) If A is given by A = 0.6 0.4 , find (I - A) -1 0.2 0.3 Answer: 3.5 2 1 2 200) Solve the matrix equation X = AX + D for the unknown matrix X = x , where A = 9 3 , D = 30 . y 2 3 40 Answer: x = 6, y = -26 0.3 0.5 0 x 10 201) Let A = 0 0.2 0.4 , X = y , and D = 20 . Solve the matrix equation (I - A)X = D. 0.1 0.1 0.2 z 30 Answer: I - A=

0.7 -0.5 0 1.5 1 0.5 50 -1 , (I A) , X = = 0 0.8 -0.4 0.1 1.4 0.7 50 0.2 0.3 1.4 50 -0.1 -0.1 0.8

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 202) Find the solution X for the matrix equation X - AX = D. A) X = D(A - I)-1 B) X = (A - I)-1 D C) X = D(I - A)-1 D) X = (I - A) -1 D E) none of these Answer: D 203) Find the solution X for the matrix equation XA + XB =C. A) X = (B + A) -1 C B) X = (A + B)-1 C C) X = C(B + A) -1 D) X = C(A + B)-1 E) none of these Answer: D

Solve the problem. 204) The economy of a small country can be regarded as consisting of three industries, I, II, and III, whose input-output matrix is I II III I 0.20 0.01 0.30 A = II 0.30 0.10 0.02 III 0.05 0.40 0.10 Suppose x, y, and z represent the output of industries I, II, and III, respectively. An algebraic expression for the amount of output from industry III that can be exported is A) z + ( 0.30x + 0.02y + 0.10z). B) z - (0.30x + 0.02y + 0.10z). C) z + (0.05x + 0.40y + 0.10z). D) z - (0.05x + 0.40y + 0.10z). E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 205) An economy consists of steel and coal. The steel industry consumes $ 0.25 of coal and $ 0.02 of steel to produce $1 of steel. The coal industry requires $0.04 of coal and $0.01 of steel to produce $1 of coal. (a) If the total production in millions of dollars of steel and coal is given by the matrix X = 2 , find the 3 amounts of steel and coal used in production. (b) Compute the amounts of steel and coal available which can be consumed or exported. Answer: (a)

0.02 0.01 0.25 0.04

2 = 0.07 , that is $70,000 steel, $620,000 coal. 3 0.62

(b) $1.93 million steel, $2.38 million coal 206) A large insurance company has a data processing division and an actuarial division. For each $1 worth of output the data processing division needs $0.20 worth of data processing and $0.40 worth of actuarial science. For each $1 worth of output, the actuarial division needs $0.30 worth of data processing and $0.10 of actuarial science. (a) Write the input-output matrix for the insurance company. (b) The company estimates a demand of 5 million dollars for the data processing division and 2 million for the actuarial division. Let x be the production level of the data processing division and let y be the production level of the actuarial division. Write a matrix equation to fit this problem. (c) At what levels should each division produce to meet the demand? Answer: (a) DPD AD (b)

DPD AD 0.20 0.30 0.40 0.10

x = 0.20 0.30 y 0.40 0.10

x + 5 y 2

207) The economy of Mongovia is composed of three industries: I, II, and III. Suppose that in order to produce $1 worth of industry I, it takes $0.02 worth of I, $0.20 worth of II and $0.30 worth of III. To produce $1 worth of industry II, its takes $0.05 worth of I, $0.02 worth of I, and $0.20 worth of III. To produce $1 worth of industry III, it takes $0.10 worth of I, $0.15 worth of II, and $0.02 worth of III. (a) Write the input-output matrix for this economy. (b) How much should each industry produce to allow for consumption at these levels: $3 million for I, $2 million for II, and $3 million for III? Answer: (a)

I II III I 0.02 0.05 0.10 II 0.20 0.02 0.15 III 0.30 0.20 0.02

(b) $3.75 million worth of industry I, $3.56 million worth of industry II, $4.93 million worth of industry III. 208) An economy consists of steel and coal. The steel industry consumes $ 0.25 of coal and $ 0.02 of steel to produce $1 of steel. The coal industry requires $0.04 coal and $0.01 steel to produce $1 coal. (a) What is the input-output matrix for this economy? (b) What is the amount of coal needed for a steel production level of $30 million? Answer: (a)

steel coal steel 0.02 0.01 coal 0.25 0.04 (b) $7.5 million

209) The economy of a small country can be regarded as consisting of three industries, I, II, III, whose input-output matrix is I II III I 0.2 0.3 0.05 A = II 0.01 0.1 0.4 III 0.3 0.02 0.1 If x, y, and z represent the outputs of industries I, II, and III respectively, write an algebraic expression for the amount of output from industry II that can be exported. Answer: y - (0.01x + 0.1y + 0.4z) 210) An economy consisting of agriculture (I) and manufacturing (II) has the following input-output matrix. I II I 0.1 0.3 A= II 0.3 0.4 How many units of agriculture and manufacturing should be produced in order to meet a demand for 15 units from I and 9 units from II? Answer: 26 units of agriculture and 28 of manufacturing

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 211) Suppose the following matrix represents the input-output matrix of a primitive economy that depends on two basic goods, yams and pigs. How much of each commodity should be produced to get 12 bushels of yams and 14 pigs? (It is not necessary to round to a whole-number quantity of pigs.) yams pigs yams 1/4 1/6 pigs 1/2 0 A) 33.75 bushels of yams and 18.75 pigs B) 21.5 bushels of yams and 29.25 pigs C) 21.5 bushels of yams and 24.75 pigs D) 28.5 bushels of yams and 18.75 pigs Answer: C 212) Suppose the following matrix represents the input-output matrix of a simplified economy that involves just three commodity categories: manufacturing, agriculture, and transportation. How many units of each commodity should be produced to satisfy a demand of 1300 units for each commodity? Mfg Agri Trans Mfg 0 1/4 1/3 Agri 1/2 0 1/4 Trans 1/4 1/4 0 A) 3263 units of manufacturing, 3640 units of agriculture, and 3029 units of transportation B) 3640 units of manufacturing, 3029 units of agriculture, and 3263 units of transportation C) 3237 units of manufacturing, 3666 units of agriculture, and 3029 units of transportation D) 3263 units of manufacturing, 3029 units of agriculture, and 3237 units of transportation Answer: C 213) Suppose the following matrix represents the input-output matrix of a primitive economy that depends on two basic goods, yams and pigs. How much of each commodity should be produced to satisfy a demand for 63 bushels of yams and 52 pigs? (It is not necessary to round to a whole-number quantity of pigs.) yams pigs yams 1/4 1/6 pigs 1/2 0 A) 107.5 bushels of yams and 105.75 pigs B) 153 bushels of yams and 74.25 pigs C) 107.5 bushels of yams and 129.38 pigs D) 133.5 bushels of yams and 74.25 pigs Answer: A

214) Suppose the following matrix represents the input-output matrix of a simplified economy with just three sectors: manufacturing, agriculture, and transportation. Mfg Agri Trans Mfg 0 0.25 0.33 Agri 0.50 0 0.25 Trans 0.25 0.25 0 Suppose also that the demand matrix is as follows: 504 D = 267 157 Find the amount of each commodity that should be produced. A) 1114 units of manufacturing, 1061 units of agriculture, and 732 units of transportation B) 928 units of manufacturing, 884 units of agriculture, and 610 units of transportation C) 928 units of manufacturing, 1061 units of agriculture, and 610 units of transportation D) 670 units of manufacturing, 635 units of agriculture, and 93 units of transportation Answer: B 215) Suppose the following matrix represents the input-output matrix, T, of a simplified economy.

Manufacturing Agriculture Transportation

Manufacturing Agriculture Transportation 0 1/4 1/3 1/2 0 1/4 1/4 1/4 0

The demand matrix is 1300 D = 1300 1300 Find the amount of each commodity that should be produced. A) 3224 units of manufacturing, 3668.6 units of agriculture, and 3023.8 units of transportation. B) 3263 units of manufacturing, 3029 units of agriculture, and 3224 units of transportation. C) 3640 units of manufacturing, 3029 units of agriculture, and 3263 units of transportation. D) 3263 units of manufacturing, 3640 units of agriculture, and 3023.8 units of transportation. Answer: A 216) A simplified economy is based on agriculture, manufacturing, and transportation. Each unit of agricultural output requires 0.3 unit of its own output, 0.5 of manufacturing, and 0.2 unit of transportation output. Each unit of manufacturing output requires 0.3 unit of its own output, 0.1 of agricultural, and 0.4 unit of transportation output. Each unit of transportation output requires 0.4 unit of its own output, 0.2 of agricultural, and 0.1 of manufacturing output. There is demand for 80 units of agricultural, 65 units of manufacturing, and 15 units of transportation output. How many units should each segment of the economy produce? A) Agriculture: 179; manufacturing: 251; transportation: 339 B) Agriculture: 339; manufacturing: 179; transportation: 251 C) Agriculture: 339; manufacturing: 251; transportation: 179 D) Agriculture: 251; manufacturing: 339; transportation: 179 Answer: C 54

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Indicate whether the inequality is true or false. 1) -23 ≥ -23 A) True B) False Answer: A 2) 13 ≤ -14 A) True B) False Answer: B Solve. 3) 4x - 10 ≤ 26 A) x ≥ -9 B) x ≤ -9 C) x ≤ 9 D) x ≥ 9 Answer: C 4) -6 - 2x < 0 A) x < -4 1 B) x > 3 C) x > -3 D) x < -3 Answer: C 5) Rewrite the inequality 3x - 2y ≥ 12 in standard form. 3 A) y ≤ x - 6 2 B) y ≤ C) y ≥

3 x+6 2

D) y ≥ -

3 x+6 2

E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) Rewrite the inequality 3x - 5y ≤ 2 in standard form. Answer: y ≥

3 2 x5 5

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) Which of the following points satisfy the linear inequality x - 3y ≥ 4? A) (1, 1) B) (0, 3) C) (2, -3) D) (3, 0) E) none of these Answer: C 8) Which of the following points satisfy the linear inequality 2x + 4y ≤ 7? A) (-1, 3) B) (0, 2) C) (1, 1) D) (2, 4) E) none of these Answer: C Classify the point as "above," "below," or "on" the line y = -2x + 7. 9) (0, 0) A) above B) on C) below Answer: C 10) ( 3, 3) A) above B) below C) on Answer: A 11) (-3, 4) A) on B) above C) below Answer: C 12) (2, 3) A) below B) on C) above Answer: B 13) (0, 8) A) on B) below C) above Answer: C

Decide if the given ordered pair is a solution to the inequality. 14) x + y > 5: (-1, 9) A) Yes B) No Answer: A 15) x - y ≤ 8: A) Yes B) No

(-3, -14)

Answer: B 16) 2x + 3y ≤ -6: A) Yes B) No

(6, 9)

Answer: B 17) x + 2y > 3: A) Yes B) No

(-3, 4)

Answer: A 18) y ≥ -x: (-23, 23) A) Yes B) No Answer: A 19) y <

1 x - 2: 3

(9, 0)

A) Yes B) No Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Graph the linear inequality by shading the points not satisfying it. 1 20) y ≤ x + 1 2 y 10

-10

-5

-10

Answer: y 10

-10

-5

-10

21) x ≤ 2y y 10

-10

-5

-10

Answer: y 10

-10

-5

-10

22) Rewrite the inequality x + 2y ≥ 4 in standard form, then graph it. y 10

-10

-5

-10

Answer: y ≥ -

1 x+2 2 y 10

-10

-5

-10

23) Rewrite the inequality 3x - 2y ≥ 12 in standard form, then graph it. y 10

-10

-5

-10

Answer: y ≤

3 x-6 2 y 10

-10

-5

-10

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the inequality by crossing out (i.e., discarding) the points not satisfying the inequality. 24) x ≤ -1 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: A 8

25) y ≥ 4 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: B Graph the inequality by shading the points that do not satisfy it. 26) 3x + y ≤ 6 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: A

27) x + 3y ≥ 3 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: B 28) -3x - 4y ≤ 12 y 10

-10

-5

-10

A) y 10

-10

-5

5 -5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: C

29) 2x + 4y ≥ -8 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: B 30) y ≤ -4x + 1 y 10

-10

-5

-10

A) y 10

-10

-5

5 -5

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: B

31) x ≥ 5 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: D 32) y ≤ -2 y 10

-10

-5

-10

A) y 10

-10

B) y 10

-10

-5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: C

33) y ≥ 4x y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5

-10

Answer: B 34) Which one of the following points is in the feasible set of the system of inequalities? x ≥ 0, y ≥ 0 x + y ≤ 12 2x + 5y ≤ 50 A) (1, -2) B) (1, 10) C) (-2, 0) D) (3, 2) E) none of these Answer: D 35) Which of the following points is in the feasible set of the system of inequalities? y≥0 x-y≤3 2x +y ≤ 5 A) (3, 2) B) (2, 3) C) (1, 2) D) (4, -6) E) none of these Answer: C 36) Which of the following points is in the feasible set of the system of inequalities? x≥0 x+y≤1 x - 2y ≤ 8 A) (4, 5) B) (-1, 2) C) (3, -2) D) (2, 0) E) none of these Answer: C

37) Which of the following points is in the feasible set of the system of inequalities? x+y≥3 3x - y ≥ -1 x≤3 A) (-1, 2) B) (0, 0) C) (1, 6) D) (2, 4) E) none of these Answer: D 38) Determine whether the point (7, 8) is in the feasible set of the system of inequalities: 2x + 6y ≤ 66 4x + 2y ≤ 48 x + y ≤ 14 x ≥ 0, y ≥ 0 A) Yes B) No Answer: B 39) Determine whether the point (12, 0) is in the feasible set of the system of inequalities: 2x + 6y ≤ 66 4x + 2y ≤ 48 x + y ≤ 14 x ≥ 0, y ≥ 0 A) Yes B) No Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 40) Find a system of inequalities that has the following feasible set. y

A FS

A = (0, 1), B = (4, -1), C = (0, -2), D = (-1, 0) y≤Answer:

1 x+1 2

y≤x+ 1 1 y≥ x- 2 4 23

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the feasible set of the system of linear inequalities. ≤ 41) y -2x - 2 y≥x+8 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5

-10

D) y 10

-10

-5 -5

-10

Answer: B ≤ 42) y -3x + 3 y≥x-6

y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: B 26

43)

x ≥ 2y x + 2y ≤ 3 y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5

-10

D) y 10

-10

-5 -5

-10

Answer: D ≥ 44) y 5 x ≥ -4

y 10

-10

-5

-10

A) y 10

-10

-5

-10

B) y 10

-10

-5 -5

-10

C) y 10

-10

-5 -5

-10

D) y 10

-10

-5 -5

-10

Answer: D 29

45) The feasible set of a certain linear programming problem is given by the following system of linear inequalities. Without graphing this set, determine which of the following points is not in the feasible set. x + 3y ≤ 6 x- y≤2 -5x + y ≤ 2 A) (3, 1) B) (-1, 2) C) (1, 1) D) (0, 0) E) none of these Answer: B 46) The feasible set of a certain linear programming problem is given by the following system of linear inequalities. Without graphing this set, determine which of the following points is not in the feasible set. 4x - 3y ≥ 15 2x + 6y ≥ 7 ≤6 y A) (7, 3) B) (1, -3) C) (8, -1) D) (8, 4) E) none of these Answer: B 47) The feasible set of a certain linear programming problem is given by the following system of linear inequalities. Without graphing this set, determine which of the following points is not in the feasible set. x + 2y ≤ 6 x- y≥5 ≤3 x A) (0, 1) B) (2, -3) C) (3, -4) D) (-1, -8) E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the given point is in the feasible set of a linear programming problem with constraints. Justify your answer. x + y ≤ 10 2x + 3y ≥ 12 x ≥ 0, y ≥ 0 48) (1, 2) Answer: No; x = 1, y = 2, does not satisfy the second inequality. 49) (3, 7) Answer: Yes; x = 3, y = 7, satisfies all four inequalities.

Determine whether the given point is in the feasible set of a linear programming problem with constraints. Justify your answer. x - 3y ≤ 20 4x + 5y ≥ 11 x ≥ 0, y ≥ 0 50) (8,3) Answer: Yes; x = 8, y = 3, satisfies all the second inequalities. 51) (3, -7) Answer: No; x = 3, y = -7 does not satisfy the last inequality.. 52) Does (3, 2) satisfy the following system of equations? x ≥ 0, y ≥ 0 x + y ≤ 10 1 x+ y≤ 6 2 3y + 2x ≤ 12 Answer: Yes 53) If (2, y) satisfies 3x + y ≤ 10, what is the largest value that y can take? Answer: 4 54) If (-3, y) satisfies 2x + y ≥ 5, what is the smallest value that y can take? Answer: 11 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A candy merchant sells two variety bags of candy. Each pound of variety bag A contains 60% caramels and 40 % chocolates and sells for $8 a pound. Each pound of variety bag B contains 45% caramels and 55% chocolates and sells for $10 a pound. The merchant has available 400 pounds of caramels and 300 pounds of chocolate. The merchant will try to sell the amount of each blend that maximizes her income. Let x be the number of pounds of variety bag A and y be the number of pounds of variety bag B. 55) Since the merchant above has available 300 pounds of chocolates, one inequality that must be satisfied is in the situation above is A) 0.60x + 0.40y ≤ 300 B) 0.60x + 0.45y ≤ 300 C) 0.45x + 0.55y ≤ 300 D) 0.40x + 0.55y ≤ 300 E) none of these Answer: D 56) In the situation above, the objective function is A) 8x + 10y B) 0.45x + 0.55y C) 0.60x + 0.40y D) 400x + 300y E) none of these Answer: A

A coffee merchant sells two blends of coffee. Each pound of blend A contains 80% Mocha Java and 20 % Jamaican and sells for $2 a pound. Each pound of blend B contains 35% Mocha Java and 65% Jamaican and sells for $2.25 a pound. The merchant has available 1000 pounds of Mocha Java and 600 pounds of Jamaican. The merchant will try to sell the amount of each blend that maximizes her income. Let x be the number of pounds of blend A and y be the number of pounds of blend B. 57) Since the merchant above has available 1000 pounds of Mocha Java, one inequality that must be satisfied is A) 0.8x + 0.2y ≤ 1000. B) 0.8x + 0.35y ≥ 1000. C) 0.8x + 0.35y ≤ 1000. D) 0.35x + 0.65y ≥ 1000. E) none of these Answer: C 58) In the situation above, the objective function is A) 0.80x + 0.20y. B) 0.35x + 2y. C) 2.25x + 0.2y. D) 1000x + 600y. E) none of these Answer: E A small manufacturing plant produces three kinds of bicyclesthree-speed, five-speed and ten-speedin two factories. Factory A produces 16 three-speeds, 12 five-speeds and 30 ten-speeds in one day, while factory B produces 15 three-speeds, 18 five-speeds and 20 ten-speeds in one day. An order is received for 30 three-speeds, 40 five-speeds, and 50 ten-speeds. It costs $1200 a day to operate factory A and $3000 a day to operate factory B. The manufacturer chooses the number of days to operate each factory in order to minimize cost. 59) In the situation above, the variables are A) x = the number of three-speeds produced y = the number of five-speeds produced z = the number of ten-speeds produced B) x = the total cost to operate factory A y = the total cost to operate factory B C) x = the number of days spent producing three-speeds y = the number of days spent producing five-speeds z = the number of days spent producing ten-speeds D) x = the number of days factory A is operated y = the number of days factory B is operated E) none of these Answer: D 60) In the situation above, the objective function is A) 30x + 40y + 50z. B) 3000x + 1200y. C) 58x + 53y. D) 1200x + 3000y. E) none of these Answer: D

61) In the situation above, which of the following inequalities must be satisfied? A) x + y ≥ 120 B) 30x + 20y ≤ 50 C) 16x + 15y ≥ 30 D) x + y + z ≥ 120 E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 62) Two kinds of crated cargo, A and B, are to be shipped by truck. Each crate of cargo A is 50 cubic feet in volume and weighs 200 pounds, whereas each crate of cargo B is 10 cubic feet volume and weighs 360 pounds. The shipping company charges $75 per crate for cargo A and $100 per crate for cargo B. The truck has a maximum load limit of 7,200 pounds and 1,000 cubic feet. (a) Fill in the chart below. A B Truck Load Limit Volume Weight Charge (b) Let x be the number of crates of cargo A and y the number of crates of cargo B shipped by one truck. Using the chart in (a), give two inequalities that x and y must satisfy because of the truck's load limits. (c) Give the inequalities that x and y must satisfy because they cannot be negative. (d) Give an expression for the total charges from shipping x crates of cargo A and y crates of cargo B. (e) Graph the feasible set of this problem. y

Answer: (a) Volume Weight Charge

A 50 200 75

B 10 360 100

Truck Load Limit 1,000 7,200

50x + 10y ≤ 1000 200x + 360y ≤ 7200 (c) x ≥ 0, y ≥ 0 (d) [charges] = 75x + 100y (e) (b)

63) A small manufacturing plant produces two kinds of bicycles1-speed and 10-speed. The plant can manufacture up to 400 of the 1-speed bicycles and up to 500 of the 10-speed bicycles per month, but it can only manufacture 650 of both kinds in all every month. It takes 40 man-hours to manufacture a 1-speed bicycle and 80 man-hours for the 10-speed; there are 44,000 man hours available per month. The profit of the 1-speed bicycle is $20 and the profit of the 10-speed bicycle is $30. (a) Let x and y represent the number of 1-speed and 10-speed bicycles manufactured per month, respectively. Give all the inequalities that x and y must satisfy. (b) Express the profit as a linear function of x and y. (c) Graph the feasible set for this problem. y

Answer: (a) x ≤ 400, y ≤ 500, x ≥ 0, y ≥ 0, x + y ≤ 650, 40x + 80y ≤ 44,000 (b) [profit] = 20x + 30y (c)

y 500 400 300 200

100

200

300

400

500 x

64) A television manufacturing company has two factoriesI and IIthat manufacture color and monochrome TV sets. Each day, factory I produces 50 color and 20 monochrome TV sets at a cost of $3,000. Each day, factory II produces 30 color and 24 monochrome TV sets at a cost of $2,700. An order is received for 2700 color and 1800 monochrome TV sets. (a) Let x and y represent the number of days factory I and factory II must operate, respectively, to fill the order. Give all constraints on x and y in the form of linear inequalities. (b) Write the total daily operating cost as a function of x and y. (c) Graph the feasible set for this problem. y

Answer: (a) 50x + 30y ≥ 2700 20x + 24y ≥ 1800 x ≥ 0, y ≥ 0 (b) [cost] = 3000x + 2700y (c) 100

80 60

40 20

65) A furniture-finishing plant finishes two kinds of tablesA and B. Table A requires 8 minutes of staining and 10 minutes of varnishing, whereas table B requires 12 minutes of staining and 9 minutes of varnishing. The staining facility is available at most 480 minutes per day and the varnishing facility is available at most 450 minutes per day. The profit is $5 on each A-table and $3 on each B-table. (a) Fill in the chart: A B Available Time Staining Varnishing Profit (b) Let x be the number of A-tables and y the number of B-tables finished per day. Give all the inequalities that x and y must satisfy. (c) Express the total daily profit as a linear function of x and y. (d) Graph the feasible set for this problem. y

Answer: (a) Staining Varnishing Profit

A 8 10 5

B 12 9 3

Available Time 480 450

(b) 8x + 12y ≤ 480 10x + 9y ≤ 450 x ≥ 0, y ≥ 0 (c) [profit] = 5x + 3y (d)

40 30 20 FS 10

50 x

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the system of inequalities that describes the possible solutions to the problem. 66) The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. The company can produce up to 40 rings each day, using up to 100 total man-hours of labor. It takes 2 man-hours to make one VIP ring and 5 man-hours to make one SST ring. Let x represent the number of VIP rings, and let y represent the number of SST rings. Find the system of inequalities that describes this company's ring production. A) x ≤ 40, y ≤ 40 5x + 2y ≤ 100 x ≥ 0, y ≥ 0 B) x ≤ 40, y ≤ 40 2x + 5y ≤ 100 x ≥ 0, y ≥ 0 C) x + y ≤ 40 5x + 2y ≤ 100 x ≥ 0, y ≥ 0 D) x + y ≤ 40 2x + 5y ≤ 100 x ≥ 0, y ≥ 0 Answer: D

67) The Pen-Ink Company manufactures two ballpoint pens: silver and gold. The silver requires 13 minutes in a grinder and 6 minutes in a bonder. The gold requires 9 minutes in a grinder and 2 minutes in a bonder. The grinder can be run no more than 460 minutes per day and the bonder no more than 200 minutes per day. Let x represent the number of silver pens, and let y represent the number of gold pens. Find the system of inequalities that represents this company's daily production of silver and gold pens. A) 13x + 6y ≤ 460 9x + 2y ≤ 200 x ≥ 0, y ≥ 0 B) x + y ≤ 460 x + y ≤ 200 x + y ≤ 660 x ≥ 0, y ≥ 0 C) 0 ≤ 19x + 11y ≤ 660 x ≥ 0, y ≥ 0 D) 13x + 9y ≤ 460 6x + 2y ≤ 200 x ≥ 0, y ≥ 0 Answer: D 68) A certain area of forest is populated by two species of animals, which scientists refer to as A and B for simplicity. The forest supplies two kinds of food, referred to as F1 and F2. For one year, an animal of species A requires 1.4 units of F1 and 1.2 units of F2 . For one year an animal of species B requires 2.1 units of F1 and 1.8 units of F2 . The forest can normally supply at most 913 units of F1 and 481 units of F2 per year. Let a represent the number of animals of species A and b represent the number of animals of species B. A) 1.4a + 2.1 b ≤ 913 1.2 a + 1.8b ≤ 481 a ≥ 0, b ≥ 0 B) 1.4a + 2.1 b ≤ 481 1.2 a + 1.8b ≤ 913 a ≥ 0, b ≥ 0 C) 1.4a + 1.2 b ≤ 913 2.1 a + 1.8b ≤ 481 a ≥ 0, b ≥ 0 D) 1.4a + 2.1 b ≥ 913 1.2 a + 1.8b ≥ 481 a ≥ 0, b ≥ 0 Answer: A

69) Consider the feasible set (FS) below of a certain linear programming problem.

What is the maximum value of the objective function 6x + 3y? A) 12 B) -6 C) 20 D) 24 E) none of these Answer: E 70) Consider the feasible set (FS) below of a certain linear programming problem.

What is the maximum value of the objective function 5x - y? A) 30 B) -40 C) 19 D) 0 E) none of these Answer: C

71) Consider the feasible set (FS) below of a certain linear programming problem.

At what point is the objective function 5x - y maximized? A) point D B) point C C) point B D) point E E) point A Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 72) The feasible set (FS) for a certain linear programming problem is shown below. Determine the values for x and y that maximize 3x + 4y. What is the maximum value? y

(3, 5)

(5, 3) (0, 2)

(0, 0)

(7, 0)

Answer: x = 3, y = 5; the maximum value is 29. 73) The feasible set (FS) for a certain linear programming problem is shown below. Determine the values for x and y that maximize 4x + 3y. What is the maximum value? y

(3, 5)

(5, 3) (0, 2)

(0, 0)

(7, 0)

Answer: x = 5, y = 3; the maximum value is 29.

74) For the feasible set (FS) below, find the coordinates of the point that minimizes 4x + 5y. Find the minimum value of 4x + 5y.

Answer: The minimal value of 4x + 5y is 16 at x =

18 8 ,y= . 7 7

75) A feasible set has these vertices : (1, -2), (4, 0), (7, 6), (4, 4). State the coordinates of the vertex that minimizes the objective function 3x - 2y. Answer: x = 4, y = 4 76) Given a feasible set with vertices (0, 0), (7, 12), (3, 4), (0, 3), and (8, 0), find the maximum of 3x +4y and where it occurs on the feasible set. Answer: maximum value is 69; occurs at x = 7, y = 12) 77) Given same feasible set with vertices (0, 0), (7, 12), (3, 4), (0, 3), and (8, 0), find the maximum of 4x +3y and where it occurs on the feasible set. Answer: maximum value is 64; occurs at x = 7, y = 12 78) A linear programming problem results in a feasible set whose consecutive vertices are (0, 0), (6,0), (4, 4), (0, 3). For which values of x and y is the objective function 3x + 4y a maximum? Answer: x = 4, y = 4 79) A linear programming problem results in a feasible set whose consecutive vertices are (0, 0), (0, 10) and (10, 0). For which values of x and y is the objective function 3x + 4y a maximum? Answer: x = 0, y = 10 80) A linear programming problem results in a feasible set whose consecutive vertices are (0, 0), (6,0), (4, 4), (0, 3). For which values of x and y is the objective function 3x + 4y a minimum? Answer: x = 0, y = 0 A feasible set has vertices (0, 5), (1, 1), (3, 8), and (9, 4). State the vertex that maximizes the objective function. 81) x + y Answer: (9, 4) 82) x + 2y Answer: (3, 8)

83) 2x + y Answer: (9, 4) 84) Find the maximum value of 2x + 3y and the point at which it occurs subject to the constraints y ≥ 3x - 9 y≤ x+ 5 y ≤ -2x +11 x ≥ 0, y ≥ 0 Answer: maximum is 25; occurs at x = 2, y = 7 85) Find the minimum value of 7x + 4y and the point at which it occurs subject to the constraints y ≤ -x + 10 y ≥ -2x + 11 3y ≤ -x + 18 4y ≥ -x + 16 Answer: minimum is 40; occurs at x = 4, y =3 86) Maximize 2x + 3y subject to the constraints x + 3y ≤ 120 7 x + y ≤ 78 2 . x ≥ 0,

x ≤ 20 y≥0

Answer: maximum is 132; occurs at x = 12, y = 36 87) Maximize the objective function 20x + 30y subject to the constraints: x + y ≤ 650 40x + 80y ≤ 44,000 x ≤ 400 y ≤ 500 x ≥ 0, y ≥ 0 Answer:

maximum value = 17,500 when x = 200 and y = 450

88) Maximize the objective function 3x + y subject to the following constraints. Solve graphically. 2x + y ≤ 25 x + y ≤ 20 x ≤ 12 x ≥ 0, y ≥ 0 y

Answer: 37 at (12, 1) A = (0,20) B = (5, 15) C = (12,1) D = (12, 0)

89) Minimize the objective function 3x + 2y subject to the following constraints. Solve graphically. 2x + y ≥ 30 2x + 5y ≥ 50 x + y ≥ 20 x ≥ 0, y ≥ 0 y

Answer: 50 at (10, 10)

90) Minimize the objective function 2x + 3y subject to the following constraints. Solve graphically. 2x + y ≥ 30 2x + 5y ≥ 50 x + y ≥ 20 x ≥ 0, y ≥ 0 y

Answer:

130 50 10 at ( , ) 3 3 3

91) Maximize the objective function 7x + 3y subject to the following constraints. x + 2y ≤ 40 3x + 2y ≤ 72 x ≤ 20 x≥ 8 y≥ 0 Answer: The maximum value is 158 and occurs at the point (20, 6). y 40

20 B

A 10

C 10

40 x

A = (8, 16), B = (16, 12), C = (20, 6)

92) Consider the linear programming problem below. y ≥ 2, x ≥ 0 x-y≤0 Maximize x + 5y subject to 1 y≤- x+ 9 2 (a) Graph the system of inequalities and outline the boundary of the feasible set. Label the feasible set FS. Also, label the vertices of the feasible set. (b) Find the maximum value of the objective function, and where it occurs. y

Answer: (a) y 10 A B 5

C 5

10 x

A = (0, 9), B = (6, 6), C = (2, 2), D = (0, 2) (b) The maximum is 45 and occurs at (0, 9).

Solve the problem. 93) North American Housing is a builder of two types of modular homes, a rambler and a spilt-level. The rambler requires 400 worker-days of carpentry, 10 worker-days of painting and sells for a profit of $3000. The split-level requires 300 worker-days of carpentry, 20 worker-days of painting and sells for a profit of $4000. North America Housing has available 48,000 worker-days of carpentry and 1600 worker-days of painting. North America Housing must also produce at least twice as many ramblers as split-levels. The builder will choose the number of each type of house it builds in order to maximize its profit. (a) Define the variables. (b) Write the system of linear inequalities used in solving the problem. (c) Write an algebraic expression for the objective function. Answer: (a) x = number of ramblers, y = number of split levels (b) 400x + 300y ≤ 48,000 10x + 20y ≤ 1600 x ≥ 2y x ≥ 0, y ≥ 0 (c) 3000x + 4000y 94) A broker has up to $20,000 to invest. He can purchase municipal bonds yielding 8% of the amount invested and Treasury bonds yielding 10% of the amount invested. The broker's client insists that no more than $8000 be invested in municipal bonds and that at least three times as much be invested in municipal bonds as Treasury bonds. How should the broker invest her client's money in order to maximize the yield? Answer: Invest $2667 in treasury bonds and $8000 in municipal bonds. 95) A manufacturer produces two items, A and B. A maximum of 2000 units can be produced per day. The cost is $3 per unit for A and $5 per unit for B. The daily production budget is $7500. If the manufacturer makes a profit of $1.75 per unit for A and $2.50 per unit for B, how many units of each should be produced to maximize profit? Answer: Produce 1250 units of A, 750 units of B. 96) Two kinds of crated cargo, A and B, are to be shipped by truck. Each crate of cargo A is 50 cubic feet in volume and weighs 200 pounds, whereas each crate of cargo B is 10 cubic feet in volume and weighs 360 pounds. The shipping company charges $75 per crate for cargo A and $100 per crate for cargo B. The truck has a maximum load limit of 7200 pounds and 1000 cubic feet. How many crates of each cargo should be shipped on each truck in order to satisfy the load limits and yield the greatest charges? Compute the corresponding charge. Answer: 18 crates of cargo A, 10 crates of cargo B; $2350

97) A television manufacturing company has two factories, I and II, that manufacture color and monochrome TV sets. Each day, factory I produces 50 color and 20 monochrome TV sets at a cost of $3000. Each day, factory II produces 30 color and 24 monochrome TV sets at a cost of $2700. An order is received for 2700 color and 1800 monochrome TV sets. For how many days should each factory operate in order to fill the order at the least cost? (Solve graphically.) Also compute the corresponding cost. y

Answer:

Factory I should operate 18 days. Factory II should operate 60 days. The corresponding minimum cost = $216,000. 98) A small manufacturing plant produces two kinds of bicycles, three-speed and ten-speed, in two factories. Factory A produces 16 three-speeds and 20 ten-speeds in one day, whereas factory B produces 12 three-speeds and 20 ten-speeds in one day. An order is received for 96 three-speeds and 140 ten-speeds. It costs $1000 a day to operate factory A and $800 a day to operate factory B. How many days should the manufacturer operate each factory in order to fill the order at minimum cost? For the linear programming problem above: (a) Define the variables. (b) Write the system of linear inequalities that must be satisfied. (c) Write the cost function. (d) Determine the feasible set. Label each boundary with its equation. (e) Solve the problem.

Answer: (a) x = number of days the manufacturer operates factory A y = number of days the manufacturer operates factory B (b) 16x + 12y ≥ 96 20x + 20y ≥ 140 x ≥ 0, y ≥ 0 (c) 1000x + 800y (d)

(e) The manufacturer should operate factory A for 3 days and factory B for 4 days in order to fill the order at a minimum cost of $6200. 99) A furniture-finishing plant finishes two kinds of tables, A and B. Table A requires 8 minutes of staining and 9 minutes of varnishing, whereas table B requires 12 minutes of staining and 6 minutes of varnishing. The staining facility is available at most 480 minutes per day and the varnishing facility is available at most 360 minutes per day. The plant has to finish at least as many B-tables as half the number of A-tables finished. The profit is $5 on each A-table and $3 on each B-table. (a) Let x be the number of A-tables and y the number of B-tables finished per day. Give all the inequalities that x and y must satisfy. (b) Graph the feasible set for this problem. (c) Find the values of x and y that maximize the profit, and the corresponding profit.

Answer: (a) 8x + 12y ≤ 480 9x + 6y ≤ 360 y ≥ 0.5x x ≥ 0, y ≥ 0 (b) y 40

30 B 20

FS C

10 D 10

A = (0, 40), B = (24, 24), C = (30, 15), D = (0, 0) (c) x = 30, y = 15; maximum profit = $195 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 100) A certain area of forest is populated by two species of animals, which scientists refer to as A and B for simplicity. The forest supplies two kinds of food, referred to as F1 and F2. For one year, species A requires 1.45 units of F1 and 1 units of F2. Species B requires 2.4 units of F1 and 1.9 units of F2 . The forest can normally supply at most 859 units of F1 and 491 units of F2 per year. What is the maximum total number of these animals that the forest can support? A) 200 animals B) 338 animals C) 931 animals D) 491 animals Answer: D

101) A company manufactures two ballpoint pens, silver and gold. The silver requires 4 min in a grinder and 6 min in a bonder. The gold requires 5 min in a grinder and 7 min in a bonder. The grinder can be run no more than 53 hours per week and the bonder no more than 54 hours per week. The company makes a $3 profit on each silver pen sold and $12 on each gold. How many of each type should be made each week to maximize profits? A) Silver pens: 218 Gold pens: 461 B) Silver pens: 462 Gold pens: 218 C) Silver pens: 1 Gold pens: 462 D) Silver pens: 0 Gold pens: 461 Answer: C 102) Wally's Warehouse sells trash compactors and microwaves. Wally has space for no more than 62 trash compactors and microwaves together. Trash compactors weigh 31 pounds and microwaves weigh 51 pounds. Wally is limited to a total of 6900 pounds for these items. The profit on a microwave is $110 and on a compactor $50. How many of each should Wally stock to maximize profit potential? A) 0 trash compactors, 61 microwaves B) 1 trash compactor, 61 microwaves C) 0 trash compactors, 62 microwaves D) 62 trash compactors, 1 microwave Answer: C

Refer to the figure to solve the problem. 103) A linear programing problem has the objective function [cost] = 16x + 12y, which is to be minimized. The following graph shows the feasible set and the straight line of all combinations of x and y for which the cost is $72. Give the linear equation (in standard form) of the line of constant cost c. Then, by inspection, find the vertex of the feasible set that gives the optimal solution. y

feasible set B

y=A) y = -

3 12 x+ ; B 4 c

B) y = -

4 c x+ ; B 3 12

C) y =

3 x + 6; C 4

D) y =

4 c x; C 3 12

4 x+6 3

Answer: B

104) A linear programing problem has the objective function [cost] = 1050 - 7x - 6y, which is to be minimized. The following graph shows the feasible set and the straight line of all combinations of x and y for which the cost is $840. Give the linear equation (in standard form) of the line of constant cost c. Then, by inspection, find the vertex of the feasible set that gives the optimal solution. y

y=-

feasible set E

7 x + 35 6

A) y = 175 -

6 c x- ; C 7 7

B) y = 175 +

6 c x+ ; B 7 7

C) y = 35 -

7 c x- ; C 6 6

D) y = 175 -

7 c x- ; B 6 6

Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. For the feasible set shown below, find an objective function in the form kx + y, where k is a positive constant, which will have its maximum value at the given point. (0, 10) y

(5, 7)

(0, 0)

(7, 3)

(8, 0)

105) (0, 10) Answer: Answers will vary; any value satisfying 0 < k ≤ 0.6 is correct. Possible answer: 0.5x + y

106) (5, 7) Answer: Answers will vary; any value satisfying 0.6 ≤ k ≤ 2 is correct. Possible answer: x + y 107) (7, 3) Answer: Answers will vary; any value satisfying (2 ≤ k ≤ 3) is correct. Possible answer: 1.5x + y 108) (8, 0) Answer: Answers will vary; any value satisfying k ≥ 3 is correct. Possible answer: 4x + y For the feasible set shown below, find an objective function in the form ax + by, where a and b are positive constants, which will have its minimum value at the given point. y

(0, 7)

(1, 4) (3, 2)

(6, 0)

109) (0, 7) Answer: Answers will vary; any values satisfying b <

a are correct. 3

Possible answer: 4x + y 110) (1, 4) Answer: Answers will vary; any values satisfying b < a < 3b are correct. Possible answer: 2x + y 111) (3, 2) Answer: Answers will vary; any values satisfying

2 b < a < b are correct. 3

Possible answer: 5x + 6y 112) (6, 0) Answer: Answers will vary: any values satisfying 0 < a < Possible answer: x + 2y

2 b are correct. 3

For the feasible set shown below, find an objective function in the form kx + y, where k is a positive constant, which will have its maximum value at the given point. y (0, 6) (3, 5)

(5, 3) FS (6, 0) x

113) (0, 6) Answer: Answers will vary; any value satisfying 0 ≤ k ≤

1 is correct. 3

Possible answer: 0.2x + y 114) (6, 0) Answer: Answers will vary; any value satisfying k > 3 is correct. Possible answer: 4x + y MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Consider the feasible set shown below that corresponds to a certain linear programming problem. Three of the boundary lines are labeled with their slopes.

115) The objective function 2x + 3y is a maximum at what point? A) A. B) B. C) C. D) D. E) O. Answer: C

116) The objective function 7x + 2y is a maximum at what point? A) A. B) B. C) C. D) D. E) O. Answer: A 117) The objective function x + 2y is a maximum at what point? A) A. B) B. C) C. D) D. E) O. Answer: A Consider the feasible set shown below that corresponds to a certain linear programming problem. Three of the boundary lines are labeled with their slopes.

118) The objective function, x + 2y, assumes its minimum value at what point? A) A. B) B. C) C. D) D. E) O. Answer: B 119) The objective function, 9x + 2y, assumes its minimum value at what point? A) A. B) B. C) C. D) D. E) O. Answer: D 120) The objective function, 2x + 9y, assumes its minimum value at what point? A) A. B) B. C) C. D) D. E) O. Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 121) Consider the following feasible set of minimization of x + y. Where must x + y have its minimum?

Answer: B 122) Consider the following feasible set (FS) for the maximization of 3x + 2y. (a) What is the maximum of 3x + 2y? (b) Does this maximum occur at a unique point? y

(0, 3)

(3, 3)

(0, 0)

(5, 0)

Answer: (a) 15 (b) No, anywhere along the line segment connecting (3, 3) to (5, 0). MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. New cars are transported from docks in Baltimore and New York to dealerships in Pittsburgh and Philadelphia. The dealership in Pittsburgh needs 20 cars and the dealership in Philadelphia needs 15 cars. It costs $60 to transport a car from Baltimore to Pittsburgh, $45 to transport a car from Baltimore to Philadelphia, $65 to transport a car from New York to Pittsburgh, and $40 to transport a car from New York to Philadelphia. There are 30 cars on the docks in Baltimore and there are 18 cars on the docks in New York. The number of cars sent from each dock to each dealership is chosen to minimize total transportation costs. 123) If x represents the number of cars sent from Baltimore to Philadelphia and y represents the number of cars sent from New York to Pittsburgh, then the number of cars sent from Baltimore to Pittsburgh is given by A) 30 - x. B) 20 - x. C) 30 - y. D) 20 - y. E) none of the above Answer: D

124) If x represents the number of cars sent from Baltimore to Philadelphia and y represents the number of car sent from New York to Pittsburgh, then the objective function is A) 45x + 65y. B) 5x + 5y + 1800. C) -15x + 15y + 180. D) 105x + 105y. E) none of these Answer: B 125) If x represents the number of cars sent from Baltimore to Philadelphia and y represents the number of cars sent from New York to Pittsburgh, which of the following inequalities must be satisfied? A) x - y ≤ 10 B) x - y ≥ 10 C) -x + y ≥ 3 D) -x + y ≤ 0 E) none of these Answer: A There is exactly $10,000 in a trust fund that is to be invested among three types of bonds, A, B, and C, which yield 5%, 6%, and 7%, respectively, on the investment. The total yield must be at least $600, no less than $3000 may be invested in B bonds, and no more than $2000 may be invested in A bonds. 126) If x and y represent the amounts invested in A and B bonds, then the amount invested in C bonds is A) 10,000 + x - y. B) x + y - 1000. C) 10,000 - x - y. D) 10,000 - x + y. E) none of these Answer: C 127) If x and y represent the amounts invested in A and B bonds, which of the following inequalities must be satisfied? A) y ≤ 3000 B) 0.05x + 0.06y ≥ 600 C) x + y ≤ 5000 D) 2x + y ≤ 1000 E) none of these Answer: D 128) If x and y represent the amounts invested in A and B bonds, then the maximum possible yield is A) $600. B) $630. C) $700. D) $670. E) none of these Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 129) Tee-Tops Inc. has warehouses in San Francisco and Oakland and some stores in Berkeley and San Jose. The Berkeley store needs 4000 tee shirts and the San Jose store needs 7000 tee shirts. The Oakland warehouse has 9000 tee shirts, whereas the San Francisco warehouse has 8000 tee shirts. The cost of shipping a tee shirt from Oakland to Berkeley is $0.05, from Oakland to San Jose $0.15, from San Francisco to Berkeley $0.03, and from San Francisco to San Jose $0.12. The number of tee shirts that are shipped from each warehouse to each store is chosen in order to minimize shipping costs. (a) Define the variables. (b) Write the system of linear inequalities used in solving the problem. (c) Write an algebraic expression for the objective function. Answer: (a) x = the number of tee shirts shipped from Oakland to Berkeley y = the number of tee shirts shipped from Oakland to San Jose (b) x + y ≤ 9000 3000 ≤ x + y 4000 ≥ x, 7000 ≥ y x ≥ 0, y ≥ 0 (c) shipping costs = 2x + 3y + 96,000 130) A small-appliance manufacturer has plants in Baltimore and Philadelphia, each of which produces toasters and blenders. The Baltimore plant can produce at most 800 appliances in one day at a cost of $12 per toaster and $15 per blender. The Philadelphia plant can produce at most 500 appliances in one day at a cost of $10 per toaster and $20 per blender. A rush order is received for 600 toasters and 300 blenders. The manufacturer will choose the number of toasters and blenders produced at each plant in order to minimize costs. (a) Name in words the four quantities that must be determined. (b) Express the four quantities in (a) algebraically using only the two variables, x and y. (c) Write the complete system of inequalities (in terms of x and y) needed to solve the problem. (d) Write the objective function in terms of x and y. Answer: (a) the number of toasters produced in Baltimore, the number of blenders produced in Baltimore, the number of toasters produced in Philadelphia, the number of blenders produced in Philadelphia (b) x, y, 600 - x, 300 - y, respectively x + y ≤ 800, (600 -x) + (300 - y) ≤ 500, (c) 300 - y ≥ 0, 600 - x > 0, x ≥ 0, y ≥ 0 (d) 12x + 15y + 10(600 - x) + 20(300 - y) = 12,000 + 2x - 5y

131) Mr. and Mrs. Adams have $20,000 to invest in low-risk, medium-risk, and high-risk stocks. They decide that at least $10,000 must be invested in low-risk stocks, at least $1000 must be invested in medium-risk stocks, and no more than $5000 can be invested in high-risk stocks. The expected yields on the stocks are 6% for low-risk, 8% for medium-risk, and 10% for high-risk. Mr. and Mrs. Adams will choose the amount to invest in each type of stock in order to maximize the yield on their investment. (a) Name in words the three quantities that must be determined. (b) Write the three quantities in (a) algebraically using only the variables x and y. (c) Write the complete set of inequalities needed to solve the problem. (d) Write the objective function. Answer: (a) amount invested in low-risk stocks, amount invested in medium-risk stocks, amount invested in high-risk stocks (b) x, y, 20,000 - x - y, respectively x≥ 0 y ≥ 0 ≥ (c) 20,000 - x - y 0 x ≥ 10,000 y ≥ 1000 20,000 - x - y ≤ 5000 (d) yield = 0.06x + 0.08y + 0.10(20,000 - x - y) = 2000 - 0.04x - 0.02y 132) Ms. Jones installs and demonstrates smoke detectors. There are two smoke detectors that she installs, types A 1 1 and B. Type A requires 1 hour to install and hour to demonstrate. Type B requires 1 hours to install and 3 2 1 hour to demonstrate. Company rules require Ms. Jones to work a minimum of 20 hours a week as an installer 5 and a maximum of 15 hours a week as a demonstrator. She gets paid $8 an hour for installing and $5 an hour for demonstrating. For each week, Ms. Jones chooses to install and demonstrate the number of alarms of each type so that her earnings are maximized. (a) Define the variables x and y. (b) Write the complete system of inequalities needed to solve the problem. (c) Write the objective function. Answer: (a) x = the number of type A smoke detectors installed and demonstrated y = the number of type B smoke detectors installed and demonstrated x ≥ 0, y ≥ 0 3 x + y ≥ 20 2 (b) 1 1 x + y ≤ 15 3 5 (c) earnings = 8(x +

3 1 1 29 y) + 5( x + y) = x + 13y 2 3 5 3

133) Mango Skates has outlet stores in Gilroy and Newark. Their A-Line model is produced at factories in Akron and Birmingham. The cost of shipping a pair of A-Line skates from Akron to Gilroy is $9; from Akron to Newark, $8; from Birmingham to Gilroy is $7; and from Birmingham to Newark, $5. Suppose that the Gilroy outlet orders 120 pairs and the Newark store orders 90 pairs. The factory in Akron can produce 100 pairs in time for shipping, and the factory in Birmingham can produce 150 pairs in time for shipping. (a) Find the number of pairs shipped from each warehouse to each outlet to minimize the shipping costs. (b) What is the minimum shipping cost? Answer: (a) 60 pairs from Birmingham to Gilroy; 60 pairs from Akron to Gilroy; 90 pairs from Birmingham to Newark; 0 pairs from Akron to Newark (b) $1410 60

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the initial simplex tableau for the linear programming problem. 1) Maximize 8x + 13y subject to 2x + 3y ≤ 350 5x + y ≤ 500 x ≥ 0, y ≥ 0 A) x y u v M 2 3 1 0 0 350 5 1 0 1 0 500 -8 -13 0 0 1 800 B)

x y u v M 2 3 1 0 0 350 5 1 0 1 0 500 0 -8 -13 0 0 1

x y u v M 2 3 1 0 0 350 5 1 0 1 0 500 8 13 0 0 1 0

x y u v M 2 3 1 0 0 350 5 1 0 1 0 500 8 13 0 0 1 800

Answer: B

2) Maximize 2x + y + 20 subject to x + 3y ≤ 10 2x - y ≤ 8 x ≥ 0, y ≥ 0 A) x y u v M 1 3 1 0 0 10 2 -1 0 1 0 8 -2 -1 0 0 1 0 B)

x y u v M 1 3 1 0 0 10 2 -1 0 1 0 8 2 1 0 0 1 0

x y u 1 3 1 2 -1 0 -2 -1 0

v M 0 0 10 1 0 8 0 1 -20

x y u 1 3 1 2 -1 0 -2 -1 0

v M 0 0 10 1 0 8 0 1 20

Answer: D 3) Maximize 3x - y subject to the following constraints. 6x - 5y ≤ 24 x + 9y ≥ 63 x ≥ 0, y ≥ 0 A) x y u v M 6 -5 1 0 0 24 1 9 0 0 1 -63 3 -1 0 0 1 0 B)

x y u 6 -5 1 -1 -9 0 -3 1 0

v M 0 0 24 1 0 -63 0 1 0

x y u 6 -5 1 1 9 0 -3 1 0

v M 0 0 24 1 0 63 0 1 0

x y u 6 -5 1 -1 -9 0 3 1 0

v M 0 0 24 1 0 -63 0 1 0

Answer: B 2

4) Minimize -x + 7y subject to the following constraints. x+ y≤ 5 3x + 6y ≤ 24 x ≥ 0, y ≥ 0 A) x y u v M 1 1 1 0 0 5 3 6 0 1 0 24 1 -7 0 0 1 0 B)

x y u v M 1 1 1 0 0 -5 3 6 0 1 0 -24 1 -7 0 0 1 0

x 1 3 -1

y u v M 1 1 0 0 -5 6 0 1 0 -24 7 0 0 1 0

x 1 3 -1

y u v M 1 1 0 0 5 6 0 1 0 24 7 0 0 1 0

Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) Maximize 10x - 7y subject to: 7x + 10y ≤ 100 3x + 4y ≤ 55 x ≥ 0, y ≥ 0 Answer:

x y u v M 7 10 1 0 0 100 3 4 0 1 0 55 0 -10 7 0 0 1

6) Maximize 20x - 50y + 10z + 30 subject to: x + y + z ≤ 250 x + 2z ≤ 125 2y + 3z ≤ 100 x ≥ 0, y ≥ 0, z ≥ 0 Answer: x y z 1 1 1 1 0 2 0 2 3 50 -20 -10

u 1 0 0 0

v w M 0 0 0 250 1 0 0 125 0 1 0 100 0 0 1 30

7) Find the linear programming problem corresponding to the following initial simplex tableau. x y u v M 1 2 1 0 0 22 3 -1 0 1 0 55 -5 -4 0 0 1 30 x + 2y ≤ 22 Answer: Maximize 5x + 4y + 30 subject to 3x - y ≤ 55 x ≥ 0, y ≥ 0 8) By introducing suitable slack variables, formulate the following linear programming problem in terms of a system of linear equations. Maximize 5x + 3y subject to the constraints: x + 3y ≤ 6 x+ y ≤ 4 y≤2 x≥0 y≥0 Answer: Among all the solutions of the system of linear equations x + 3y + u =6 x + y +v =4 y + w =2 -5x - 3y + M = 0, find one for which x ≥ 0, y ≥ 0 , u ≥ 0, v ≥ 0, w ≥ 0, and M is maximum. 9) By introducing suitable slack variables, formulate the following linear programming problem in terms of a system of linear equations. Maximize 3x + 2y + 5z subject to the constraints: 4x - 7z ≤ 40 x + y + 6z ≤ 80 x ≥ 0, y ≥ 0, z ≥ 0 Answer: Among all the solutions of the system of linear equations 4x - 7z + u = 40 x + y + 6z +v = 80 -3x - 2y - 5z +M= 0 find one for which x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0, and M is maximum. Explain why the given linear programming problem is not in standard form. x - y ≤ 10 10) Maximize -2x + 4y subject to 2x + y ≤ -5 x ≥ 0, y ≥ 0 Answer: The second constraint is not in the form Linear polynomial ≤ [nonnegative constant. x-y ≤ 4 11) Minimize x + 2y subject to 2x + y ≤ 12 x ≥ 0, y ≥ 0 Answer: The objective function is to be minimized. 12) Maximize x + 2y subject to

x+ y≤ 8 3x + y ≤ 10

Answer: The variables are not constrained to be non negative.

13) Maximize x + 3y subject to

3x + y ≥ 24 x + 7y ≤ 5 x ≥ 0, y ≥ 0

Answer: The first constraint is not in the form Linear polynomial ≤ [nonnegative constant.

14) Minimize 3x + 4y subject to

x + y ≤ 10 2x + 3y ≥ 1 x ≥ 0, y ≥ 0

Answer: 1. The objective function is to be minimized. 2. Not all constraints have the form: Linear polynomial ≤ [nonnegative constant]. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use slack variables to convert the constraints into linear equations. 15) Maximize 2x + 8y subject to: x + 5y ≤ 15 9x + 4y ≤ 25 with: x ≥ 0, y ≥ 0 A) x + 5y + u ≤ 15 9x + 4y + v ≤ 25 -2x - 8y + M ≥ 0 Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0 B) x + 5y + u = 15 9x + 4y + v = 25 -2x - 8y + M = 0 Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0 C) x + 5y = u + 15 9x + 4y = v + 25 -2x - 8y = M Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0 D) x + 5y + u = 15 9x + 4y + v = 25 2x + 8y + M = 0 Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0 Answer: B

3x + 5y 7x + 9y ≤ 30 x + 6y ≤ 40 with: x ≥ 0, y ≥ 0 A) 7x + 9y + u = 30 x + 6y + u = 40 -3x - 5y + M = 0 Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0 B) 7x + 9y + u ≤ 30 x + 6y + v ≤ 40 3x + 5y + M ≤ 0 Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0 C) 7x + 9y + u = 30 x + 6y + v = 40 -3x - 5y + M = 0 Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0 D) 7x + 9y = u + 30 x + 6y = v + 40 3x + 5y + M = 0 Maximize M given that x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0

16) Maximize subject to:

Answer: C x + 2y + 3z x + 4y + 8z ≤ 40 7x + y + 4z ≤ 50 with: x ≥ 0, y ≥ 0, z ≥ 0 A) x + 4y + 8z = u + 40 7x + y +4z = v + 50 -x - 2y - 3z = M Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0 B) x + 4y + 8z = u - 40 7x + y +4z = v - 50 -x - 2y - 3z + M = 0 Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0 C) x + 4y + 8z + u = 40 7x + y + 4z + v = 50 -x - 2y - 3z + M = 0 Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0 D) x + 4y + 8z + u = 40 7x + y + 4z + v = 50 x + 2y + 3z + M = 0 Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0

17) Maximize subject to:

Answer: C

4x + 12y + 7z 5z + 3y + z ≤ 12 x + 2y + z ≤ 24 with: x ≥ 0, y ≥ 0, z ≥ 0 A) 5x + 3y + z = u - 12 x + 2y + z = v - 24 4x + 12y + 7z = M Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0 B) 5x + 3y + z = u + 12 x + 2y + z = v + 24 -4x - 12y - 7z = M Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0 C) 5x + 3y + z + u = 12 x + 2y + z + v = 24 -4x - 12y - 7z + M = 0 Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0 D) 5x + 3y + z + u = 12 x + 2y + z + u = 24 4x + 12y + 7z + M = 0 Maximize M given that x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0

18) Maximize subject to:

Answer: C Introduce slack variables as necessary, and write the initial simplex tableau for the problem. 19) Find x ≥ 0 and y ≥ 0 such that 2x + 5y ≤ 15 3x + 3y ≤ 19 and 4x + y is maximized.

x 2 3 -4 B) x 2 3 4 C) x 2 3 4 D) x 2 3 -4

y 5 3 -1 y 5 3 1 y 5 3 1 y 5 3 -1

u 1 0 0 u 1 0 0 u 1 0 0 u 1 0 0

v 0 1 0 v 0 1 0 v 0 1 0 v 0 1 0

M 0 0 1 M 0 0 1 M 0 0 1 M 0 0 1

15 19 0 15 19 0 19 15 0 19 15 0

Answer: A

20) Find x ≥ 0 and y ≥ 0 such that x + y ≤ 94 3x + y ≤ 173 and 2x + y is maximized.

x 1 3 -2 B) x 1 3 -2 C) x 1 3 2 D) x 1 3 2

y 1 1 -1 y 1 1 -1 y 1 1 1 y 1 1 1

u 1 0 0 u 1 0 0 u 1 0 0 u 1 0 0

v 0 1 0 v 0 1 0 v 0 1 0 v 0 1 0

M 0 94 0 173 1 0 M 0 173 0 94 1 0 M 0 173 0 94 1 0 M 0 94 0 173 1 0

Answer: A 21) Find x ≥ 0 and y ≥ 0 such that 2x + y ≤ 27 3x + 5y ≤ 73 and 4x + 2y is maximized.

x 2 3 -4 B) x 2 3 4 C) x 2 3 -4 D) x 2 3 4

y 1 5 -2 y 1 5 2 y 1 5 -2 y 1 5 2

u 1 0 0 u 1 0 0 u 1 0 0 u 1 0 0

v 0 1 0 v 0 1 0 v 0 1 0 v 0 1 0

M 0 73 0 27 1 0 M 0 73 0 27 1 0 M 0 27 0 73 1 0 M 0 27 0 73 1 0

Answer: C

22) Find x ≥ 0 and y ≥ 0 such that 5x + 10y ≤ 101 10x + 15y ≤ 139 and 2x + 5y is maximized.

x 5 10 -2 B) x 5 10 -2 C) x 5 10 2 D) x 5 10 2

y 10 15 -5 y 10 15 -5 y 10 15 5 y 10 15 5

u 1 0 0 u 1 0 0 u 1 0 0 u 1 0 0

v 0 1 0 v 0 1 0 v 0 1 0 v 0 1 0

M 0 0 1 M 0 0 1 M 0 0 1 M 0 0 1

101 139 0 139 101 0 101 139 0 139 101 0

Answer: A

23) Find x ≥ 0 and y ≥ 0 such that x + y ≤ 20 3x + 2y ≤ 40 2x + 3y ≤ 60 and x + 2y is maximized.

x 1 3 2 -1 B) x 1 3 2 1 C) x 1 3 2 1 D) x 1 3 2 -1

y u v w M 1 1 0 0 0 40 2 0 1 0 0 20 3 0 0 1 0 60 -2 0 0 0 1 0 y u v w M 1 1 0 0 0 40 2 0 1 0 0 20 3 0 0 1 0 60 2 0 0 0 1 0 y u v w M 1 1 0 0 0 20 2 0 1 0 0 40 3 0 0 1 0 60 2 0 0 0 1 0 y u v w M 1 1 0 0 0 20 2 0 1 0 0 40 3 0 0 1 0 60 -2 0 0 0 1 0

Answer: D Determine the particular solution corresponding to the simplex tableau. 24) x y u v M 2 3 1 0 0 350 5 1 0 1 0 500 0 -8 -13 0 0 1 A) x = 2, y = 3, u = 350, v = 500, M = 0. B) x = 0, y = 0, u = 350, v =500, M = 0. C) x = 0, y = 0, u = -8, v = -13, M = 0. D) x = 2, y = 3, u = 0, v = 0, M = 0. E) none of these Answer: B

25) x y u v M 0 -3 1 1 0 2 1 2 0 0 0 6 0 -4 0 5 1 10 A) x = 2, y = 6, u = 10, v = 0, M = 0. B) x = 6, y = 0, u = 2, v = 0, M = 10. C) x = 6, y = 0, u = 6, v = 0, M = 10. D) x = 2, y = 6, u = 0, v = 0, M = 10. E) none of these Answer: B x 26) 0 2 1

y z u v M 0 -2 0 1 3 3 1 0 0 0 1 4 0 1 1 0 4 8 A) x = 8, y = 4, z = 8, u = 8, v = 3, M = 4. B) x = 3, y = 4, z = 8, u = 0, v = 0, M = 0. C) x = 0, y = 4, z = 0, u = 8, v = 3, M = 0. D) x = 3, y = 0, z = 8, u = 0, v = 0, M = 4. E) none of these

Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. x 27) 2 3 -6

y 0 1 0

z u 1 1 0 5 0 -3

v M 2 0 1 4 0 5 1 1 50

Answer: x = 0, y = 5, z = 1, u = 0, v = 0, M = 50 x 28) 1 0 0

y z 0 -3 1 2 0 -4

u 1 0 5

v M 2 0 2 4 0 6 1 1 8

Answer: x = 2, y = 6, z = 0, u = 0, v = 0, M = 8 29) x

u v M 4 1 0 4 7 7

0 1

1 7

2 0 7

17 7

1 1 26 7

1 0 0 0

Answer: x = 2, y = 4, u = 0, v = 0, M = 26

30)

x y 2 0 3

u v M 6 1 0 10

4 1

22 0 0 3

7 0 2

1 0 1 121

Answer: x = 0, y =

5 4

5 , u = 0, v = 10, M = 121 4

31) Determine by inspection a particular solution for the following system of linear equations: x + 2y + 6u = 12 2y - 6u + v = 2 - 2u +w = 5 y+ u + M = 17 Answer: x = 12, y = 0 , u = 0, v = 2, w = 5, M = 17 32) Consider the linear programming problem. Maximize M = 4x + 5y subject to 2x + 3y ≤ 9 x+ y≤4 x ≥ 0, y ≥ 0 (a) Introducing suitable slack variables, rewrite it as a linear system of equations and state the equivalent problem. (b) Set up the initial simplex tableau. (c) Determine the particular solution that results from setting all group-I variables equal to zero. Answer: (a) Among all the solutions of the system of linear equations 2x + 3y + u =9 x+ y +v =4 -4x - 5y +M=0 find one for which x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0, and M is maximum. (b)

x y u 2 3 1 1 1 0 -4 -5 0

v M 0 0 9 1 0 4 0 1 0

33) Consider the linear programming problem. Maximize M = 7x + 2y + z subject to x + y + z ≤ 12 ≤ 6 x - 2y x + 3y - 2z ≤ 10 x ≥ 0, y ≥ 0, z ≥ 0 (a) Introducing suitable slack variables, rewrite it as a linear system of equations and state the equivalent problem. (b) Set up the initial simplex tableau. (c) Determine the particular solution that results from setting all group-I variables equal to zero. Answer: (a) Among all solutions of the system of linear equations x+ y+ z+u = 12 x - 2y +v = 6 x + 3y - 2z w + = 10 -7x - 2y - z +M = 0 find one for which x ≥ 0, y ≥ 0, z ≥ 0, u ≥ 0, v ≥ 0, w ≥ 0, and M is maximum. (b) x y z 1 1 1 1 -2 0 1 3 -2 -7 -2 -1

u 1 0 0 0

v w M 0 0 0 12 1 0 0 6 0 1 0 10 0 0 1 0

(c) x = 0, y = 0, z = 0, u = 12, v = 6, w = 10, M = 0 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the solutions that can be read from the simplex tableau. 34) x y z u v M 3 4 0 3 1 0 12 1 5 1 7 0 0 23 -3 4 0 1 0 1 15 A) x, y, u = 0, z = 23, v = 12, M = 15 B) x, y, z = 0, u = 23, v = 12, M = 15 C) x, y, u = 0, x = 23, v = 12, M = 15 D) x, y, u = 0, z = 12, v = 23, M = 15 Answer: A 35)

x y z u v M 2 0 3 1 1 0 8 1 1 2 4 0 0 13 4 0 1 8 0 1 24 A) u = 8, y = 13, M = 24; x, z, u = 0 B) v = 8, y = 13, M = 24; x, z, v = 0 C) v = 8, y = 13, M = 24; x, z, u = 0 D) u = 8, y = 13, M = 24; x, z, v = 0 Answer: C

36)

x y z u v M 1 4 1 0 5 0 5 0 1 3 1 2 0 8 0 0 0 0 4 1 5 A) z = 5, u = 8, M = 5; x, y, v = 0 B) x = 5, y = 8, M = 5; z, u, v = 0 C) z = 5, y = 8, M = 5; x, u, v = 0 D) x = 5, u = 8, M = 5; y, z, v = 0 Answer: D

37)

x y u v w M 1 0 1 3 0 0 5 3 0 0 1 1 0 17 3 1 0 2 0 0 29 2 0 0 3 0 1 18 A) x = 5, v = 17, y = 29, M = 18; u, w = 0 B) x = 5, w = 17, y = 29, M = 18; u, v = 0 C) u = 5, w = 17, y = 29, M = 18; x, v = 0 D) u = 5, v = 17, y = 29, M = 18; x, w = 0 Answer: C

38)

x y u v w M 0 3 0 1 1 0 19 0 4 1 0 1 0 8 1 5 0 0 1 0 12 0 -3 0 0 1 1 16 A) w = 19, u = 8, x = 12, M = 16; y, v = 0 B) v = 19, w = 8, x = 12, M = 16; y, u = 0 C) v = 19, u = 8, x = 12, M = 16; y, w = 0 D) v = 19, u = 8, w = 12, M = 16; x, y = 0 Answer: C

x y u v M -2 1 1 0 0 2 39) The result of pivoting the simplex tableau 6 4 0 -1 0 6 about -2 (first row, first column) is 1 -2 0 0 1 0 A) x y u v M 1 1 1 0 0 0 2 2 0

7 3 0 2

3 1 0 84 1 0 1 0 2

B) x

y u 1 1 1 2 2 6 1

0 0

0 -1 0 42 0 0 1 0

4 -2

C) x y 1 1 0 4 0 -2

v M

u v M 1 0 0 0 0 -1 0 42 0 0 1 0

D) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 40) Pivot the following simplex tableau about the element 2. x y u v M 4 2 1 0 0 10 1 1 0 1 0 21 -5 -6 0 0 1 17 Answer:

x y 2 1

u v M 1 0 0 5 2

-1 0 -

1 1 0 16 2

7 0

3 0 1 47

41) Pivot the following simplex tableau about the element 2 (first column). x y u v M 4 2 1 0 0 25 0 5 0 1 0 7 2 -3 0 0 1 12 Answer:

x 0 0

y u 8 1 5 0 3 1 0 2

v M 0 -2 1 1 0 7 1 0 6 2

42) Pivot the following simplex tableau about the element 3 (second column). x y u v M 8 3 1 0 0 12 6 7 0 1 0 63 -1 -2 0 0 1 100 Answer:

x 8 3 -

y 1

u v M 1 0 0 3

38 3

0 -

7 1 0 3

13 3

2 0 1 108 3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 43) Pivot the simplex tableau about the following elements (a) 5 (first row, first column), (b) 3 (first row, second column), (c) 5 (second row, first column), and (d) 2 (second row, second column) to find the particular solution corresponding to the new tableau. Which of the pivot operations increases M the most? x y u v M 5 3 1 0 0 30 5 2 0 1 0 30 -1 -5 0 0 1 0 A) 5 (first row, first column) B) 2 (second row, second column) C) 3 (first row, second column) D) 5 (second row, first column) Answer: B x y u v M 2 1 0 5 0 20 44) In the simplex tableau 4 0 1 8 0 32 , what is the next pivot element? -3 0 0 2 1 -9 A) -3 in the third row, first column. B) 8 in the second row, fourth column. C) 1 in the first row, second column. D) 4 in the second row, first column. E) none of these Answer: D x y z u v M 45) In the simplex tableau 5 12 6 1 0 0 12 , what is the next pivot element? 15 10 0 1 0 0 5 4 -2 0 0 0 1 0 A) 12 in the first row, second column B) 5 in the first row, first column C) 15 in the second row, first column D) 10 in the second row, second column E) none of these Answer: D

x y z u 46) In the simplex tableau 0 12 6 12 1 5 -1 0 0 -2 -3 9 A) -1 B) 4 C) 5 D) 6 E) none of these

v M 1 0 24 , what is the next pivot element? 0 0 8 0 1 5

Answer: D x y u 47) In the simplex tableau 1 10 -5 0 6 2 0 -2 3 A) -2 B) -5 C) 10 D) 2 E) none of these

v w M 0 1 0 -10 , what is the next pivot element? 1 7 1 12 0 4 0 -3

Answer: B x y 1 10 48) In the simplex tableau 2 -5 0 -2 A) -2 B) 2 C) -5 D) 10 E) none of these

u v M 1 0 0 10 0 1 0 -10 , what is the next pivot element? 0 0 1 -3

Answer: D x y u v M 1 6 1 0 0 10 49) In the simplex tableau 3 4 0 1 0 -10 , what is the next pivot element? 0 5 0 0 1 -6 A) 3 B) 4 C) 6 D) 10 E) none of these Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. For the simplex tableau given, determine the next pivot element and pivot about that element. 50) x y u v M 1 2 8 0 0 6 0 4 10 1 0 20 0 -3 2 0 1 2 Answer: Pivot about 2. x y u v M 1 1 4 0 0 3 2 -2 0 -6 1 0 8 3 0 14 0 1 11 2 For the given simplex tableau, determine the next pivot element and pivot about that element. 51) x y u v M 6 0 -3 1 0 30 0 1 5 0 0 10 -1 0 -4 0 1 -40 Answer: Pivot about 5. x y u v M 3 6 0 1 0 36 5 0

1 1 0 0 5

-1

4 0 0 1 -32 5

52) x y u v M 6 2 1 0 0 10 1 30 1 0 6 -4 -12 0 0 1 0 Answer: Pivot about 3. x y u v M 16 -2 0 1 0 6 3 3 1 1 0 3

1 0 3

0 0 0

4 1 24

53)

x y u 0 8 -2 1 3 -1 0 -4 0

v M 0 1 3 0 0 -12 1 0 -2

Answer: Pivot about -1. x y u v M -2 2 0 0 1 27 -1 -3 1 0 0 12 0 -4 0 1 0 -2 54) x y z 0 11 2 0 -6 8 1 3 3 0 -8 -1

u 1 0 0 0

v w M 0 3 0 12 1 16 0 -7 0 1 0 9 0 7 1 -6

Answer: Pivot about 11. x y z u v 2 1 0 1 0 11 11 0

100 11

27 3 0 11 11

5 11

w M 3 0 11

6 194 1 11 11

0 -

5 11

2 11

63 11

8 101 0 11 11

30 11

x y u 3 2 1 55) Consider the simplex tableau 2 3 0 -5 -3 0 particular solution. Answer:

12 11

v M 0 0 21 1 0 12 . Find the next simplex tableau and its corresponding 0 1 22

y u v M 5 3 0 1 0 3 2 2 1

3 0 2

1 0 2

9 0 2

5 1 52 2

x = 6, y = 0, u = 3, v = 0, M = 52 Use the simplex method to solve. 56) Maximize 2x + 6y subject to the following constraints. 3x + 6y ≤ 45 x + y ≤ 10 x ≥ 0, y ≥ 0 Answer: x = 0, y =

15 , M = 45 2

57) Maximize 4x - 5y + 10 subject to the following constraints. x+y≤3 y≤2 x ≥ 0, y ≥ 0 Answer: x = 3, y = 0, M = 22 58) Maximize -4x + 5y subject to the following constraints. x+y≤4 1 x-y≤3 2 x ≥ 0, y ≥ 0 Answer: x = 0, y = 4, M = 20 59) Maximize 5x + 6y subject to the following constraints. x-y≤2 2 x+y≤8 3 x ≥ 0, y ≥ 0 Answer: x = 6, y = 4, M = 54 60) Maximize 6x + 7y+300 subject to the following constraints. 2x +3y ≤ 400 x + y ≤ 150 ≥ x 0, y ≥ 0 Answer: x = 50, y = 100, M = 1300 61) Maximize 2x + y subject to the following constraints. 3 x + y ≤ 38 2 5x + y ≤ 80 x ≥ 0, y ≥ 0 Answer: x = 12, y = 20, M = 44 62) Maximize 10x + y subject to the following constraints. 5y + x ≤ 25 y + 2x ≤ 14 x ≥ 0, y ≥ 0 Answer: x = 7, y = 0, M = 70 63) Maximize M = 4x - 2y subject to the following constraints. x + 2y ≤ 7 -5x + 8y ≤ 3 x ≥ 0, y ≥ 0 Answer: x = 7, y = 0, M = 28

64) Maximize M = 3x + 2y subject to the following constraints. x + y ≤ 20 2x + y ≤ 25 ≤ 10 x x ≥ 0, y ≥ 0 Answer: x = 5, y = 15, M = 45 65) Maximize M = 3x + 3y subject to the following constraints. x + y ≤ 650 x + 2y ≤ 1100 y ≤ 500 x ≥ 0, y ≥ 0 Answer: x = 200, y = 450, M = 1950 or x = 650, y = 0, M = 1950 66) Maximize 10x + 8y subject the following constraints. 2x + 3y ≤ 10 x+ y≥ 1 x ≥ 0, y ≥ 0 Answer: x = 5, y = 0, maximum = 50 67) Maximize -4x + y subject to the following constraints. x≤3 y≥1 4x - y ≥ 7 x ≥ 0, y ≥ 0 Answer: x = 2, y = 1, maximum = -7 or x = 3, y = 5, maximum = -7 68) Maximize M = 3x + 2y subject to the following constraints. 5x + 2y ≤ 100 5x + 18y ≤ 180 x + y≥5 x ≥ 0, y ≥ 0 Answer: Maximum = 64 when x = 18, y = 5. 69) Maximize M = 3x + 2y + 5z subject to the following constraints. x - 2z ≤ 10 x + y + 6z ≤ 80 x ≥ 0, y ≥ 0, z ≥ 0 Answer: x = 10, y = 70, z = 0, M = 170 70) Maximize 3x + y - z subject to the following constraints. x+y+z≤4 -x + y + z ≥ 1 x ≥ 0, y ≥ 0, z ≥ 0 Answer: Maximum = 7 when x =

3 5 , y = , z = 0. 2 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 71) Consider the simplex tableau that is the final one in a problem to maximize x + 2y + 3z. x y z u v M 2 1 0 -3 8 0 2 3 0 1 5 2 0 5 1 0 0 6 5 1 19 What is the maximum value of x + 2y + 3z? A) -19. B) 19. C) 0. D) -12. E) none of these Answer: B 72) Consider the simplex tableau that is the final one in a problem to maximize x + 2y + 3z. x y z u v M 2 1 0 -3 8 0 2 3 0 1 5 2 0 5 1 0 0 6 5 1 19 What are the values of x, y and z when the maximum value of x + 2y + 3z occurs? A) x = 19, y = 2, z = 5. B) x = 2, y = 1, z = 0. C) x = 0, y = 2, z = 5. D) x = 2, y = 5, z = 0. E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 73) Using the following initial simplex tableau, perform as many pivoting operations as needed until a maximum for the objective function M is reached. Give the final values of all the variables and that of the objective function. x y u v M 1 1 1 0 0 4 2 -1 0 1 0 4 -6 -2 0 0 1 0 Answer: x =

8 4 56 , y = , u = 0, v = 0, M = 3 3 3

Solve the problem. 74) Big Round Cheese Company has on hand 45 pounds of Cheddar and 49 pounds of Brie each day. It prepares two Christmas packagesthe "Holiday" box, which has 5 pounds of Cheddar and 2 pounds of Brie, and the "Noel" box, which contains 2 pounds of Cheddar and 7 pounds of Brie. Profit on each Holiday assortment is $6, profit on each Noel assortment is $8. The initial and final tableaux are as follows: x y u v M 7 2 1 0 0 7 31 31 x y u v M 2 5 5 2 1 0 0 45 0 1 0 5 31 31 2 7 0 1 0 49 26 28 -6 -8 0 0 1 0 0 0 1 82 31 31 (initial) (final) Here, x = the number of Holiday boxes per day and y = the number of Noel boxes per day. (a) Give the optimal production schedule and the resulting profit. (b) How much excess Cheddar and excess Brie remain each day if this plan is followed? Answer: (a) x = 7, y = 5, M = $82 (b) none 75) A stereo manufacturer makes three types of stereo systems, I, II, and III, with profits of $20, $30, and $40, respectively. No more than 100 type-I systems can be made per day. Type-I systems require 5 man-hours, and the corresponding numbers of man-hours for types II and III are 10 and 15, respectively. If the manufacturer has available 2000 man-hours per day, determine the number of units from each system that must be manufactured in order to maximize profit. Compute the corresponding profit. Answer: 100 of type-I, 150 of type-II, and none of type-III systems; maximum profit = $6500 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 76) Jerry builds rocking chairs and porch swings and sells them a a local flea market. The lumber requirements for each type of item are summarized in the table below. Rocking Chair Porch Swing 2×4's 1 5 2×6's 1 2 1×4's 2 3 Currently, Jerry has 100 2×4's, 55 2×6's, and 100 1×4's in stock. Rocking chairs sell for $40 and porch swings sell for $65. How many pieces of each type of lumber are left over after Jerry has built the number of chairs and swings that maximize his revenue? A) 4 2×4's, 0 2×6's, and 5 1×4's. B) 0 2×4's, 0 2×6's, and 15 1×4's. C) 5 2×4's, 0 2×6's, and 4 1×4's. D) 15 2×4's, 0 2×6's, and 0 1×4's. Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Write the initial simplex tableau for the linear programming problem. 77) Maximize x + y + 3z subject to the following constraints. ≤2 x+y x z + ≤5 y+ z≤1 x ≥ 0, y ≥ 0, z ≥ 0 Answer: x y z 1 1 0 1 0 1 0 1 1 -1 -1 -3

u 1 0 0 0

v w M 0 0 0 1 0 0 0 1 0 0 0 1

2 5 1 0

78) Maximize 3x -2y subject to the following constraints. -x + y ≤ 3 3x + y ≤ 10 y≤2 x ≥ 0, y ≥ 0 Answer: x -1 3 0 -3

y u 1 1 1 0 1 0 2 0

v w M 0 0 0 3 1 0 0 10 0 1 0 2 0 0 1 0

79) Maximize x + 2y + 30 subject to the following constraints. x + 3y ≤ 10 2x + y ≤ 12 x ≥ 0, y ≥ 0 Answer:

x y u 1 3 1 2 1 0 -1 -2 0

v M 0 0 10 1 0 12 0 1 30

80) Maximize x +3y + 40 subject to the following constraints. 2x - 5y ≤ 20 2x + y ≤ 16 x ≥ 0, y ≥ 0 Answer:

x y u 2 -5 1 2 1 0 -1 -3 0

v M 0 0 20 1 0 16 0 1 40

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 81) The linear programming problem, Minimize 5x - y subject to

-2x - 2y ≤ 12 -3x + 2y ≥ 0 , is equivalent to the linear x ≥ 0, y ≥ 0

programming problem: A) Maximize 5x - y subject to

-2x - 2y ≤ 12 3x - 2y ≤ 0 x ≥ 0, y ≥ 0

B) Maximize 5x - y subject to

-2x - 2y ≤ 12 3x + 2y ≥ 0 x ≥ 0, y ≥ 0

C) Maximize 5x - y subject to

-2x - 2y ≤ 12 -3x + 2y ≥ 0 x ≥ 0, y ≥ 0

-2x - 2y ≤ 12 D) Maximize 5x - y subject to -3x + 2y ≤ 0 x ≥ 0, y ≥ 0 E) none of these Answer: E SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 82) Rewrite the following linear programming problem as a maximization problem whose constraints have the form [polynomial] ≤ [constant]. x + y ≤ 25 Minimize 3x - y subject to 3x + 2y ≥ 6 x ≥ 0, y ≥ 0 x + y ≤ 25 Answer: Maximize y - 3x subject to -3x - 2y ≤ -6 x ≥ 0, y ≥ 0 Write the initial simplex tableau for the linear programming problem. 83) Minimize 3x + 4y subject to the following constraints. x + y ≤ 10 2x + 3y ≥ 1 x ≥ 0, y ≤ 0 Answer:

x y u 1 1 1 -2 -3 0 3 4 0

v M 0 0 10 1 0 -1 0 1 0

84) Minimize 5x - 6y subject to the following constraints. 3 x+y≤5 2 1 x+y≤3 2 x ≥ 0, y ≥ 0 Answer:

x y u v M 3 1 1 0 0 5 2 1 2

1 0 1 0 3

5 -6 0 0 1 0 85) Minimize 5x - 6y + 10 subject to the following constraints. 3x + y ≤ 7 4x + y ≤ 5 x ≥ 0, y ≥ 0 Answer:

x y u v M 3 1 1 0 0 7 4 1 0 1 0 5 5 -6 0 0 1 10

86) Minimize 2x + 3y subject to the following constraints. 3x + 2y ≥ 10 x - 5y ≤ 20 x ≥ 0, y ≥ 0 Answer:

x y u v M -3 -2 1 0 0 -10 1 -5 0 1 0 20 2 3 0 0 1 0

Use the simplex method to solve. 87) Minimize -x + 2y subject to the following constraints. 2x + 3y ≤ 6 2x + y ≤ 14 x ≥ 0, y ≥ 0 Answer: x = 3, y = 0, minimum = -3 88) Minimize 3x + 2y subject to the following constraints. x + 2y ≥ 20 3x + y ≥ 25 x ≥ 0, y ≥ 0 Answer: x = 6, y = 7, minimum = 32 89) Minimize 3x + 2y subject to the following constraints. x + y ≥ 10 x - y ≤ 15 x ≥ 0, y ≥ 0 Answer: Minimum = 20, when x = 0, y =10.

90) Minimize 3x + 2y subject to the following constraints. 4x + 3y ≥ 24 2x + y ≥ 10 x ≥ 0, y ≥ 0 Answer: Minimum = 17 when x = 3, y = 4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 91) Consider the following simplex tableau that is the final tableau in a problem to minimize -x + 2y. x y u v w M 1 0 3 0 0 0 10 0 0 1 0 1 0 0 0 1 -6 0 0 0 3 0 0 8 1 0 0 7 0 0 5 0 0 1 4 What is the minimum value of -x + 2y? A) 4 B) 0 C) -4 D) -10 E) none of these Answer: C 92) Consider the following simplex tableau that is the final tableau in a problem to minimize -x + 2y. x y u v w M 1 0 3 0 0 0 10 0 0 1 0 1 0 0 0 1 -6 0 0 0 3 0 0 8 1 0 0 7 0 0 5 0 0 1 4 What are the values of x and y when the minimum value of -x + 2y occurs? A) x = 10, y = 3 B) x = -20, y = 6 C) x = 10, y = 0 D) x = -10, y = -3 E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 93) The following simplex tableau occurred while minimizing 4x - 2y. x y u v M 3 1 5 1 0 0 2 2 2 0 1 -2 7 0 0 2

1 0

3 1 -4

Find the minimum and where it occurs. Answer: Minimum = 4 when x =

5 ,y=3 2

94) It is desired to minimize 3x - 5y. The final simplex tableau is x y u v M 1 0 6 -2 0 10 0 1 5 3 0 8 0 0 1 4 1 -10 (a) What is the minimum value of 3x - 5y? (b) What values of x and y yield the minimum value of 3x - 5y? Answer: (a) minimum = 10 (b) x = 10, y = 8 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 95) Consider the following simplex tableaux. I. II. x y u v M x y u v M 1 0 -3 1 0 2 1 0 4 2 0 0 0 1 2 0 0 4 0 1 8 -3 0 0 0 0 -5 3 1 10 0 0 12 6 1 0 III.

x y u v M 3 1 0 0 2 2 1 0 1 0 4 10 6 0 0 1 3 8

IV.

x 2 3 6

y 1 0 0

u v M 0 3 0 2 1 1 0 1 8 0 1 -5

Which of these tableaux have a solution that corresponds to a minimum for the associated linear programming problem? A) II, III, and IV only B) II and IV only C) II and III only D) IV only E) I only Answer: A

96) The linear programming problem, Maximize -3x - 4y subject to

-x + y ≤ 0 -3x + 4y ≤ 0 , is equivalent to the linear x ≥ 0, y ≥ 0

programming problem: A) Minimize 3x + 4y subject to

x-y≤0 3x - 4y ≥ 0 x ≥ 0, y ≥ 0

B) Minimize 3x + 4y subject to

x-y≥0 3x - 4y ≥ 0 x ≥ 0, y ≥ 0

C) Minimize 3x + 4y subject to

x-y≥0 3x + 4y ≥ 0 x ≥ 0, y ≥ 0

D) Minimize 3x + 4y subject to

x-y≥0 3x - 4y ≤ 0 x ≥ 0, y ≥ 0

E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the simplex method to solve. 97) A manufacturing company makes three brands of television sets A, B, and C in two different plants I and II. Plant I has a production capacity of 40 brand-A sets, 40 brand-B sets, and 20 brand-C sets per day, and a daily operating cost of $3000. Plant II has a production capacity of 20 brand-A sets, 60 brand-B sets, and 20 brand-C sets per day, and a daily operating cost of $2000. How many days should each plant be operated to fill an order for 600 brand-A, 1000 brand-B, and 400 brand-C sets at the minimum cost? Answer: Each plant should operate 10 days; minimum cost = $50,000. 98) An electronic store sells VCR brands X, Y and Z. The profit per VCR for brand X is $30, for Brand Y is $60 and for Brand Z is $50. The total space in the back room allotted to all brands is sufficient for 600 VCRs, and inventory is delivered once a month. At least 100 customers per month demand Brand X, at least 200 will demand brand Y or Z and at least 50 per month will demand brand Z. How can the electronic store satisfy all these constraints and maximize profit. Answer: Stock 100 of Brand X, 450 of Brand Y, and 50 of Brand Z. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A toy making company has at least 300 squares of felt, 700 oz of stuffing, and 230 ft of trim to make dogs and dinosaurs. A dog uses 1 square of felt, 4 oz of stuffing, and 1 ft of trim. A dinosaur uses 2 squares of felt, 3 oz of stuffing, and 1 ft of trim. 99) It costs the company $1.21 to make each dog and $1.46 for each dinosaur. What is the company's minimum cost? A) $219 B) $242 C) $243 D) $267 Answer: D 29

100) It costs the company $1.30 to make each dog and $1.18 for each dinosaur. What is the company's minimum cost? A) $248 B) $284 C) $177 D) $228 Answer: A Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill #1 contains 4 units of A and 3 of B. Pill #2 contains 1 unit of A, 2 of B, and 4 of C. Pill #3 contains 10 units of A, 1 of B, and 5 of C. 101) Pill #1 costs 5 cents, pill #2 costs 10 cents, and pill #3 costs 1 cent. How many of each pill must Larry take to minimize his cost? A) P1 = 2, P2 = 0, P3 = 6 B) P1 = 1, P2 = 0, P3 = 9 C) P1 = 0, P2 = 0, P3 = 12 D) P1 = 3, P2 = 0, P3 = 4 Answer: C A workshop of Peter's Potters makes vases and pitchers. Profit on a vase is $3, profit on a pitcher is $4. Each vase requires 1 hour of labor, each pitcher requires 1 hour of labor. Each item requires 1 unit of time in the kiln. Labor is limited to 4 2 hours per day and kiln time is limited to 6 units per day. Initial and final tableaux are shown in finding the production plan that will maximize profits: (x = the number of vases and y = the number of pitchers made per day). x y u v M x y u v M 1 1 1 0 0 4 0 1 2 -1 0 2 2 1 0 -2 2 0 4 1 1 0 1 0 6 0 0 2 2 1 20 -3 -4 0 0 1 0 (initial) (final) 102) How much surplus labor is there when the optimal plan is in effect? A) 4 hours B) 2 hours C) 1 hour D) 0 hours E) none of these Answer: D 103) How much surplus kiln time is available when the optimal plan is in effect? A) 4 units B) 0 units C) 2 units D) 1 unit E) none of these Answer: B

104) If labor were increased by one hour a day, profits could be increased to A) $21. B) $22. C) not increased D) $24. E) none of these Answer: B 105) If labor were increased by one hour a day, the optimal production plan would require how many vases? A) 3 B) 1 C) 2 D) 4 E) none of these Answer: C 106) If kiln time were decreased by one unit a day, determine the optimal production schedule. A) x = 3, y = 4 B) x = 4, y = 2 C) x = 2, y = 3 D) x = 6, y = 1 E) none of these Answer: C In a linear programming problem in standard form, the initial and final tableaux are given below. x y u v M 5 3 1 0 0 20 x y u vM 2 2 1 3 1 0 0 50 1 1 0 1 0 10 1 5 0 1 0 70 2 2 -6 -24 0 0 1 0 0 0 3 3 1 360 (initial) (final) 107) We wish to add h units to the first resource (originally at level 50). What are the limits on the size of h so that the solution can still be read from the final tableau? A) -5 ≤ h ≤ 120 B) -120 ≤ h ≤ 8 C) 5 ≤ h ≤ 8 D) -8 ≤ h ≤ 20 E) none of these Answer: D 108) Given that x ≥ 0 and y ≥ 0, if h units were added to the first resource, wha is the maximum value of the objective function? A) 360 B) 360 + 3h C) 360 + h h5 D) 360 + 5 E) none of these Answer: B

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 109) Big Round Cheese Company has on hand 45 pounds of Cheddar and 49 pounds of Brie each day. It prepares two Christmas packagesthe "Holiday" box, which has 5 pounds of Cheddar and 2 pounds of Brie, and the "Noel" box, which contains 2 pounds of Cheddar and 7 pounds of Brie. Profit on each Holiday assortment is $6, profit on each Noel assortment is $8. Determine the number of boxes of each assortment that will maximize profits for Big Round Cheese Company given the initial and final tableaux: x y u v M 7 2 1 0 0 7 31 31 x y u vM 2 5 5 2 1 0 0 45 0 1 0 5 31 31 2 7 0 1 0 49 26 28 -6 -8 0 0 1 0 0 0 1 82 31 31 (initial) (final) Here, x = the number of Holiday boxes per day and y = the number of Noel boxes per day. One day Big Round gets a total of 80 pounds of Brie. How should the production schedule change to maximize profit? What is the new profit? Answer: x = 5, y = 10, M = $110 110) A furniture manufacturer makes TV tables and bookcases in the same workshop. Each item brings in a profit of $18. Each bookcase requires 2 hours of labor 3 units of teak wood. Each TV table requires 6 hours of labor and 1 unit of teak wood. Resources in the workshop are restricted to 6 units of teak wood a day and 12 hours of labor per day. The initial and final tableaux are given below with x = the number of TV tables and y = the number of bookcases produced each day. x y u v M 3 1 3 0 1 0 8 16 2 x y u vM 1 3 3 1 3 1 0 0 6 1 0 0 8 16 2 6 2 0 1 0 12 9 9 -18 -18 0 0 1 0 0 0 1 54 2 4 (initial)

(final)

(a) What is the optimal production schedule and maximum daily profit? (b) If the amount of available teak wood can be increased to 10 units per day, how does the production schedule change? How do maximum profits change? (c) If the amount of teak changes by h units in a day, how would the optimal production schedule and maximum profit change? (d) What restrictions should be placed on the size of h to make the analysis in (c) valid? Answer: (a) x =

3 3 , y = , M = $54 2 2

(b) x = 1, y = 3: Make 1 TV table and 3 bookcases per day; M = $72 (c) x =

3 1 3 3 9 - h, y = + h, and M = 54 + h. 2 8 2 8 2

(d) -4 ≤ h ≤ 12

111) Consider a linear programming problem given in standard form where the final simplex tableau for maximizing 3x + 5y is shown below. x y u v M 4 1 0 1 0 4 7 7 1 7

2 0 7

17 7

1 1 26 7

1 0 0 0

If, in the above (primal) problem, the resource given in the first constraint (and associated with the slack variable u) were increased by 7 units, determine the new values of x and y and the new maximum for the objective function. Answer: Maximum = 43 when x = 1, y = 8 112) Let the final simplex tableau for a linear programming problem be given by x y

u v 1 0 5

w M 2 0 3

0 1 -

3 0 5

5 3

0 0

6 4 1 5 3

0 0

1 0

7 3

1 112

What will the new output be if u increases by 3 units? 3 1 3 Answer: x = 12 , y = 5 , v = 35 , M = 115 5 5 5 Find the transpose of the given matrix. 113) 2 1 -1 3 Answer: 2 -1 1 3 114) [2 2 3] 2 Answer: 2 3 6 0 115) 2 0 1 4 Answer: 6 2 1 0 0 4 116) 1 2 Answer: [1 2] 33

117) 2 3 8 0 5 6 2 0 Answer: 3 5 8 6 1 118) 3 7 9 Answer: [1 3 7 9] 119) [7 4 11] Answer:

7 4 11

120) 8 -2 6 4 Answer:

8 6 -2 4

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the transpose of the matrix. 121) 5517 1786 A) 51 57 18 76 B) 1786 5517 C) 15 75 81 67 D) 35 51 76 18 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 122) Is it true that the transpose of the transpose of a matrix is the original matrix? Answer: Yes. 34

123) Give an example of a matrix that is its own transpose. Answer: Any symmetric n x n matrix will do. For example, 2 1 or 2 0 . 1 3 0 3 124) Rewrite the following linear programming problem using matrix notation. x + y ≥ 200 2x + y ≥ 300 Minimize 3x + 4y subject to 2x + 5y ≥ 500 x ≥ 0, y ≥ 0 Answer: Minimize [3

1 1 4] x subject to 2 1 y 2 5

200 x ≥ x ≥ 0 300 and y y 0 500

125) Give the matrix formulation of the following linear programming problem. 7x + 6y + 3z ≤ 45 + 2z ≤ 22 Maximize 3x + 5y + 4z subject to x 3x + 2y + 10z ≤ 37 x ≥ 0, y ≥ 0, z ≥ 0 x 7 6 3 Answer: Maximize [3 4 5] y subject to 1 0 2 z 3 2 10

x 45 y ≤ 22 z 37

126) Rewrite the following linear programming problem without using matrices. Minimize CX subject to AX ≥ BT, X ≥ 0, where x 0 C = [1 1 3], B = [20 10], A = 1 1 1 , X = y , and 0 = 0 . 0 1 2 z 0 Answer: Minimize x + y + 3z subject to the constraints x + y + z ≥ 20 y + 2z ≥ 10 x ≥ 0, y ≥ 0, z ≥ 0 127) Rewrite the following linear programming problem without using matrices. Maximize CX subject to AX ≤ B, X ≥ 0, where 20 6 1 C = [40 50], B = 22 , A = 1 2 , X = 1 , and 0 = 0 . 2 0 33 1 4 Answer: Maximize 40x +50 y subject to the constraints 6x + y ≤ 20 y + 2z ≤ 22 x + 4y ≤ 33 x ≥ 0, y ≥ 0

5 2 25 128) Let A = 4 3 , B = 17 , C = [2 3], X = x y 4 6 30 (a) Give the linear programming problem whose matrix formula is "Maximize CX subject to AX ≤ B and X ≥ 0." (b) Give the initial simplex tableau that corresponds to the problem in part (a). 5x + 2y ≤ 25 ≤ Answer: (a) Maximize 2x + 3y subject to 4x + 3y 17 4x + 6y ≤ 30 x ≥ 0, y ≥ 0 (b) x y u v w M 5 2 1 0 0 0 25 4 3 0 1 0 0 17 4 6 0 0 1 0 30 -2 -3 0 0 0 1 0 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. x+ y≥6 129) To solve the linear programming problem, Minimize 50x + 70x subject to 3x + 5y ≥ 24 ,we can use the simplex x ≥ 0, y ≥ 0 method on its dual. What is the objective function of the dual problem? A) u + 3v B) u + 5v C) 70u + 50v D) 6u + 24v E) none of these Answer: D Determine the dual problem of the given linear programming problem. 5x + y ≤ 80 130) Maximize 4x + 2y subject to 3x + 2y ≤ 76 x ≥ 0, y ≥ 0 5u + 3v ≥ 4 A) Minimize 80u + 76v subject to the constraints u + 2v ≥ 2 x ≥ 0, y ≥ 0 5u + 3v ≤ 4 B) Minimize 80u + 76v subject to the constraints u + 2v ≥ 2 x ≥ 0, y ≥ 0 5u + 3v ≤ 4 C) Minimize 80u + 76v subject to the constraints u + 2v ≤ 2 x ≥ 0, y ≥ 0 5u + 2v ≥ 4 D) Minimize 80u + 76v subject to the constraints u + 3v ≥ 2 x ≥ 0, y ≥ 0 E) none to the above Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 131) Minimize 3x + 4y subject to

x + y ≥ 10 2x + 3y ≥ 30 x ≥ 0, y ≥ 0

Answer: Maximize 10u + 30v subject to

u + 2v ≤ 3 u + 3v ≤ 4 u ≥ 0, v ≥ 0

2x + y ≤ 17 ≤ 132) Maximize 9x + 15y subject to x + 2y 12 x + 3y ≤ 16 x ≥ 0, y ≥ 0 Answer: Minimize 17u + 12v + 16w subject to

2u + v + w ≥ 9 u + 2v + 3w ≥ 15 u ≥ 0, v ≥ 0, w ≥ 0

x + 2y ≥ 4 -x + y≥5 133) Minimize 8x + 10y subject to 3x + 4y ≥ 2 x ≥ 0, y ≥ 0 Answer: Maximize 4u + 5v + 2w subject to

u - v + 3w ≤ 8 2u + v + 4w ≤ 10 u ≥ 0, v ≥ 0, w ≥ 0

-u + 8v ≤ 3 ≥ 134) Minimize 3x + 5y + z subject to 8x + y + 9z 10 x ≥ 0, y ≥ 0, z ≥ 0 -2u + 8v ≤ 3 4u + v ≤ 5 Answer: Maximize -7u + 10v subject to 6u + 9v ≤ 1 u ≥ 0, v ≥ 0 x+ y≤7 135) Maximize 3x + 10y subject to 5x + 7y ≤ 20 x ≥ 0, y ≥ 0 u + 5v ≥ 3 Answer: Minimize 7u + 20v subject to u + 7v ≥ 10 u ≥ 0, v ≥ 0

136) Minimize 2x + 5y + z subject to

6x + 5y - 2z ≥ 4 3x + z≥1 x ≥ 0, y ≥ 0, z ≥ 0

6u + 3v ≤ 2 ≤5 5u Answer: Maximize 4u + v subject to -2u + v ≤ 1 u ≥ 0, v ≥ 0

≤6 x - 2y + z ≤9 137) Maximize 10x + 12y + 10z subject to 3x y + 3z ≤ 12 x ≥ 0, y ≥ 0, z ≥ 0 ≥ 10 u + 3v ≥ 12 w -2u + Answer: Minimize 6u +9 v + 12w subject to v + 3w ≥ 10 u ≥ 0, v ≥ 0, w ≥ 0

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. State the dual problem. Use y1 , y2 , y3 and y4 as the variables. Given: y1 ≥ 0, y2 ≥ 0, y3 ≥ 0, and y4 ≥ 0. 138) Maximize subject to:

z = 3x1 + 2x2 x1 + x2 ≤ 14 2x1 + x2 ≤ 28 x1 ≥ 0, x2 ≥ 0

A) Minimize subject to: B) Minimize subject to: C) Minimize subject to: D) Minimize subject to:

w = 28y1 + 14y2 2y1 + y2 ≥ 3 y1 + y2 ≥ 2 w = 14y1 + 28y2 y1 + 2y2 ≤ 3 y1 + y2 ≤ 2 w = 28y1 + 14y2 2y1 + y2 ≤ 3 y1 + y2 ≤ 2 w = 14y1 + 28y2 y1 + 2y2 ≥ 3 y1 + y2 ≥ 2

Answer: D

139) Maximize

z = x1 + 2x2 + 3x3

subject to:

6x1 + 3x2 + x3 ≤ 18 4x1 + 7x2 + 3x3 ≤ 35 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

A) Minimize w = 35y1 + 18y2 subject to: 4y1 + 6y2 ≥ 1 7y1 + 3y2 ≥ 2 3y1 + y2 ≥ 3 B) Minimize w = 35y1 + 18y2 subject to: 4y1 + 6y2 ≤ 1 7y1 + 3y2 ≤ 2 C) Minimize subject to:

D) Minimize subject to:

3y1 + y2 ≤ 3 w = 18y1 + 35y2 6y1 + 4y2 ≥ 1 3y1 + 7y2 ≥ 2 y1 + 3y2 ≥ 3 w = 18y1 + 35y2 6y1 + 4y2 ≤ 1 3y1 + 7y2 ≤ 2 y1 + 3y2 ≤ 3

Answer: C 140) Maximize subject to:

z = 8x1 + 7x2 9x1 + 3x2 ≤ 191 x1 + x2 ≤ 25 2x1 + 6x2 ≤ 91 x1 ≥ 0, x2 ≥ 0

A) Minimize

w = 191y1 + 25y2 + 91y3

subject to: 9y1 + y2 + 2y3 ≥ 8 3y1 + y2 + 6y3 ≥ 7 B) Minimize

w = 191y1 1 + 25y2 + 91y3

subject to: 9y1 + y2 + 2y3 ≤ 8 3y1 + y2 + 6y3 ≤ 7 C) Minimize subject to: D) Minimize subject to:

w = 91y1 + 25y2 + 191y3 2y1 + y2 + 9y3 ≥ 8 6y1 + y2 + 3y3 ≥ 7 w = 91y1 + 25y2 + 191y3 2y1 + y2 + 9y3 ≤ 8 6y1 + y2 + 3y3 ≤ 7

Answer: A

141) Minimize

w = 6x1 + 3x2

subject to:

3x1 + 2x2 ≥ 35 2x1 + 5x2 ≥ 43 x1 ≥ 0, x2 ≥ 0

A) Maximize

z = 35y1 + 43y2

subject to:

3y1 + 2y2 ≤ 6 2y1 + 5y2 ≤ 3

B) Maximize

z = 35y1 + 43y2

subject to:

3y1 + 2y2 ≥ 6 2y1 + 5y2≥ 3

C) Maximize

z = 43y1 + 35y2

subject to:

2y1 + 3y2 ≥ 6 5y1 + 2y2 ≥ 3

D) Maximize

z = 43y1 + 35y2

subject to:

2y1 + 3y2 ≤ 6 5y1 + 2y2 ≤ 3

Answer: A 142) Minimize

w = 2x1 + 3x2 + x3

subject to:

x1 + 3x2 + 2x3 ≥ 35 2x1 + 4x2 + 3x3 ≥ 50 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

A) Maximize

z = 35y1 + 50y2

subject to:

y1 + 2y2 ≤ 2 3y1 + 4y2 ≤ 3

B) Maximize subject to:

2y1 + 3y2 ≤ 1 z = 35y1 + 50y2 y1 + 2y2 ≥ 2 3y1 + 4y2 ≥ 3

2y1 + 3y2 ≥ 1 C) Maximize z = 50y1 + 35y2 subject to:

2y1 + y2 ≤ 2 4y1 + 3y2 ≤ 3

3y1 + 2y2 ≤ 1 D) Maximize z = 50y1 + 35y2 subject to:

2y1 + y2 ≥ 2 4y1 + 3y2 ≥ 3 3y1 + 2y2 ≥ 1

Answer: A

143) Minimize

w = 11x1 + 7x2 + 4x3 + 5x4

subject to:

11x1 + 5x2 + 7x3 + 4x4 ≥ 25 16x1 + 5x2 + 11x3 + 7x4 ≥ 30 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0

A) Maximize

z = 25y1 + 30y2

subject to:

11y1 + 16y2 ≤ -11 5y1 + 5y2 ≤ -7 7y1 + 11y2 ≤ -4 4y1 + 7y2 ≤ -5

B) Maximize

z = -25y1 - 30y2

subject to:

11y1 + 16y2 ≤ 11 5y1 + 5y2 ≤ 7 7y1 + 11y2 ≤ 4 4y1 + 7y2 ≤ 5

C) Maximize

z = 25y1 + 30y2

subject to:

11y1 + 16y2 ≤ 11 5y1 + 5y2 ≤ 7 7y1 + 11y2 ≤ 4 4y1 + 7y2 ≤ 5

D) Maximize

z = 25y1 + 30y2

subject to:

11y1 + 16y2 ≥ 11 5y1 + 5y2 ≥ 7 7y1 + 11y2 ≥ 4 4y1 + 7y2 ≥ 5

Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine the dual in matrix form of the given problem. x 1 2 3 x 4 x 0 144) Maximize [-2 5 4] y subject to -5 1 4 y ≤ 3 and y ≥ 0 z 9 2 7 z 6 z 0 u 1 -5 9 Answer: Minimize [4 3 6] v subject to 2 1 2 w 3 4 7 x 4 -6 9 145) Minimize [5 -8 -6] y subject to 3 5 -1 z 8 2 13

u v w

≤

u -2 5 and v 4 w

≥

0 0 0

5 u -8 and v w -6

≥

x 8 x 0 y ≥ 7 and y ≥ 0 z 2 z 0

u 4 3 8 Answer: Maximize [8 7 2] v subject to -6 5 2 w 9 -1 13

u v w

≤

0 0 0

146) A linear programming maximization problem with four variables and three nontrivial inequalities has the number 27 as the maximum value of the objective function. (a) How many variables and nontrivial inequalities will the dual problem have? (b) What is the minimum value of the objective function of the dual problem? Answer: (a) 3 variables, 4 nontrivial inequalities (b) 27 147) Determine the dual problem. Solve either the original problem or its dual by the simplex method and then give the solutions to both. 2x + 4y ≥ 10 Minimize 20x + 30y subject to 5x + y ≥ 16 x ≥ 0, y ≥ 0 Answer: Original: x = 3, y = 1, M = 90 (minimum) 65 10 Dual: u = ,v= , M = 90 (maximum) 9 9 148) Consider a linear programming problem given in standard form where the final simplex tableau for maximization is shown below. x y u v M 2 -1 1 0 0 5 7 7 0 1 -

3 7

5 0 14 7

0 0

2 7

11 1 28 7

If x and y are the original variables and u and v are the slack variables, what is the solution to the problem and to its dual? Answer: The values of the original variables are 5 and 14, and the dual variables are

2 11 and . 7 7

149) Consider a linear programming problem given in standard form where the final simplex tableau for maximizing 3x + 5y shown below. x y u v M 4 1 0 1 0 4 7 7 1 7

2 0 7

17 7

1 1 26 7

1 0 0 0

If x and y are the original variables and u and v are the slack variables, what is the solution to the dual problem? Answer: The values of the dual variables are

17 1 and and the objective function has a minimum of 26. 7 7

150) Consider the following linear programming problem. 2x + 3y ≤ 9 Maximize M = 5x + 6y subject to x + y≤4 x ≥ 0, y ≥ 0 The associated final simplex tableau is as follows: x y u v M 0 1 1 -2 0 1 1 0 -1 3 0 3 0 0 1 3 1 21 (a) Give the solution to the problem. (b) Give the solution to its dual. Answer: (a) x = 3, y = 1, M = 21 (maximum) (b) u = 1, v = 3, M = 21 (minimum) 151) Consider the following linear programming problem. x - 2z ≤ 10 Maximize M = 3x + 2y + 5z subject to x + y + 6z ≤ 80 x ≥ 0, y ≥ 0, z ≥ 0. The associated final simplex tableau is as follows: x y z u v M 1 0 -2 1 0 0 10 0 1 8 -1 1 0 70 0 0 5 1 2 1 170 (a) Give the solution to the problem. (b) Give the solution to its dual. Answer: (a) x = 10, y = 70, z = 0, M = 170 (maximum) (b) u = 1, v = 2, M = 170 (minimum)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 152) A workshop of Peter's Potters makes vases and pitchers. Profit on a vase is $3, profit on a pitcher is $4. Each 1 vase requires hour of labor, each pitcher requires 1 hour of labor. Each item requires 1 unit of time in the kiln. 2 Labor is limited to 4 hours per day and kiln time is limited to 6 units per day. Initial and final tableaux are shown in finding the production plan that will maximize profits: (x = the number of vases and y = the number of pitchers made per day). x y u v M x y u v M 1 1 1 0 0 4 0 1 2 -1 0 2 2 1 0 -2 2 0 4 1 1 0 1 0 6 0 0 2 2 1 20 -3 -4 0 0 1 0 (initial) (final) Peter's Pottery wishes to add mugs to its line. Each mug requires

1 1 hour of labor and unit of kiln time. What 4 2

minimum profit must be associated with each mug to make production of mugs worthwhile? A) $1.50 B) $2.50 C) $3.50 D) $1.00 E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. A furniture manufacturer makes TV tables and bookcases in the same workshop. Each item brings in a profit of $18. Each bookcase requires 2 hours of labor 3 units of teak wood. Each TV table requires 6 hours of labor and 1 unit of teak wood. Resources in the workshop are restricted to 6 units of teak wood a day and 12 hours of labor per day. The initial and final tableau are given below with x = the number of TV tables and y = the number of bookcases produced each day. x y u v M 3 1 3 0 1 0 8 16 2 x y u vM 1 3 3 1 3 1 0 0 6 1 0 0 8 16 2 6 2 0 1 0 12 9 9 -18 -18 0 0 1 0 0 0 1 54 2 4 (initial) (final) 153) Write the dual problem to the above primal problem and give a short interpretation of the dual and its solution. u + 6v ≥ 18 Answer: Dual problem: Minimize 6u +12v subject to 3u+ 2v ≥ 18 u ≥ 0, v ≥ 0 Interpretation: The minimum profit the manufacturer would expect for this business is $54 per day. He 9 9 should get u = $ = $4.50 profit per unit of wood and v = $ = $2.25 profit per hour of labor. 2 4 154) The manufacturer thinks he can produce a desk chair in the same workshop. The chair uses 2 units of teak wood and 5 hours of labor. At what profit would it be worthwhile to include desk chairs in the production line? Answer: $20.25

Big Round Cheese Company has on hand 45 pounds of Cheddar and 49 pounds of Brie each day. It prepares two Christmas packagesthe "Holiday" box, which has 5 pounds of Cheddar and 2 pounds of Brie, and the "Noel" box, which contains 2 pounds of Cheddar and 7 pounds of Brie. Profit on each Holiday assortment is $6, profit on each Noel assortment is $8. Determine the number of boxes of each assortment that will maximize profits for Big Round Cheese Company yields the initial and final tableaux: x y u v M 7 2 1 0 0 7 31 31 x y u vM 2 5 5 2 1 0 0 45 0 1 0 5 31 31 2 7 0 1 0 49 26 28 -6 -8 0 0 1 0 0 0 1 82 31 31 (initial) (final) Here, x = the number of Holiday boxes per day and y = the number of Noel boxes per day. 155) State and interpret the dual problem giving its solution. Answer: Minimize 45u + 49v subject to Solution: u =

5u + 2v ≥ 6 2u + 7v ≥ 8 u ≥ 0, v ≥ 0

26 28 ,v= , M = $82 31 31

Interpretation: Big Round Cheese Company should expect a minimum profit of $82 per day. The 26 28 ≈ $0.84 profit per pound of Cheddar and $ ≈ $0.90 profit per pound of Brie. company should get $ 31 31 156) Big Round is contemplating adding a third assortment, the "Santa" box, to its offerings. The Santa box would contain 3 pounds of Cheddar and 5 pounds of Brie. At what profit on each Santa box would this be worthwhile? Answer: $7.04

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) What is the set of A ∪ A′? A) A B) A ∩ A′ C) U D) ∅ E) none of these Answer: C 2) What is the set of A ∩ A′? A) A ∪ A′ B) U C) A D) ∅ E) none of these Answer: D Consider the sets: U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6, 8}, and B = {1, 2, 3, 5, 7}. 3) Which of the following statements is true? A) A ∩ B is the subset of A B) A ∩ B = U C) A ∩ B = ∅ D) A is a subset if A ∩ B E) none of these Answer: A 4) What is the set of A ∪ B′? A) A B) {2} C) ∅ D) {4, 6, 8} E) none of these Answer: A 5) What is the set of A′ ∪ B? A) {2} B) {4, 6, 8} C) ∅ D) {1, 3, 5,7} E) none of these Answer: D Consider the sets: U ={all professors}, A ={female professors}, and B ={professors under 40 years of age}. 6) What is the set of A′ ∩ B? A) female professors under 40 years of age} B) {male professors who are 40 or older} C) {professors who are male or under 40} D) {male professors under 40 years of age} E) none of these Answer: D

7) What is the set of (A ∪ B)′? A) {professors who are male or 40 or older} B) {professors who are male and under 40} C) {professors who are female or under 40} D) {professors who are male and 40 or older} E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Suppose that A = {1, 3, 5, 7, 9}, B = {1, 3, 7}, and U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. List the elements of the indicated set. 8) B′ Answer: {2, 4, 5, 6, 8, 9} 9) A ∪ B Answer: {1, 3, 5, 7, 9} 10) A ∩ B′ Answer: {5, 9} 11) U ∪ A′ Answer: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} 12) A ∪ A′ Answer: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} Let U = {1, 3, 5, 7, 9, 11}, A = {1, 5, 9, 11}, B = {3, 5, 7}, and C = {1, 3, 11}. List the elements of the indicated set. 13) A ∩ B Answer: {5} 14) U ∩ B Answer: B 15) A ∩ (B ∪ C)′ Answer: {9} 16) A′ ∪ B ∪ C Answer: {1, 3, 5, 7,11} Suppose that U = {a, b, c, d, e}, Α = {a, e}, B = {a, b}, and C = {a, b, c}. List the elements of the indicated set. 17) A ∩ B ∩ C Answer: {a} 18) A ∪ B ∪ C Answer: {a, b, c, e} 19) (A ∪ B) ∩ C′ Answer: {e}

Let U = {cars}, A = {cars manufactured in the United States}, B = {cars with automatic transmissions}, and C = {convertibles}. Describe the indicated set in words. 20) A′ Answer: {cars manufactured outside the United States} 21) A′ ∩ C Answer: {convertibles manufactured outside the United States} 22) Β ∪ C Answer: {cars which have automatic transmission or are convertibles} Let U = {students}, A = {male students}, B = {varsity students}, and C = {Dean's list students}. Describe the indicated set using set-theoretic notation. 23) {female varsity students} Answer: A′ ∩ B 24) {student who are male or on the Dean's list} Answer: A ∪ C 25) {male varsity students who are not on the Dean's list} Answer: A ∩ B ∩ C′ Let U = {all voters}, A = {those who voted in the last election}, B = {those who voted Democrat in the last election}, and C = {voters under 30 years of age}. Describe the indicated set. 26) A ∩ C Answer: {those voters under 30 who voted in the last election} 27) B ∩ C′ Answer: {those voters 30 years old or older who voted Democrat in the last election} 28) A′ ∪ C Answer: {those voters who did not vote in the last election or are under 30 years old} Let U = {all people}, A = {all American citizens}, B = {people who do not live in their country of citizenship}, C = {German citizens}, and D = {all people living in the United States}. Describe the indicated set using set-theoretic notation. 29) {German citizens living in the United States} Answer: C ∩ D 30) {American citizens living abroad} Answer: A ∩ B 31) {Those people who are non-American or do not live in their country of citizenship} Answer: A′ ∪ B 32) {Those people with dual American-German citizenship} Answer: A ∩ C

Simplify the expression. 33) U ∩ S ∩ S′ Answer: ∅ 34) U ∪ S′ Answer: S′ 35) U ∪ ∅ Answer: ∅ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide whether the statement is true or false. 36) 9 ∈ {18, 27, 36, 45, 54} A) True B) False Answer: B 37) 5 ∉ {10, 15, 20, 25, 30} A) True B) False Answer: A 38) {5, 10, 15, 20} ∩ {5, 15} = {5, 10, 15, 20} A) True B) False Answer: B 39) {4, 8, 12, 16} ∪ {4, 12} = {4, 8, 12, 16} A) True B) False Answer: A 40) {9, 8, 11} ∪ {11, 9, 8} = {9, 11} A) True B) False Answer: B 41) {15, 8, 12} ∪ ∅ = {15, 8, 12} A) True B) False Answer: A 42) {5, 8, 11} ∩ ∅ = {5, 8, 11} A) True B) False Answer: B

43) {3, 5, 7} ∩ {4, 6, 8} = {3, 5, 7, 4, 6, 8} A) True B) False Answer: B 44) {10, 4, 3} ∪ {10, 4, 3} = ∅ A) True B) False Answer: B 45) {3, 5, 7} ∩ {4, 6, 7} = {7} A) True B) False Answer: A 46) {5, 7, 9} ∪ {6, 8, 9} = {5, 7, 9, 6, 8} A) True B) False Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 47) List all the subsets of {1, 2, 3}. Answer: ∅ , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} 48) List all the subsets of {1, 2, 3, 4} which do not include 3. Answer: ∅ , {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4} 49) List all the subsets of {a, e, i, o, u} that do not contain i. Answer: ∅ , {a}, {e}, {o}, {u}, {a, e}, {a, o}, {a, u}, {e, o}, {e, u}, {o, u}, {a, e, o}, {a, e, u}, {a, o, u}, {e, o, u} {a, e, o, u} 50) List all the subsets of {+, - , ×, ÷}. Answer: ∅ , { + }, { - }, { × }, { ÷ }, {+, - }, { +, ×}, { +, ÷}, { -, ×}, {-, ÷ }, { ×, ÷}, { +, -, ×}, {+, -, ÷ }, { +, ×, ÷}, { -, ×, ÷ }, {+, -, ×, ÷} MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 51) A set D is a subset of set C provided that A) at least one element of C is not an element of D. B) at least one element of D is not an element of C. C) every element of C is an element of D. D) every element of D is an element of C. E) none of these Answer: D

Insert "⊆ " or "⊈ " in the blank to make the statement true. 52) {2, 4, 6} {1, 2, 3, 4, 6} A) ⊆ B) ⊈ Answer: A 53) {3, 4, 5, 6, 8} __ {4, 6, 8} A) ⊈ B) ⊆ Answer: A 54) ∅ __ {6, 8, 10} A) ⊆ B) ⊈ Answer: A 55) {0, 9} { 7, 9} A) ⊆ B) ⊈ Answer: B 56) {12, 32, 37} A) ⊆ B) ⊈

{9, 32, 37, 47}

Answer: B 57) {b, d, j, i} A) ⊈ B) ⊆

{b, d, j, i, n}

Answer: B 58) {i, d, c} A) ⊆ B) ⊈

{i, d, c}

Answer: A Find the number of subsets of the set. 59) {3, 4, 5} A) 7 B) 6 C) 3 D) 8 Answer: D

Solve the problem. 60) Jose is applying to college. He receives information on 6 different colleges. He will apply to all of those he likes. He may like none of them, all of them, or any combination of them. How many possibilities are there for the set of colleges that he applies to? A) 16 B) 6 C) 60 D) 64 Answer: D 61) If n(S) = 10, n(R ∪ S) = 16, and n(R ∩ S) = 2, find n(R). A) 6 B) 8 C) 4 D) 2 E) none of these Answer: B 62) If n(S) = 12, n(R ∪ S) = 18, and n(R ∩ S) = 3, find n(R). A) 4 B) 8 C) 10 D) 9 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 63) Find n(S ∪ T), given that n(S) = 10, n(T) = 6, and n(S ∩ T) = 5. Answer: 11 64) Find n(A ∪ B) if n(A) = 12, n(Β) = 5, and n(A ∪ B) = 17. Answer: 0 65) Find n(A ∪ B) if n(A) = 12, n(Β) = 5, and n(A ∩ B) = 3. Answer: 14 66) If S and T are sets such that n(S ∪ T) = 13, n(S) = 7, and n(S ∩ T) = 3, find n(T). Answer: 9 67) Find n(S ∪ T), given that n( S) = 350, n(Τ) = 400, and n(S ∩ T) = 175. Answer: 575

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the inclusion-exclusion principle to answer the question. 68) If n(A) = 5, n(B) = 11, and n(A ∩ B) = 3; what is n(A ∪ B)? A) 14 B) 16 C) 12 D) 13 Answer: D 69) If n(A) = 16, n(B) = 45, and n(A ∪ B) = 53; what is n(A ∩ B)? A) 4 B) 24 C) 10 D) 8 Answer: D 70) If n(B) = 60, n(A ∩ B) = 11, and n(A ∪ B) = 105; what is n(A)? A) 56 B) 58 C) 54 D) 45 Answer: A 71) If n(A) = 15, n(A ∪ B) = 43, and n(A ∩ B) = 11; what is n(B)? A) 40 B) 39 C) 28 D) 38 Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 72) In a survey of 1000 people, it was found that 720 watched television news programs daily and 434 read a newspaper daily. How many of those surveyed used both media to receive news? Answer: 154 73) At a certain university, 540 students are math and computer science majors. If 300 are majoring in math and 120 are majoring in both, how many are majoring in computer science? Answer: 360 74) At a certain business, 300 employees can type or use the required software. If 155 employees can type and 175 can use the software, how many can both type and use the software? Answer: 30 75) Of the 80 applicants for a job that requires fluency in French or Spanish, 40 indicated that they are fluent in French, and 50 indicated that they are fluent in Spanish. How many are fluent in both? Answer: 10

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 76) Results of a survey of fifty students indicate that 30 like red jelly beans, 29 like green jelly beans, and 17 like both red and green jelly beans. How many of the students surveyed like no green jelly beans? A) 17 B) 21 C) 38 D) 30 Answer: B 77) Mrs. Bollo's second grade class of thirty students conducted a pet ownership survey. Results of the survey indicate that 8 students own a cat, 15 students own a dog, and 5 students own both a cat and a dog. How many of the students surveyed own a cat or a dog? A) 13 B) 15 C) 5 D) 18 Answer: D 78) A local television station sends out questionnaires to determine if viewers would rather see a documentary, an interview show, or reruns of a game show. There were 600 responses with the following results: 180 were interested in an interview show and a documentary, but not reruns. 24 were interested in an interview show and reruns but not a documentary 84 were interested in reruns but not an interview show. 144 were interested in an interview show but not a documentary. 60 were interested in a documentary and reruns. 36 were interested in an interview show and reruns. 48 were interested in none of the three. How many are interested in exactly one kind of show? A) 268 B) 288 C) 278 D) 298 Answer: B

79) In which Venn diagram does the shaded portion represent S ∪ T? A)

Answer: B 80) In which Venn diagram does the shaded portion represent A′ ∩ B? A)

Answer: D

81) In which Venn diagram does the shaded portion represent (A ∪ B′)′? A)

Answer: D 82) In which Venn diagram does the shaded portion represent R′ ∩ (S ∪ T)? A)

Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 83) Draw a two-circle Venn diagram and shade the portion corresponding to the set B′. Answer:

84) Draw a two-circle Venn diagram and shade the portion corresponding to the set A′ ∪ B. Answer:

85) Draw a two-circle Venn diagram and shade the portion corresponding to the set (A ∪ B) ∩ B′. Answer:

86) Draw a two-circle Venn diagram and shade the portion corresponding to the set (A ∪ B)′. Answer:

87) Draw a three-circle Venn diagram and shade the portion corresponding to the set (A ∪ B) ∩ C′. Answer:

88) Draw a three-circle Venn diagram and shade the portion corresponding to the set A ∪ (B ∪ C′). Answer:

89) Draw a three-circle Venn diagram and shade the portion corresponding to the set (A ∪ B′) ∩ C. Answer:

90) Let A, B, and C be subsets of the universal set U. Draw a three-circle Venn diagram and shade the portion corresponding to the set (A ∪ B ∪ C)′. Answer:

91) Give a set-theoretic expression that describes the shaded portion of the Venn diagram shown below.

Answer: (A ∪ B) ∩ C′ 92) Give a set-theoretic expression that describes the shaded portion of the Venn diagram shown below.

Answer: (A ∪ B) ∩ C 93) Give a set-theoretic expression that describes the shaded portion of the Venn diagram shown below.

Answer: R′ ∩ (S ∪ T) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. By drawing a Venn diagram, simplify the expression to involve at most one union and the complement symbol applied only to R, S, and T. 94) (S ∩ T') ∪ (R ∩ S' ∩ T') ∪ (T ∩ S ∩ R) A) T ∪ (S ∩ R') B) T' ∪ (S ∩ R) C) R' ∪ (S ∩ T) D) (T' ∩ R) ∪ (S ∩ R) Answer: B 95) (S ∩ R) ∪ (S ∩ T) ∪ (S ∩ T' ∩ R') ∪ (T ∩ R') A) S' ∪ (R ∩ T) B) S ∪ (R' ∩ T) C) R' ∪ (T ∩ S') D) (R' ∩ S) ∪ (T' ∩ S) Answer: B Solve the problem. 96) Assume that the universal set U is the set of all students in a math class, who recently took a test. All students received an A, B, or C. Let A be the set of students who got an A, B the set of students who scored a B, and C the set of students who scored a C. Let D be the set of men, and E the set of women. Let F be the set of students who are business majors. Describe in words the following set. E ∩ A ∩ F' A) Women who got an A and are not business majors. B) Men who got an A and are business majors. C) Students who are women, or got an A, or are not business majors. D) Women who got an A and are business majors. Answer: A

97) Assume that the universal set U is the set of all students in a math class, who recently took a test. All students received an A, B, or C. Let A be the set of students who got an A, B the set of students who scored a B, and C the set of students who scored a C. Let D be the set of men, and E the set of women. Let F be the set of students who are English majors. Describe in words the following set. (D ∩ F) ∪ C' A) Male English majors or students who did not get a C. B) Men who did not get a C or English majors who did not get a C. C) Male English majors who did not score a C. D) Male English majors who scored a C. Answer: A 98) Simplify the expression (S′ ∩ T)′ using De Morgan's Laws. A) S ∩ T′ B) S ∪ T′ C) S′ ∩ T D) S′ ∪ T E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 99) Simplify the expression (A′ ∪ B′)′ using De Morgan's Laws. Answer: A ∩ B 100) Simplify the expression (S ∪ T′)′ using De Morgan's Laws. Answer: S′ ∩ T 101) Simplify the expression (A′ ∩ B)′ using De Morgan's Laws. Answer: A ∪ B′. 102) Simplify the expression S ∪ (S′ ∩ T′)′ using De Morgan's Laws. Answer: S ∪ T MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 103) Are (R ∪ S′) ∩ T and ((R′ ∩ S) ∪ T)′ equal? A) Yes B) No Answer: B 104) Are (R ∪ S′) ∩ T′ and ((R′ ∩ S) ∪ T)′ equal? A) Yes B) No Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 105) State De Morgan's Laws. Answer: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 106) Suppose n(U) = 100, n(A) = 24, n(B) = 60, and n(A ∩ B) = 10. Find n(A ∩ B′). A) 40 B) 100 C) 14 D) 26 E) none of these Answer: C 107) Suppose n(U) = 100, n(A) = 24, n(B) = 60, and n(A ∩ B) = 10. Find n(A′ ∩ B′). A) 90 B) 26 C) 100 D) 6 E) none of these Answer: B 108) Consider the Venn diagram below. Find n((A ∩ B) ∪ C).

A) 8 B) 13 C) 31 D) 21 E) none of these Answer: C 109) Consider the Venn diagram below. Find n(A′ ∩ B ∩ C).

A) 21 B) 2 C) 5 D) 8 E) none of these Answer: B

110) Consider the Venn diagram below. Find n(A ∪ B)′.

A) 28 B) 40 C) 50 D) 39 E) none of these Answer: A 111) Consider the Venn diagram below. Find n(A ∩ B)′.

A) 23 B) 39 C) 40 D) 43 E) none of these Answer: D A total of 37,451 Ph.D.'s were earned in 1991. Out of the 1164 Ph.D.'s in business and management, 292 were earned by women. Women earned 13,782 Ph.D.'s. 112) How many men earned Ph.D's in business and management? A) 13,490 B) 23,669 C) 872 D) The number cannot be determined. Answer: C 113) How many men earned Ph.D's in fields other than business and management? A) 23,669 B) 22,797 C) 13,490 D) The number cannot be determined. Answer: B Suppose n(U) = 200, n(A) = 80, n(B) = 65, n(C) = 90, n(A ∩ B) = 48, n(A ∩ C) = 62, n(B ∩ C) = 45, and n(A′ ∩ B′ ∩ C′) = 80. 114) Find n(A ∩ B ∩ C). A) 120 B) 20 C) 40 D) impossible to determine Answer: B

115) Find n(A ∪ B ∪ C). A) 80 B) 345 C) 120 D) impossible to determine Answer: C A survey of 100 bank customers revealed that 58 of them have a savings account, 63 of them have a checking account, 22 of them have a savings account and a loan, 16 of them have a checking account and a loan, 27 of them have only a checking account, 12 have a savings account and a checking account and a loan. Assume that every customer has at least one of the services. 116) How many customers have a loan? A) 50 B) 11 C) 37 D) 38 E) none of these Answer: C Out of 30 job applicants, 11 are female, 17 are college graduates, 7 are bilingual, 3 are female college graduates, 2 are bilingual women, 6 are bilingual college graduates, and 2 are bilingual female college graduates. 117) How many college graduates are male? A) 14 B) 18 C) 4 D) 10 E) none of these Answer: A 118) How many female college graduates are not bilingual? A) 3 B) 19 C) 1 D) 13 E) none of these Answer: C 119) How many women who are not college graduates are bilingual? A) 2 B) 9 C) 0 D) 11 E) none of these Answer: C

120) How many applicants are either female or bilingual? A) 15 B) 18 C) 2 D) 16 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Let n(A ∪ B) = 50, n(Β) = 20, n(Α) = 45, and U = A ∪ B. 121) Find n(A ∩ B). Answer: 15 122) Find n(A ∩ B′). Answer: 30 123) Find n(A′ ∪ B′). Answer: 35 124) Let n(U) = 100, n(A) = 24, n(A ∪ B) = 74, and n(A ∩ B) = 10. (a) Determine n(B). (b) Determine n(B′). (c) Determine n(A′ ∪ B′). Answer: (a) 60 (b) 40 (c) 90 125) Let n(U) = 100, n(A) = 20, n(A ∪ B) = 70, and n(A ∩ B) = 5. (a) Draw a Venn diagram displaying the given sets and the number of elements in each basic region. (b) Determine n(B). (c) Determine n(A′ ∪ B′). Answer: (a)

(b) 55 (c) 95 126) Let A and B be subsets of the universal set U. Draw an appropriate Venn diagram and use the following data to determine the number of elements in each basic region. n(U) = 90, n(A′) = 42, n(A′ ∩ B) = 30, n(A ∩ B) = 10 Answer:

127) Let n(U) = 53, n(A) = 20, n(B) = 15, n(C) = 27, n(Α ∩ B) = 10 , n(Α ∩ C) = 7, n(B ∩ C) = 8, and n(Α ∩ B ∩ C) = 6. (a) Draw a Venn diagram displaying the given data and the number of elements in each basic region. (b) Determine n(A ∩ B ∩ C′). (c) Determine n((A ∪ B ∪ C)′). Answer: (a)

(b) 4 (c) 10 128) Let n(U) = 100, n(A) = 48, n(B) = 47, n(C) = 32, n(Α ∩ B) = 22, n(Α ∩ C) = 10, n(B ∩ C) = 14, and n(Α ∩ B ∩ C) = 4. (a) Draw a Venn diagram displaying the given data and the number of elements in each basic region. (b) Determine n(A ∩ B ∩ C′). (c) Determine n((A ∪ B)′). Answer: (a)

(b) 18 (c) 27 129) Let A, B, and C be subsets of the universal set U. Draw a Venn diagram and use the data given below to label the number of elements in each basic region. n(A′) = 29, n(A ∪ B) = 28, n(B) = 16, n(C) = 20, n(A ∩ B) = 8, n(B ∩ C) = 9. Answer:

130) Let R, S, and T be subsets of the universal set U. Draw a Venn diagram and use the data given below to label the number of elements in each basic region. n(U) = 68, n(R ∪ S ∪ T) = 56, n(R ∩ T′) = 30, n(R ∩ S) = 12, n(R ∩ T) = 5, n(S ∩ T′) = 15, n(S′ ∩ T) = 9, and n(R ∩ S ∩ T) = 4. Answer:

131) Draw a Venn diagram and label each basic region with the number of elements in it if n(U) = 30, n(Α) = 10, n(B) = 7, n(C) = 13, n(A ∩ B) = 3, n(B ∩ C) = 1, n(Α ∩ C) = 4, and n(Α ∩ B ∩ C) = 1. Answer:

132) Draw a Venn diagram based on the information below and label each basic region with the number of elements in it. n(U) = 50, n(A) = 25, n(B) = 11, n(C) = 27, n(A ∩ B) = 8, n(A ∩ C) = 12, n(B ∩ C) = 6, and n(A ∩ B ∩ C) = 5. Answer:

A survey of 100 bank customers revealed that 58 of them have a savings account, 63 of them have a checking account, 22 of them have a savings account and a loan, 16 of them have a checking account and a loan, 27 of them have only a checking account, 12 have a savings account and a checking account and a loan. Assume that every customer has at least one of the services. 133) Determine the number of customers who have checking or savings accounts but no loans. Answer: 63 134) Determine the number of customers who have a loan but do not have a savings account. Answer: 15 135) Determine the number of customers who have a checking account but no savings account. Answer: 31 A survey was made to determine the popularity of hockey, baseball and football. Of those surveyed, 35% watched hockey, 58% watched baseball, and 47% watched football. 15% watched hockey and baseball, 20% watched baseball and football, and 22% watched hockey and football. 7% watched all three sports. 136) What percentage of those surveyed only watched baseball? Answer: 30% 137) What percentage watched hockey and football, but not baseball? Answer: 15% 138) What percentage watched none of the three sports? Answer: 10%

Solve the problem. 139) A certain jewelry store has 1000 customers, each of whom buys gold or diamonds. Suppose that 645 buy gold and 240 buy gold and diamonds. (a) How many customers buy only diamonds? (b) How many buy something other than gold or diamonds? Answer: (a) 355 (b) 0 140) Of 95 students, 62 are foreign, and of those 62 foreign students, 23 are female. If 45 of the students are male, how many female students are not foreign? Answer: 27 141) In 1993-1994, a total of 1059 doctoral degrees were awarded in the mathematical sciences. Out of the 831 men who received such degrees, 345 were United States citizens. The number of women recipients who were not United States citizens was 104. How many women among the United States citizens received doctoral degrees? Answer: 124 142) A certain clinic has 400 patients, each of whom is being treated for heart disease or diabetes. Suppose that 300 are treated for heart disease and 75 are treated for heart disease and diabetes. (a) How many patients are treated for diabetes? (b) How many patients are treated only for heart disease? Answer: (a) 100 (b) 225 143) Suppose that 500 patrons at a restaurant ordered either steak or french fries. If 200 ordered steak and 300 ordered french fries, how many ordered steak and french fries? Answer: 0 144) One hundred male college students were surveyed. Twenty of the students were members of a fraternity, 5 were honors students, 77 were neither members of a fraternity nor honors students, and 18 were members of a fraternity but not honors students. (a) Draw a Venn diagram displaying the given data and the number of elements in each basic region. (b) How many honors students do not belong to a fraternity? (c) How many fraternity members are honors students? Answer: (a)

(b) 3 (c) 2

145) A grand jury questioned 70 persons. Forty were lawyers, 23 were women, and 21 were under indictment. There were 12 female lawyers, 5 indicted women, and 14 indicted lawyers. Two of the indicted lawyers were women. (a) Draw a Venn diagram displaying the given data and the number of elements in each basic region. (b) How many of the lawyers were not under indictment? (c) How many of the indicted women were not lawyers? Answer: (a)

(b) 26 (c) 3 146) Blood samples of 100 people were tested. The A, B and Rh antigens were found in the blood of 32, 33, and 74 people, respectively. None of the antigens was found in 13 samples. Ten samples contained the A and Rh antigens only, 9 contained the B and Rh antigens only, 6 contained the B antigen only and 3 contained the A and B antigens only. (a) Draw a Venn diagram displaying the given data and the number of elements on each basic region. (b) How many samples contained the A and B antigens? (c) How many samples do not contain the Rh antigen? Answer: (a)

(b) 18 (c) 26 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a Venn diagram to answer the question. 147) At East Zone University (EZU) there are 681 students taking College Algebra or Calculus. 205 are taking College Algebra, 537 are taking Calculus, and 61 are taking both College Algebra and Calculus. How many are taking Algebra but not Calculus? A) 144 B) 83 C) 620 D) 476 Answer: A 148) At East Zone University (EZU) there are 708 students taking College Algebra or Calculus. 401 are taking College Algebra, 380 are taking Calculus, and 73 are taking both College Algebra and Calculus. How many are taking Calculus but not Algebra? A) 255 B) 307 C) 328 D) 635 Answer: B 22

149) A local television station sends out questionnaires to determine if viewers would rather see a documentary, an interview show, or reruns of a game show. There were 800 responses with the following results: 240 were interested in an interview show and a documentary, but not reruns; 32 were interested in an interview show and reruns, but not a documentary; 112 were interested in reruns but not an interview show; 192 were interested in an interview show but not a documentary; 80 were interested in a documentary and reruns; 48 were interested in an interview show and reruns; 64 were interested in none of the three. How many are interested in exactly one kind of show? A) 374 B) 394 C) 364 D) 384 Answer: D 150) A survey of 100 families showed that 35 had a dog; 28 had a cat; 10 had a dog and a cat; 42 had neither a cat nor a dog, and in addition did not have a parakeet; 0 had a cat, a dog, and a parakeet. How many had a parakeet only? A) 10 B) 15 C) 20 D) 5 Answer: D 151) A survey of a group of 114 tourists was taken in St. Louis. The survey showed the following: 63 of the tourists plan to visit Gateway Arch; 48 plan to visit the zoo; 11 plan to visit the Art Museum and the zoo, but not the Gateway Arch; 12 plan to visit the Art Museum and the Gateway Arch, but not the zoo; 17 plan to visit the Gateway Arch and the zoo, but not the Art Museum; 9 plan to visit the Art Museum, the zoo, and the Gateway Arch; 16 plan to visit none of the three places. How many plan to visit the Art Museum only? A) 13 B) 62 C) 98 D) 37 Answer: A

152) A survey of 149 college students was done to find out what elective courses they were taking. Let A = the set of those taking art; B = the set of those taking basket weaving; and C = the set of those taking canoeing. The study revealed the following information: n(A) = 45; n(B) = 55; n(C) = 40; n(A ∩ B) = 12; n(A ∩ C) = 15; n(B ∩ C) = 23; n(A ∩ B ∩ C) = 2. How many students were not taking any of these electives? A) 67 B) 57 C) 59 D) 10 Answer: B Use a Venn Diagram and the given information to determine the number of elements in the indicated region. 153) n(U) = 60, n(A) = 35, n(B) = 16, and n(A ∩ B) = 9. Find n(A ∪ B)'. A) 18 B) 51 C) 9 D) 42 Answer: A 154) n(A) = 33, n(B) = 24, n(A ∩ B) = 4, n(A' ∩ B') = 7. Find n(U) A) 57 B) 60 C) 53 D) 64 Answer: B 155) n(U) = 147, n(A) = 48, n(B) = 68, n(A ∩ B) = 19, n(A ∩ C) = 22, n(A ∩ B ∩ C) = 10, n(A' ∩ B ∩ C') = 39, and n(A' ∩ B' ∩ C') = 36. Find n(C). A) 26 B) 14 C) 30 D) 46 Answer: D 156) n(A) = 85, n(B) = 93, n(C) = 87, n(A ∩ B) = 17, n(A ∩ C) = 19, n(B ∩ C) = 13, n(A ∩ B ∩ C) = 11, and n(A' ∩ B' ∩ C') = 171. Find n(U) A) 398 B) 408 C) 227 D) 312 Answer: A 157) n(A ∪ B ∪ C) = 85, n(A ∩ B ∩ C) = 12, n(A ∩ B) = 26, n(A ∩ C) = 23, n(B ∩ C) = 21, n(A) = 61, n(B) = 42, and n(C) = 40. Find n(A' ∩ B ∩ C) A) 9 B) 8 C) 11 D) 10 Answer: A 24

158) n(U) = 72, n(A) = 34, n(B) = 23, n(C) = 33, n(A ∩ B) = 7, n(A ∩ C) = 8, n(B ∩ C) = 8, and n(A ∩ (B ∩ C)) = 5. Find n(A ∩ (B ∪ C)'). A) 24 B) 5 C) 28 D) 4 Answer: A 159) n(U) = 76, n(A) = 34, n(B) = 25, n(C) = 34, n(A ∩ B) = 7, n(A ∩ C) = 7, n(B ∩ C) = 7, and n(A ∩ (B ∩ C)) = 3. Find n(((A ∪ B) ∪ C)'). A) 3 B) 23 C) 24 D) 1 Answer: D 160) n(U) = 83, n(A) = 24, n(B) = 34, n(C) = 36, n(A ∩ B) = 4, n(A ∩ C) = 6, n(B ∩ C) = 4. Find n(A ∩ (B ∩ C)). A) 2 B) 3 C) 19 D) 17 Answer: B Solve the problem. 161) How many five digit license plates can be formed allowing repetition of letters and numbers including 0? A) 36 ∙ 35 ∙ 34 B) 265 C) 10 ∙ 26 D) 365 E) none of these Answer: D 162) How many four-digit codes can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if the last digit cannot be 0 or 1 or 2 and repetition of digits is allowed? A) 7200 B) 720 C) 700 D) 7000 E) none of these Answer: D 163) How many three-letter words can be formed allowing repetition of letters? A) 26 ∙ 25 ∙ 24 B) 3 26 C) 263 D) 3 ∙ 26 E) none of these Answer: C

164) Toss a coin 7 times. What is the number of sequences of heads or tails? A) 7 2 B) 48 C) 2 ∙ 7 D) 2 7 E) none of these Answer: D 165) How many three-digit codes can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if the last digit cannot be 0 or 1 and repetition of digits is allowed? A) 512 B) 504 C) 800 D) 720 E) none of these Answer: C 166) An exam contains 5 multiple-choice questions, each having 4 possible answers. In how many different ways can the exam be completed? A) C(5, 4) B) 5 4 C) 5 ∙ 4 ∙ 3 ∙ 2 D) 4 5 E) none of these Answer: D 167) Ten cars can park in ten parking spaces. In how many ways can they park if only one car is allowed per space? A) 1010 B) 2 10 C) 10 D) 10 ∙ 9∙ 8 ∙ 7∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 E) none of these Answer: D 168) Eight horses are entered in a race. In how many ways can they cross the finish line if ties are not allowed? A) 2 8 B) 8 ∙ 7∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 C) 8 D) 8 8 E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 169) How many different outfits can be selected from 3 suits, 5 shirts, and 2 hats? Answer: 30 170) How many different outfits can be selected from 5 suits, 7 shirts, 4 ties and 6 hats? Answer: 840

171) How many 4-letter words can be made from the letters of "MISSISSIPPI" if the words must begin with an S and letters cannot be repeated? Answer: 6 172) A communication signal can be transmitted from station A to one of four switching offices, after which it is sent to one of 10 possible repeaters and delivered to station B. How many different ways are there to send the signal from A to B? Answer: 40 173) How many four-digit sequences can be formed allowing no repetition of digits? Answer: 10 ∙ 9 ∙ 8 ∙ 7 = 5040 174) How many six-digit sequences can be formed allowing no repetition of digits? Answer: 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5= 151,200 175) Assuming that you can get a customized license plate where there it must consist of a distinct letter followed by any combination of four letters or digits, how many different custom license plates can be printed? Answer: 26 ∙ 36 ∙ 36 ∙ 36 ∙ 36 = 43,670,016 176) In how many ways can 100 runners win first, second, and third place? Answer: 100 ∙ 99 ∙ 98 = 970,200 177) Area codes in the United States are three-digit numbers where the first digit is not a 0 or a 1. Assuming that the middle digit can be any number, how many codes are possible? (Some of these are not used.) Answer: 8 ∙ 10 ∙ 10 = 800 178) A license plate contains three distinct letters followed by three digits. How many different license plates can be printed? Answer: 26 ∙ 25 ∙ 24 ∙ 10 ∙ 10 ∙ 10 = 15,600,000 179) Area codes in the United States are three-digit numbers where the first digit is not a 0 or a 1, and the middle digit is either 0 or 1. How many codes are possible? (Some of these are not used.) Answer: 8 ∙ 2 ∙ 10 = 160 180) In a small computer network, the address of a given station is an eight-digit word where each digit is either 0 or 1. How many stations with different addresses can be accommodated? Answer: 2 8 = 256 181) Toss a coin four times. How many possible sequences of heads or tails are there? Answer: 16 182) How many six-letter words can be formed from the letters in "STABLE" if the first two letters must be vowels and repetition is allowed? Answer: 2 2 ∙ 6 4 = 5184 183) Consider an alphabet with 30 distinct letters. How many three-letter words can be formed allowing repetition of letters? Answer: 27,000

184) An exam consists of four multiple-choice questions; each question has four choices. Assuming all the questions are answered, in how many ways can the test be completed? Answer: 256 185) How many five-digit codes are there with no repeated digits? Answer: 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 = 30,240 186) A mathematics professor wishes to select from among 15 different textbooks, each having four different possible computer software packages, with two versions each to accommodate two computer brands available. How many different options does the professor have? Answer: 120 187) A college student wants to select one foreign language course from among four possible courses, one math course from among seven possible courses, and one economics course from among five possible courses. In how many different ways can she select the three courses? Answer: 140 188) An organization can elect a president and a different secretary in 210 different ways. How many members does the organization have? Answer: 15 189) An organization can elect a president and a different vice-president in 350 different ways. How many members does the organization have? Answer: 26 190) The letters of the word "MATH" are arranged in a random order. (a) How many different arrangements are possible? (b) How many start with a vowel? (c) How many start with "A" and end with "H"? Answer: (a) 4! = 24 (b) 6 (c) 2 191) The letters of the word "PENCIL" are arranged in a random order. (a) How many different arrangements are possible? (b) How many start with a vowel? (c) How many start with "E" and end with "C"? Answer: (a) 6! = 720 (b) 240 (c) 24 192) How many four-letter words can be made from the letters of "MISSISSIPPI," if (a) letters can be repeated? (b) letters cannot be repeated? Answer: (a) 4 4 = 256 (b) 4! = 24

193) An experiment consists of four trials of tossing two indistinguishable coins and recording the combination of heads and tails. (a) How many outcomes are possible per trial? (b) How many complete experiments are possible? Answer: (a) 3 (b) 81 194) You are to create a secret four-digit code for your ATM card using the digits 0 to 9. (a) How many different codes can be made? (b) If you are only allowed to use a digit once, how many different codes can be made? Answer: (a) 104 = 10,000 (b) 10 ∙ 9 ∙ 8 ∙ 7 = 5040 195) You are to create a 5 digit password for your e-mail account using the digits 0 to 9 and the letters A though E. (a) How many different codes can be made? (b) If you are only allowed to use a digit or letter once, how many different codes can be made? Answer: (a) 155 = 759,375 (b) 15 ∙ 14 ∙ 13 ∙ 12∙ 11= 360,360 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 196) Suppose there are 8 roads connecting town A to town B and 4 roads connecting town B to town C. In how many ways can a person travel from A to C via B? A) 64 ways B) 12 ways C) 16 ways D) 32 ways Answer: D 197) A shirt company has 4 designs that can be made with short or long sleeves. There are 5 color patterns available. How many different types of shirts are available from this company? A) 11 types B) 20 types C) 9 types D) 40 types Answer: D 198) A person ordering a certain model of car can choose any of 7 colors, either manual or automatic transmission, and any of 4 audio systems. How many ways are there to order this model of car? A) 66 ways B) 56 ways C) 64 ways D) 52 ways Answer: B

199) At a lumber company that sold shelves, a customer could choose from 5 types of wood, 2 different widths and 4 different lengths. How many different types of shelves could be ordered? A) 28 types B) 40 types C) 50 types D) 11 types Answer: B 200) A restaurant offers 4 possible appetizers, 12 possible main courses, and 4 possible desserts. How many different meals are possible at this restaurant? (Two meals are considered different unless all three courses are the same). A) 182 meals B) 64 meals C) 20 meals D) 192 meals Answer: D 201) A restaurant offered salads with 7 types of dressings and one choice of 2 different toppings. How many different types of salads could be offered? A) 9 types B) 49 types C) 4 types D) 14 types Answer: D Evaluate. 202) 4! A) 48 B) 6 C) 24 D) 12 Answer: C 203)

18! 17! A) 9 B) 36 C) 18 D) 17 Answer: C

204)

7! 5! A) 7 B) 42 7 C) 5 D) 2! Answer: B

Decide whether the situation involves permutations or combinations. 205) An arrangement of 4 people for a picture. A) Permutation B) Combination Answer: A 206) A committee of 2 delegates chosen from a class of 28 students to bring a petition to the administration. A) Permutation B) Combination Answer: B 207) A selection of a chairman and a secretary from a committee of 19 people. A) Permutation B) Combination Answer: A 208) A sample of 5 items taken from 63 items on an assembly line. A) Permutation B) Combination Answer: B 209) A blend of 2 spices taken from 8 spices on a spice rack. A) Permutation B) Combination Answer: B Evaluate the permutation. 210) P(8, 5) A) 8 B) 336 C) 1 D) 6720 Answer: D 211) P(6, 0) A) 1440 B) 0.5 C) 720 D) 1 Answer: D 212) P(4, 4) A) 3 B) 0 C) 24 D) 1 Answer: C

213) P(n, n - 1) A) n - 1 B) n! C) 1 D) n Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 214) Calculate P(6, 4). Answer: 360 215) Calculate P(3, 3). Answer: 6 Solve the problem. 216) How many different signals may be formed from five different colored flags if a signal consists of three different colored flags arranged in a particular order? Answer: P(5, 3) = 5 ∙ 4 ∙ 3 = 60 217) How many different lines can 6 people form? Answer: P(6, 6) = 6! = 720 218) How many ways can 7 of 10 books be arranged on a shelf? Answer: P(10, 7) = 10 ∙ 9 ∙ 8 ∙ 7∙ 6 ∙ 5 ∙ 4 = 604,800 219) How many permutations are there of the letters in the word "CATS" taken three at a time. Answer: 24 220) List all the permutations of the letters in the word "CATS" taken two at a time. Answer: CA AC TC SC CT AT TA SA CS AS TS ST MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 221) How many different three-digit numbers can be written using digits from the set {3, 4, 5, 6, 7} without any repeating digits? A) 120 B) 10 C) 60 D) 20 Answer: C 222) How many different three-number "combinations" are possible on a combination lock having 24 numbers on its dial? Assume that no numbers repeat. (Combination locks are really permutation locks.) A) 255,020 B) 12,144 C) 1,530,100 D) 510,040 Answer: B 32

223) There are 12 horses in a race. In how many ways can the first three positions of the order of the finish occur? (Assume there are no ties.) A) 1324 B) 1320 C) 220 D) 218 Answer: B 224) A license plate is to consist of 2 letters followed by 4 digits. Determine the number of different license plates possible if repetition of letters and numbers is not permitted. A) 3,275,980 B) 3,276,000 C) 3,276,140 D) 6,760,000 Answer: B 225) A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 9 different flags available, how many possible signals can be flown? A) 27 B) 84 C) 729 D) 504 Answer: D Evaluate the combination. 226) C(13, 6) A) 617,760 B) 8,648,640 C) 1716 D) 10,080 Answer: C 227) C(5, 0) A) 60 B) 1 C) 30 D) 120 Answer: B 228) C(23, 1) A) 23! - 10 B) 23 C) 23! D) 2 Answer: B

229) C(24, 24) A) 2 B) 24! - 5 C) 24! D) 1 Answer: D 230) C(s, 10) A) s! s! B) 10! C)

s! 10!(s - 10)!

s! (s - 10)!

Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 231) Calculate C(4, 4) + C(4, 3) + C(4, 2) + C(4, 1) + 1. Answer: 1 + 4 + 6 + 4 + 1 = 16 Solve the problem. 232) How many different selections of three newspapers can be made from a set of 10 newspapers? Answer: C(10, 3) = 120 233) How many two-letter combinations can be made from the letters in the set {a, b, c , d}? List all of them. Answer: C(4, 2) = 6 {a, b}, {a, c }, {a, d}, {b, c }, { b, d}, {c, d} 234) A pizza parlor offers five toppings on its pizza. How many different pizzas are there with three or four toppings? Answer: C(5, 4) + C(5, 3) = 5 + 10 = 15 235) A pizza parlor offers five toppings on its pizza. How many different pizzas are there with two, three or four toppings? Answer: C(5, 4) + C(5, 3) + C(5, 2) = 5 + 10 + 10 = 25 236) A computer word consists of eight digits. In how may different ways can it have a two-digit error or a four-digit error? Answer: C(8, 2) + C(8, 4) = 28 + 70 = 98 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 237) How many ways can a committee of 3 be selected from a club with 12 members? A) 110 ways B) 220 ways C) 6 ways D) 1320 ways Answer: B

238) In how many ways can a student select 6 out of 10 questions to work on an exam? A) 24 ways B) 1,000,000 ways C) 5040 ways D) 210 ways Answer: D 239) How many ways can a committee of 6 be selected from a club with 10 members? A) 210 ways B) 1,000,000 ways C) 60 ways D) 151,200 ways Answer: A 240) If the police have 7 suspects, how many different ways can they select 5 for a lineup? A) 42 ways B) 2520 ways C) 21 ways D) 35 ways Answer: C 241) In how many ways can a group of 9 students be selected from 10 students? A) 10 ways B) 90 ways C) 1 way D) 9 ways Answer: A 242) A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking, how many ways can you get 4 apples? A) 15 ways B) 5 ways C) 10 ways D) 8 ways Answer: B 243) A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking, how many ways can you get 4 oranges? A) 5 ways B) 15 ways C) 8 ways D) 0 ways Answer: D 244) A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking, how many ways can you get exactly 3 apples? A) 20 ways B) 180 ways C) 30 ways D) 60 ways Answer: C 35

245) A coin is tossed seven times and the sequence of heads and tails is observed. What is the number of different possible outcomes? A) 14 B) 21 C) 49 D) 128 E) none of these Answer: D 246) A coin is tossed seven times and the sequence of heads and tails is observed. What is the number of different outcomes having exactly three heads? A) 35 B) 210 C) 10 D) 21 E) none of these Answer: A 247) A coin is tossed five times and the sequence of heads and tails is observed. What is the number of different outcomes having more heads than tails? A) 21 B) 12 C) 10 D) 3 E) none of these Answer: E There are 30 electric calculators, four of which need to be recharged. An instructor takes 25 of the 30 calculators to his statistics class. 248) How many samples of 25 calculators contain all four calculators that need to be recharged? A) 30 25 B) 26 21 C) 25 4 D) 30 4 E) none of these Answer: B

249) How many samples of 25 calculators contain at least one calculator that needs to be recharged? A) 4 ∙ C(29, 21) B) C(30, 25) - C(26, 25) C) C(30, 25) - C(26, 4) D) C(25, 1) ∙ C(30, 24) + C(25, 2) ∙ C(30, 23) + C(25, 3) ∙ C(30, 22) + C(25, 4) ∙ C(30, 21) E) none of these Answer: B Solve the problem. 250) How many poker hands consist of four clubs and a card of a different suit? A) 39 ∙ C(13, 4) B) 48 ∙ C(13, 4) C) 48 ∙ C(52, 4) D) 39 ∙ C(52, 4) E) none of these Answer: A 251) How many poker hands consist of three clubs and two cards of a different suit? A) 39 ∙ 38∙ C(52, 3) B) 48 ∙ 47∙ C(13, 3) C) 39 ∙ 38∙ C(13, 3) D) 48 ∙ 47∙ C(52, 3) E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 252) In how many ways can a committee of 12 senators be selected from the 100 members of the United States Senate so that no two committee members are from the same state? (Do not evaluate.) Answer: C(50, 12) ∙ 2 12 253) How many advisory committees can be formed with four professors and two students from a department with 10 professors and 50 students? Answer: C(10, 4) ∙ C(50, 2) = 257,250 254) In how many ways can you choose two groups of three people from among eight people? Answer: C(8, 3) ∙ C(5, 3) = 560 255) In how many ways can you choose two groups of four people from among eight people? Answer: C(8, 4) ∙ C(5, 4) = 350 256) How many ways can 100 balls be put into three boxes with 50 balls going into one box and 25 into each of the other two? (Do not evaluate.) Answer: C(100, 50) ∙ C(50, 25) 257) In how many ways can a jury of 12 people be chosen from an available pool of fourteen women and six men if the jury must consist of eight women and four men? Answer: C(14, 8) ∙ C(6, 4) = 45,045

258) Consider the partial map of the streets in a certain city shown below. What is the total number of routes (with no backtracking) from A to B

Answer: C(8, 3) = 56 259) In the street map shown below, how many of the routes from A to B pass through C (with no backtracking)?

Answer: 30 260) In the street map shown below, how many routes are there from A to B (with no backtracking)?

Answer: 35 261) A coin is tossed eight times and the sequence of heads and tails is observed. (a) How many different outcomes are possible? (b) How many different outcomes have exactly two tails? (c) How many different outcomes have at least two tails? Answer: (a) 2 8 = 256 (b) C(8, 2) = 28 (c) 256 - 9 = 247 38

262) A coin is tossed ten times and the sequence of heads and tails is observed. (a) How many different outcomes are possible? (b) How many different outcomes have exactly three tails? (c) How many different outcomes have at least one tail? Answer: (a) 2 10 = 1024 (b) C(10, 3) = 120 (c) 1024 - 1 = 1023 263) There are 25 quarts of milk on a supermarket shelf, four of which are spoiled. A customer buys three quarts of milk. (a) How many samples are possible? (b) How many samples contain exactly two quarts of spoiled milk? (c) How many samples contain at least two quarts of spoiled milk Answer: (a) C(25, 3) = 2300 (b) C(4, 2) ∙ C(21, 1) = 126 (c) C(4, 2) ∙ C(21, 1) + C(4, 3) ∙ C(21, 0) = 130 264) In how many ways can a committee of seven people be chosen from 15 married couples if: (a) two particular couples must be on the committee? (b) no married couples may be on the committee? Answer: (a) C(26, 3) = 2600 (b) C(15, 7)∙ 2 7 = 823,680 265) An urn contains four red balls and six white balls. A sample of four balls is selected. (a) How many samples are possible? (b) How many samples contains exactly three white balls? (c) How many samples contain four white balls? Answer: (a) C(10, 4) = 210 (b) C(6, 3) ∙ C(4, 1) = 80 (c) C(6, 4) = 15 266) Suppose an experiment consists of tossing a coin seven times and recording the sequences of heads and tails. (a) How many outcomes are possible? (b) How many outcomes have exactly three heads? (c) How many outcomes have at least three heads? Answer: (a) 2 7 = 128 (b) C(7, 3) = 35 (c) 128 - C(7, 0) - C(7, 1) - C(7, 2) = C(7, 3) + C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7) = 99 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 267) A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking, how many ways can you get 4 oranges? A) 8 B) 0 C) 5 D) 15 Answer: B

268) A bag contains 9 apples and 7 oranges. If you select 8 pieces of fruit without looking, how many ways can you get exactly 7 apples? A) 72 B) 3528 C) 252 D) 504 Answer: C 269) How many different three-digit numbers can be written using digits from the set {4, 5, 6, 7, 8} without any repeating digits? A) 10 B) 60 C) 120 D) 20 Answer: B 270) How many 5-card poker hands consisting of 3 aces and 2 kings are possible with an ordinary 52-card deck? A) 12 B) 6 C) 288 D) 24 Answer: D 271) Bob is planning to pack 6 shirts and 4 pairs of pants for a trip. If he has 13 shirts and 9 pairs of pants to choose from, in how many different ways can this be done? A) 216,274 B) 215,976 C) 216,216 D) 216,306 Answer: C 272) Evaluate 8 . 8 A) 0 B) 1 C) 2 D) 40,320 E) none of these Answer: B 273) Evaluate 9 . 1 9 A) 8 B) 362,880 C) 40,320 D) 9 E) none of these Answer: D

274) Evaluate 7 . 6 A) 7 B) 720 C) 5040 7 D) 6 E) none of these Answer: A 275) Evaluate 10 . 2 A) 5 B) 22 C) 1,814,400 D) 45 E) none of these Answer: D 276) Evaluate 12 . 0 A) 120 B) 0 C) 12 D) 1 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 277) Evaluate 9 . 4 Answer: 126 278) Evaluate 15 . 13 Answer: 105 279) Evaluate 10 . 0 Answer: 1 280) Evaluate 10 . 2 Answer: 45 281) Evaluate 1000 . 998 1000 ∙ 999 999,000 Answer: 1000 = 1000 = = = 499,500 2∙1 2 998 2

282) Evaluate 1000 . 999 1000 Answer: 1000 = 1000 = = 1000 1 999 1 283) Find the coefficient of x4 y5 in the binomial expansion of (x + y)9 . Answer: 9 = 126 5 284) Find the coefficient of a 6 b4 in (a + b)10. Answer: 10 = 210 4 285) Find the coefficient of a 7 b3 in (a + b)10. Answer: 10 = 120 3 286) Calculate: 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 0 1 2 3 4 5 6 7 8 Answer: 256 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 287) What is the fourth term in the binomial expansion of (a + b)12 ? A) 12a 4 b8 B) 495a 4b4 C) a 4b8 D) 12a 4 b4 E) none of these Answer: E 288) What is the fourth term in the binomial expansion of (a + b)10 ? A) 10a 3 b7 B) 210a 3b7 C) 120a 7b3 D) 10a 7 b3 E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 289) Determine the first three terms in the binomial expansion of (x + y)12. Answer: x12, 12x11y, 66x10y2 290) Determine the last three terms in the binomial expansion of (x + y)12. Answer: 66x2 y10, 12xy11, y12

291) Determine the middle term in the binomial expansion of (x + y)14. Answer: 3432x7y7 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A pizza parlor offers onions, green peppers, pepperoni, olives and sausage as topping for the plain cheese base. 292) How many different types of pizza can be made? A) 5 B) 32 C) 64 D) 120 E) none of these Answer: B 293) How many different types of pizza can be made without meat? A) 60 B) 16 C) 6 D) 8 E) none of these Answer: D 294) How many subsets does the set B = {1, 3, 7} have? A) 3 2 B) 7 C) 2 ∙ 3 D) 2 3 E) none of these Answer: D 295) How many subsets does the set B = {1, 3, 7,9} have? A) 4 ∙ 3 ∙ 2 B) 2 4 C) 3 4 D) 3 2 E) none of these Answer: B 296) How many subsets does {a, e, i, o, u, y} have? A) 6 2 B)

6 2

C) 2 6 D) 6! E) none of these Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 297) How many subsets of the set {a, b, c, d, e, f} do not contain the letter "d"? Answer: 32 298) How many subsets of the set {a, b, c, d, e, f, g} do not contain the letter "d"? Answer: 64 299) How many subsets of the set {3, 4, 6, 8, 9} do not contain a perfect square? Answer: 8 300) Consider the alphabet as a set of 26 letters. How many subsets of the alphabet are there? Answer: 2 26 = 67,108,864 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 301) In how many ways can a hand of five cards be dealt from an ordinary deck of 52 cards? A) 47! 52! B) 5! C)

52! 5! 47!

D) 52 ∙ 51 ∙ 50 ∙ 49 ∙ 48 E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 302) In how many ways can a selection of at least one card be made from a hand of five cards? Answer: 2 5 - 1 = 31 303) A population of crabs that eats algae lives in a bay. In the bay there are five kinds of algae. A biologist wants to find out which types of algae are eaten by the crabs. If the biologist examines the stomach contents of the crabs, how many possibilities are there for the kinds of algae he will find? Answer: 2 5 = 32

304) Prove that for n a positive integer, we have n + n + ... + n = 2 n 0 1 n Answer: Write down the binomial expansion for (1 + 1)n . n 1 ∙ 1n - 1 + n 1n (1 + 1)n = n 1 n + n 1 n - 1 ∙ 1 + n 1 n - 2 ∙ 1 2 + ... + 0 1 2 n-1 n n 1n + n 1n (1 + 1)n = n 1 n + n 1 n + n 1 n + ... + 0 1 2 n-1 n n + n (1 + 1)n = n + n + n + ... + 0 1 2 n-1 n n + n 2 n = n + n + n + ... + 0 1 2 n-1 n MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 305) Jose is applying to college. He receives information on 6 different colleges. He will apply to all of those he likes. He may like none of them, all of them, or any combination of them. How many possibilities are there for the set of colleges that he applies to? A) 60 B) 16 C) 64 D) 6 Answer: C 306) Anna goes to a frozen yogurt shop. She can choose from any of the following toppings: cashews, marshmallow cream, peanut butter chips, blackberries, and brownie bits. How many different variations of yogurt and toppings can be made? A) 32 B) 6 C) 64 D) 16 Answer: A 307) Joe goes to a mexican restaurant and order nachos. He can have just cheese or add any of the following: chicken, onion, sour cream or olives. How many different variations of nachos are possible? A) 32 B) 8 C) 16 D) 64 Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 308) Compute Answer:

8 . 4, 2, 2 8! = 420 4! 2! 2!

309) How many partitions of S = {a, b, c, d} are of the type (1, 1, 2)? Answer: 12 310) Let S be a set with 6 elements. Compute the number of ordered partitions of type: (1, 2, 3). Answer: 60 311) Let S be a set with 6 elements. Compute the number of ordered partitions of type: (2, 2, 2). Answer: 90 4 2, 2 (b) List all ordered partitions of {a, b, c, d} of type (2, 2).

312) (a) Compute

Answer: (a) 6 (b) ({a, b}, {c, d}), ({c, d} , {a, b}), ({a, c}, {b, d}), ({b, d}, {a, c}), ({a, d}, {b, c}), ({b, c}, {a, d}) 4 1, 1, 2 (b) List all ordered partitions of {a, b, c, d} of type (1, 1, 2).

313) (a) Compute

Answer: (a) 12 (b) ({a}, {b}, {c, d}), ({a}, {c}, {b, d}), ({a}, {d}, {b, c}) ({b}, {c}, {a, d}), ({b}, {d}, {a,c}), ({b}, {a}, {c,d}) ({c}, {a}, {b, d}), ({c}, {b}, {a,d}), ({c}, {d}, {a,b}) ({d}, {a}, {b, c}), ({d}, {b}, {a,c}), ({d}, {c}, {a,b}) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 314) A project manager oversees 12 employees. How many ways can the employees be divided into four groups of three? A) 720 B) 12 C) 1540 D) 15,400 E) none of these Answer: D 315) A mathematics instructor divides her 24 students into groups of four people to work on a project. In how many ways can the students be divided into these groups if each group works on a different project? 24! A) (6!)4 B)

1 24! ∙ 4! (6!)4

1 24! ∙ 6! (4!) 6

24! (4!)6

E) none of these Answer: D

316) The human resources director of a small company interviewed nine people for an entry-level position. Two were rated as highly qualified, five were rated as qualified, and two were rated as unqualified. In how many different ways could the ratings be assigned? A) 1680 B) 126 C) 280 D) 756 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 317) Ten sweepstakes prizes are awarded among ten winners: one $10,000 grand prize, two $5000 second prizes and seven $1000 third prizes. In how many ways can this be done? Answer:

10 = 360 1, 2, 7

318) A television executive wants to schedule eight commercials during four half-hours programs. In how many ways can this be done if two commercials are scheduled during each program and different programs reach different numbers of people? Answer:

8 = 2520 2, 2, 2, 2

319) A student is enrolled in six three-credit courses. She wants to earn 2 A's, 2 B's, and 2 C's. In how many ways can this be accomplished? Answer: 90 320) Try to calculate

20 . What is incorrect about this expression? 4, 5, 6, 7

Answer: 4 + 5 + 6 + 7 > 20 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 321) What is the number of unordered partitions of S = {a, b, c, d, e, f} of type (3, 3)? A) 45 B) 20 C) 90 D) 10 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 322) Let S be a set with 6 elements. Determine the number of unordered partitions of S of type: (3, 3). Answer: 10 323) Let S be a set with 6 elements. Determine the number of unordered partitions of S of type: (2, 2, 2). Answer: 15

324) List (a) All ordered partitions of {a, b, c} of type (2, 1) (b) All unordered partitions of {a, b, c, d, e, f} of type (2, 2, 2) Answer: (a) ({a, b}, {c}), ({a, c}, {b}), ({b, c}, {a}) (b) ({a, b}, {c, d}, {e, f}), ({a, c} ,{b, d}, {e, f}), ({a, b}, {c, e}, {b, f}), ({a, b}, {c, f}, {d, e}), ({a, c}, {b, e}, {d, f}), ({a, c}, {b, f}, {d, e}), ({a, d}, {b, c}, {e, f}), ({a, d}, {b, e},{c, f}), ({a, d}), ({b, f}, {c, e}), ({a, e}, {b, c}, {d, f}), ({a, e}, {b, d}, {c, f}), ({a, e}, {b, f}, {c, d}), ({a, f}, {b, c}, {d, e}), ({a, f}, {b, d}, {c, e}), ({a, f}, {b, e}, {c, d}) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 325) A mathematics instructor divides her 24 students into groups of four people to work on a project. If each group works on the same project, compute the number of ways in which the class can be divided. 24! A) (4!)6 B)

24! (6!)4

1 24! ∙ 4! (6!) 4

1 24! ∙ 6! (4!)6

E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 326) A baker places 60 doughnuts into five boxes of a dozen. In how many ways can this be done? (Do not evaluate.) Answer:

1 60! ∙ 5! (12!)5

327) In how many ways can five pairs of socks be chosen from a collection of ten socks? Answer:

1 10! ∙ = 945 5! (2!)5

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 328) Which of the following is true? A) The number of ordered partitions of s = {x, y, z} of type (1, 2) is the same as the number of permutations of {x, y, z}. B) ({1, 2}, {3}) and ({2, 1}, {3}) are different ordered partitions. C) ({1, 2}, {3}) and ({1, 3}, {2}) are different ordered partitions. D) If S = {x, y, z, w, u, v} then there are more unordered partitions than ordered partitions of S of type (2, 2, 2). E) none of these Answer: C

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The number of events associated with a sample space having n outcomes is A) 2 n . B) n 2 . C) n!. D) n. E) none of these Answer: A A light bulb manufacturer tests a light bulb by letting it burn until it burns out. The experiment consists of observing how long (in hours) the light bulb burns. Let E be the event "the bulb lasts less than 100 hours," F be the event " the bulb lasts less than 50 hours," and G be the event "the bulb lasts more than 120 hours." 2) State the event E ∪ F. A) "The bulb lasts less than 50 hours." B) "The bulb lasts more than 50 hours." C) "The bulb lasts less than 100 hours." D) "The bulb lasts between 50 and 100 hours." E) none of these Answer: C 3) State the event F′ ∩ G′. A) "The bulb lasts between 50 and 100 hours." B) "The bulb lasts 50 hours or more." C) "The bulb lasts between 120 hours or less." D) "The bulb lasts between 50 and 120 hours inclusive." E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 4) An experiment consists of sending 50 electronic mail messages from computer A to computer B and observing the number of messages that arrive with at least one error. Describe the sample space of this experiment. Answer: S = {0, 1, 2, ..., 50} 5) An experiment consists of rolling two dice and observing the sum of the dots on their uppermost faces. What is the sample space for this experiment? Answer: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} 6) A quality control process consists of selecting two transistors at random, testing them, and recording whether each is defective (D) or nondefective (N). What is the sample space for this experiment? Answer: {(D, D), (D, N), (N, D), (N, N)}

7) The Power and Light Company records the number of kilowatt-hours of electricity used by a consumer. Let E be the event "less than 500 kilowatt-hours," F be the event "more than 1000 kilowatt-hours," and G be the event "less than 700 kilowatt-hours." Describe the events: (a) E ∩ F′ (b) E ∪ G (c) F ∩ G Answer: (a) less than 500 kilowatt-hours (b) less than 700 kilowatt-hours (c) an impossible event MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 8) There are 3 balls in a hat; one with the number 3 on it, one with the number 5 on it, and one with the number 7 on it. You pick a ball from the hat at random and then you flip a coin. List the elements that make up the sample space. A) {3 5 7 H, 3 5 7 T} B) {3 H, 5 H, 7 H} C) {3 H T, 5 H T, 7 H T} D) {3 H, 3 T, 5 H, 5 T, 7 H, 7 T} Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 9) There are five finalists, A, B, C, D, E, in a lottery drawing. Three of them are to be selected to win $10,000 prizes. (a) What is the sample space for this experiment? (b) Describe the event "C and E win $10,000 prizes" as a subset of the sample space. (c) Describe the event "neither A nor B wins a $10,000 prize". Answer: (a) {(A, B, C), (A, B, D), (A, B, E), (A, C, D), (A, C, E), (A, D, E), (B, C, D), (B, C, E), (B, D, E), (C, D, E)} (b) {(A, C, E), (B, C, E), (C, D, E)} (c) {(C, D, E)} 10) An experiment consists of arranging a white ball (W), a black ball (B), and a red ball (R) in a row. (a) What is the sample space S for this experiment? (b) Describe the event E = "the white ball is in the middle," as a subset of the sample space. (c) Describe the event F = "the red ball is next to the black ball," as a subset of the sample space. (d) Describe the event E ∪ F. Answer: (a) {(W, B, R), (W, R, B), (B, W, R), (B, R, W), (R, B, W), (R, W, B)} (b) {(B, W, R), (R, W, B)} (c) {(W, B, R), (W, R, B), (B, R, W), (R, B, W)} (d) {(B, W, R), (R, W, B), (W, B, R), (W, R, B), (B, R, W), (R, B, W)} = S 11) An experiment consists of tossing a coin three times and recording the sequence of heads and tails. (a) What is the sample space? (b) Determine the event E = "More heads than tails occur." (c) Determine the event F = "The number of heads equals the number of tails." Answer: (a) {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (b) {HHH, HHT, HTH, THH} (c) ∅

12) An experiment consists of tossing a coin twelve times and observing the number of heads. (a) Determine the sample space for this experiment. (b) Determine the event E = "At least 7 tails are observed." Answer: (a) S = {0, 1, 2, ..., 12} (b) E = {0, 1, 2, 3, 4, 5} 13) An experiment consists of rolling a die and then tossing a coin and recording the outcomes. (a) What is the sample space for this experiment? (b) Describe the event "a number less than 4 on the die and tail on the coin" as a subset of the sample space. Answer: (a) {(1, H), (1, T), (2, H), (2, T), ..., (6, H), (6, T)} (b) {(1, T), (2, T), (3, T)} 14) A letter is selected at random from the word "TEXTBOOK." (a) What is the sample space for this experiment? (b) Describe the event "the letter chosen is a vowel" as a subset of the sample space. Answer: (a) {T, E, X, B, O, K} (b) {E, O} MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A die is rolled twice. Write the indicated event in set notation. 15) The first roll is a 1. A) {(1, 1), (1, 3), (1, 5)} B) {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)} C) {(1, 1), (1, 2), (1, 4), (1, 5), (1, 6)} D) {(1, 3)} Answer: B 16) The second roll is a 6. A) {(3, 6)} B) {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6)} C) {(1, 6), (3, 6), (5, 6)} D) {(1, 6), (2, 6), (4, 6), (5, 6), (6, 6)} Answer: B 17) The first roll is a 3 and so is the second. A) {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)} B) (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3)} C) {(3, 3), (1, 3), (3, 1), (3, 5), (5, 3)} D) {(3, 3)} Answer: D 18) The sum of the rolls is 6, and one roll is a 2. A) {(2, 4)} B) {(2, 4), (4, 2)} C) {(4, 2)} D) ∅ Answer: B

19) The sum of the rolls is either 6 or 7, and one roll is a 2. A) {(2, 4), (4, 2)} B) {(2, 4), (2, 5), (4, 2), (5, 2)} C) {(2, 4), (2, 5)} D) {(4, 2), (5, 2)} Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 20) Given any two events, E and F, are E ∩ F and E′ ∪ F′ mutually exclusive? Answer: yes 21) Let S = {a, b, c, d, e} be a sample space, E = {a, b, e}, and F = {b, c}. (a) Determine the events E ∪ F and E′ ∩ F′. (b) Are E ∪ F and E′ ∩ F′ mutually exclusive? Answer: (a) E ∪ F = {a, b, c, e}, E′ ∩ F′ = {d}. (b) yes 22) Let S = {1, 2, 3, 4, 5, 6} be the sample space of a certain experiment, E = {2, 4, 5}. What is the largest event F in S such that E and F are mutually exclusive? Answer: F = E′ = {1, 3, 6} MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 23) Which of the following events are mutually exclusive? A) being a college student and being a high school graduate B) being a mother and being an uncle C) living in Baltimore and working in Washington, D.C. D) being a steelworker and being a stamp collector E) none of these Answer: B 24) Two events A and B are mutually exclusive if A) A ∩ B = ∅. B) A ∪ B = ∅. C) A ∪ B = U. D) A ∩ B = U. E) none of these Answer: A Classify the type of probability as logical, empirical, or judgmental. 25) Pr(Kevin rolls four 6's when he rolls a die 8 times) A) Logical B) Empirical C) Judgmental Answer: A

26) Pr(Angela's favorite team will win the football game next week by 6 points) A) Logical B) Judgmental C) Empirical Answer: B 27) Pr(a certain battery in an appliance will last for less than 1400 hours) A) Judgmental B) Logical C) Empirical Answer: C 28) Which of the following is a valid probability distribution for a sample space S = {a, b, c, d}? A) Pr(a) = 0.3, Pr(b) = 0.1, Pr(c) = 0.2, Pr(d) = 0.5 B) Pr(a) = -0.2, Pr(b) = 0.5, Pr(c) = 0.4, Pr(d) = 0.3 C) Pr(a) = 0.5, Pr(b) = 0.2, Pr(c) = 0.1, Pr(d) = 0.3 D) Pr(a) = 0.6, Pr(b) = 0, Pr(c) = 0.3, Pr(d) = 0.1 E) none of these Answer: D 29) Two fair die are rolled. What is the probability that the numbers that appear add to 4? 1 A) 5 B)

1 6

1 12

1 36

E) none of these Answer: C 30) Two fair die are rolled. What is the probability that the numbers that appear are both 3? 1 A) 36 B)

1 3

1 6

1 2

E) none of these Answer: A

31) What is the probability of getting either a black card or an ace in one draw from an ordinary deck of 52 cards? 26 A) 52 B)

28 52

30 52

29 52

E) none of these Answer: B 32) A fair coin is tossed six times. What is the probability of obtaining no heads? 1 A) 64 B) 0 3 C) 32 D)

1 32

E) none of these Answer: A 33) A fair coin is tossed six times. What is the probability of obtaining at most five heads? 5 A) 64 B)

1 64

59 64

63 64

E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. An experiment with outcomes s1, s2 , s3 , and s4 is described by the probability table below. Outcome Probability s1 0.5 s2

0.1

0.2

34) What is Pr({s2 , s4})? Answer: 0.3

35) What is Pr({ s1 , s3, s4 })? Answer: 0.9 An experiment consists of tossing a coin three times and observing the sequence of heads and tails. Each of the eight outcomes has the same probability of occurrence. 36) Compute the probability that "T T T" is the outcome. Answer:

1 = 0.125 8

37) Compute the probability the number of heads is larger than the number of tails. Answer:

1 = 0.5 2

An experiment consists of selecting a letter at random from the letters of the word "TEXTBOOK." 38) Compute the probability that the letter selected is an "X." Answer:

1 = 0.125 8

39) Compute the probability that the letter selected is a "T." Answer:

1 = 0.25 4

40) Compute the probability that the letter selected is a vowel. Answer:

3 = 0.375 8

An analyst develops a prediction of the return on a $1000 investment after one year. Note that a return of less than $1000 is a loss and a return of more than $1000 is a profit. Return Probability Less than $1000 0.10 $1000 0.10 More than $1000, less than $1100 0.35 At least $1100, less than $1500 0.25 At least $1500, less than $2000 0.15 More than $2000 0.05 41) What is the probability that the investment gives a profit? Answer: 0.8 42) What is the probability that the investment at least doubles? Answer: 0.05 43) What is the probability that the investment gives a profit, and the total return is less than $1500? Answer: 0.60

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the probability of the given event. 44) Two fair dice are rolled. The sum of the numbers on the dice is 4. 1 A) 12 B)

11 12

C) 3 2 D) 3 Answer: A 45) Two fair dice are rolled. The sum of the numbers on the dice is 6 or 10. 4 A) 3 B)

4 9

2 9

1 60

Answer: C 46) Two fair dice are rolled. The sum of the numbers on the dice is greater than 10. A) 3 1 B) 12 C)

1 18

5 18

Answer: B 47) When a single card is drawn from a well-shuffled 52-card deck, find the probability of getting a red 5 or a red 2. 1 A) 13 B)

1 4

1 26

1 52

Answer: A

48) A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing an ace or a 9? 2 A) 13 B)

13 2

C) 10 5 D) 13 Answer: A 49) A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a face card or a 5? 2 A) 13 B)

48 52

C) 16 4 D) 13 Answer: D Solve the problem. 50) A survey of 7000 students on a college campus asked where the student's hometown is, relative to the school. The results are shown in the table below. Consider the experiment of selecting a student at random from the group surveyed and observing his or her answer. Determine the probability distribution for this experiment. Hometown location Number of students Same state 2743 Different state 3101 Different country 1156 A) Hometown location Probability Same state 0.608 Different state 0.557 Different country 0.835 B) Hometown location Probability Same state 0.392 Different state 0.443 Different country 0.165 C) Hometown location Probability Same state 39.2 Different state 44.3 Different country 16.5 D) Hometown location Probability Same state 0.608 Different state 0.443 Different country 0.165 Answer: B

51) Let E and F are mutually exclusive and Pr(E) = 0.2 and Pr(F) = 0.6. Find Pr(E ∪ F). A) 0.12 B) 0.8 C) 0 D) 1 E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 52) Let E and F be events such that Pr(E) = 0.3, Pr(F) = 0.6, Pr(E ∩ F) = 0.2. Find Pr(E ∪ F). Answer: 0.7 53) Let E and F be events such that Pr(E) = 0.7, Pr(F) = 0.5, Pr(E ∩ F) = 0.3. Find Pr(E ∪ F). Answer: 0.9 An experiment with outcomes s1, s2 , s3 , s4 , and s5 is described by the probability table below. Let E = {s1 , s4 } and F = {s1 , s2, s3 , s5 }. Outcome s1

Probability 0.05

0.10

0.20

0.50

0.15

54) Compute Pr(E). Answer: 0.55 55) Compute Pr(F). Answer: 0.5 56) Compute Pr(E ∩ F). Answer: 0.05 57) Compute Pr(E ∪ F). Answer: 1 58) Compute Pr(E′). Answer: 0.45

In a study of the ages of its employees, over a period of several years a university finds the following: Age (years) Probability 18-30 0.20 18-45 0.65 18-60 0.90 18-80 1.00 59) Find the probability associated with each of the events: 18-30 years, 31-45 years, 46-60 years, and 61-80 years. Answer: Pr(18-30 years) = 0.20 Pr(31-45 years) = 0.45 Pr(46-60 years) = 0.25 Pr(61-80 years) = 0.10 60) Find the probability that an employee selected at random is at least 46 years old. Answer: 0.35 Solve the problem. 61) The probability that a student will pass mathematics is 0.8, that she will pass physics is 0.65 and that she will pass both courses is 0.6. Find the probability that (a) she will pass at least one of the two courses. (b) she will pass physics but not mathematics. (c) she will fail both courses. Answer: (a) 0.85 (b) 0.05 (c) 0.15 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 62) When two balanced dice are rolled, there are 36 possible outcomes. Find the probability that either doubles are rolled or the sum of the dice is 4. 2 A) 9 B)

1 4

7 36

1 36

Answer: A 63) A lottery game has balls numbered 1 through 15. If a ball is selected at random, what is the probability of obtaining an even numbered ball or the number 12 ball? A) 7 5 B) 4 C)

4 5

7 15

Answer: D 11

64) A spinner has regions numbered 1 through 21. What is the probability that the spinner will stop on an even number or a multiple of 3? 10 A) 9 B)

1 3

C) 17 2 D) 3 Answer: D 65) Of the 47 people who answered "yes" to a question, 15 were male. Of the 63 people who answered "no" to the question, 11 were male. If one person is selected at random from the group, what is the probability that the person answered "yes" or was male? A) 0.319 B) 0.527 C) 0.664 D) 0.236 Answer: B 66) If the odds against an event are 2 to 5, what is the probability that the event will occur? 2 A) 7 B)

2 5

3 7

3 5

E) none of these Answer: E SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 67) If the odds against an event are 2 to 3, what is the probability that the event will occur? Answer:

3 5

68) If the odds against an event are 3 to 5, what is the probability of the event occurring? Answer: 1 -

3 5 = 3+5 8

69) If the odds against an event are a to b, what is the probability of the event occurring? Answer: 1 -

a b = a+b a+b

Solve the problem. 70) What are the odds in favor of Brown winning an election if the probability that he will lose is 0.83? Answer: 17 to 83 12

71) In the game Clue there are six equally likely suspects and six equally likely murder weapons. What are the odds that Professor Plum killed the victim with the rope? Answer: 1 to 35 (or 35 to 1 that he did not kill the victim with the rope) 72) The odds of snow tomorrow is 3 to 7. What is the probability? Answer:

3 = 0.3 = 30% 10

73) The probability of a certain horse winning the race is

1 . What are the odds? 5

Answer: 1 to 4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 74) The odds of a horse winning a race are posted as 6 to 5. Find the probability that the horse will win the race. 5 A) 11 B)

5 6

1 2

6 11

Answer: D 75) The odds of Carl beating his friend in a round of golf are 8 to 3 Find the probability that Carl will beat his friend. 8 A) 11 B)

3 8

2 3

3 11

Answer: A 76) The odds of winning a particular lottery are 1 to 2,300,000. Find the probability of winning. 1 A) 2,300,000 B)

2,300,000 2,300,001

1 2,299,999

1 2,300,001

Answer: D

77) If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is 0.02, what are the odds of an earthquake occurring? A) 49 to 1 B) 1 to 48 C) 1 to 50 D) 1 to 49 Answer: D 78) If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is 0.04, what are the odds of a direct hit? A) 1 to 24 B) 24 to 1 C) 1 to 25 D) 1 to 23 Answer: A 79) Five horses are running at a race track. Being an inexperienced bettor, you assume that every order of finish is equally likely. You bet that Son-of-a-Gun will win and that Gentle Lady will come in second. What is the probability that you will win both bets? 2 A) 5 B)

1 20

9 20

1 25

E) none of these Answer: B 80) What is the probability that a family with six children has exactly two girls? 3 A) 8 B)

1 3

15 64

1 64

E) none of these Answer: C

A student is studying mathematics and chemistry. The probability that he passes mathematics is 0.75, the probability that he fails chemistry is 0.2, and the probability that he passes mathematics but fails chemistry is 0.05. 81) Find the probability that he passes both courses. A) 0 B) 0.70 C) 0.75 D) 0.60 E) none of these Answer: B 82) Find the probability that he either passes mathematics or fails chemistry. A) 0.95 B) 0.15 C) 0.90 D) 1.0 E) none of these Answer: C A basket contains five red balls, four white balls and three blue balls. Two balls are drawn, one after the other, with the first ball replaced before the second is drawn. 83) Find the probability of drawing two white balls. 2 1 A) = 12 6 B)

1 9

4 1 = 12 3

2 9

E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 84) Find the probability of drawing a white ball and a red ball. Answer:

2 ∙ 5 ∙4 5 = 12∙12 18

85) Find the probability of drawing either at least one white ball or exactly one red ball. Answer: Pr(at least one white ball)+Pr(exactly one red ball)-Pr(at least one white ball ∩ exactly one red ball) = 35 5 55 = 72 18 72 The letters of the word "SCRAMBLE" are scrambled and arranged in a random order. 86) In the situation above, how many different arrangements are possible? Answer: 8! = 40,320 87) In the situation above, how many arrangements start with a vowel? Answer: 2 ∙ 7! = 10,080 15

5 + 9

88) In the situation above, what is the probability of an arrangement starting with a vowel? Answer: 0.25 =

1 4

89) In the situation above, what is the probability that the resulting arrangement reads "SCRAMBLE"? Answer:

1 1 ≈ 0.0000248 = 8! 40320

An urn contains six red balls and four green balls. A sample of seven balls is selected at random. 90) Find the probability that five red and two green balls are selected. Answer: 6 4 5 2 3 = 10 10 7 91) Find the probability that all the balls selected are red. Answer: 0 92) Find the probability that at least four red balls are selected. Answer: 6 4 6 4 6 4 4 3 + 5 2 + 6 1 5 = 10 6 7 Solve the problem. 93) An coin is to be tossed 10 times. What is the probability of obtaining 7 heads and 3 tails? Answer: 10 7 15 = 10 128 2 94) A coin is to be tossed 10 times. What is the probability that at least 8 heads appear? Answer: 10 10 10 + + 8 9 10 2 10

7 ≈ 0.055 128

95) If the odds in favor of an event are 4 to 7, what is the probability that the event will NOT occur? Answer:

7 11

96) A coin is to be tossed n times. If the probability of obtaining at least one head is required to be greater than 0.8, compute the smallest value for n. Answer: 3

97) A jury of 10 people is to be selected from an available pool of 12 women and 6 men. What is the probability that the numbers of men and women in the selected jury are equal? Answer: 12 ∙ 6 5 5 24 ≈ 0.109 = 18 221 10 98) Suppose that a pair of dice is tossed and the number on the uppermost faces are observed. (a) What is the probability that the sum is 11? (b) What is the probability that the sum is less than 5? Answer: (a)

1 18

(b)

1 6

99) A pair of dice is tossed as many times as needed until the sum of the dots on the uppermost faces is either 6 or 9. (a) What is the probability of obtaining the 6 before the 9? (b) What is the probability of obtaining the 9 before the 6? Answer: The game ends if the score is 6 or 9. The number of different ways for getting a 6 is 5 and the number of different ways for getting a 9 is 4. 5 (a) 9 (b)

4 9

100) An exam contains six "true or false" questions. What is the probability that a student guessing at the answers will get exactly four correct? Answer:

15 64

101) A salesman visits four cities on his route. If he can visit these cities in a random order, what is the probability that (a) he visits city A before city B? (b) he visits city A first and city B last? Answer: (a)

1 2

(b)

1 12

102) Compute the probability of obtaining three face cards when five cards are dealt from a standard 52-card deck. Answer: 12 ∙ 40 3 2 55 ≈ 0.066 = 52 833 5

103) Four people are asked to pick a month. (a) What is the probability that they pick the same month? (b) What is the probability that they pick four different months? Answer: (a) (b)

1 123

1 1728

11 ∙ 10 ∙ 9 55 ≈ 0.573 = 12 ∙ 12 ∙ 12 96

104) The figure below shows a partial map of the streets in a certain city. A tourist starts at point A and selects at random a path to point B with no backtracking.

What is the probability that he passes through point C? Answer: 3 1

∙

6 3

9 4

10 21

The figure below shows a partial map of the streets in a certain city. A tourist starts at point A and selects at random a path to point B with no backtracking.

105) Based on the figure above, compute the total number of possible routes. Answer: 9 = 126 4

106) Based on the figure above, compute the number of routes that pass through points C and D. Answer: 3 1

2 1

4 = 36 2

107) Based on the figure above, compute the probability that the tourist passes through points C and D. Answer: 3 1

2 1 9 4

4 2

2 7

The figure below shows a sketch of the streets in a portion of a certain city. A tourist starts at point A and selects at random a path to point B with no backtracking.

108) Based on the figure above, compute the probability that the tourist passes through point C. Answer:

10 21

109) Based on the figure above, compute the probability that the tourist passes through point D. Answer:

20 63

110) Based on the figure above, compute the probability that the tourist passes through point C or point D. Answer:

10 20 2 44 + = 21 63 21 63

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the probability. 111) A sample of 4 different calculators is randomly selected from a group containing 13 that are defective and 38 that have no defects. What is the probability that at least one of the calculators is defective? A) 0.295 B) 0.705 C) 0.130 D) 0.692 Answer: B

112) In a batch of 8,000 clock radios 5% are defective. A sample of 12 clock radios is randomly selected without replacement from the 8,000 and tested. The entire batch will be rejected if at least one of those tested is defective. What is the probability that the entire batch will be rejected? A) 0.0833 B) 0.540 C) 0.460 D) 0.0500 Answer: C 113) Let E and F be events such that Pr(F) = 0.4 and Pr(E ∩ F) = 0.3. Compute Pr(E′ ∩ F). A) 0.006 B) 0.10 C) 0.55 D) 0.05 E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 114) Let E and F be events such that Pr(E′) = 0.4, Pr(F) = 0.3, and Pr(E ∪ F) = 0.7. Compute Pr(E ∩ F). Answer: 0.2 Suppose that Pr(E) = 0.3, Pr(F) = 0.5, and Pr(E ∩ F) = 0.2. 115) Calculate Pr(E′). Answer: 0.7 116) Calculate Pr(E′ ∩ F′). Answer: 0.4 117) Calculate Pr(E ∩ F′). Answer: 0.1 Let E and F be events with Pr(E) = 0.4, Pr(F) = 0.6, and Pr(E ∪ F) = 0.8. 118) Find Pr(E′ ∩ F). Answer: 0.4 A basket contains five red balls, four white balls and three blue balls. Two balls are drawn, one after the other, with the first ball replaced before the second is drawn. 119) Find the probability of drawing at most one white ball. Answer: 1-

1 8 = 9 9

120) Find the probability of drawing at least one white ball. Answer: 1 -

8 ∙8 5 = 12∙12 9

Solve the problem. 121) A factory produces screws, which are packaged in boxes of 30. Four screws are selected from each box for inspection. A box fails inspection if two or more of these four screws are defective. What is the probability that a box containing two defective screws will pass inspection? Answer: 2 28 2 2 143 ≈ 0.986 1= 30 145 4 122) What is the probability that in a group of seven people two or more of them have the same birth month? (Assume that each month is equally likely.) Answer: 1 -

12 × 11 × 10 × 9 × 8 × 7 × 6 3071 ≈ 0.889 = 3456 127

123) Find the probability that among five persons, at least two were born on the same weekday. Answer: 1 -

7 × 6 × 5 × 4 × 3 2041 ≈ 0.85 = 2401 75

124) An urn contains three white balls and four red balls. Two balls are chosen at random. What is the probability that at least one of the balls is red? Answer: 3 2 6 1= 7 7 2 125) Five couples attend a group marriage counseling session. What is the probability that at least two couples were married in the same month? Answer: 1 -

12 ∙ 11 ∙ 10 ∙ 9 ∙ 8 89 ≈ 0.618 = 5 144 12

126) In a class of twenty students, what is the likelihood that two or more have the same birthday? (Ignore leap years and assume that each of the 365 days are equally likely.) Answer: Approximately 41.1% MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the complement formula to answer the question. 127) If two fair dice are rolled, what is the probability that a total showing is more than two? 8 A) 9 B) 1 35 C) 36 D)

17 18

Answer: C

128) If nine fair coins are tossed, what is the probability of obtaining at least one head and at least one tail ? A) 1 511 B) 512 C)

255 256

507 512

Answer: C 129) Events E and F are independent and Pr(F) ≠ 0. Which of the following must be true? A) Pr(E ∪ F) = Pr(E) + Pr(F) B) Pr(E|F) = 0 C) Pr(E|F) = Pr(F) D) Pr(E ∩ F) = 0 E) none of these Answer: E 130) Events E and F are independent if A) Pr(E ∩ F) = 0 B) Pr(E ∪ F) = Pr(E) + Pr(F) C) Pr(E ∩ F) = Pr(E)Pr(F) D) Pr(E ∪ F) = 0 E) none of these Answer: C 131) Events E and F are mutually exclusive and Pr(F) ≠ 0. Which of the following must be true? A) Pr(E ∩ F) = Pr(E) ∙ Pr(F) B) Pr(E|F) = Pr(E) C) Pr(E ∪ F) = Pr(E) + Pr(F) D) E and F are independent. E) none of these Answer: C 132) Which of the following is NOT true: Pr(A)*Pr(B|A) A) Pr(A|B)= assuming that both Pr(A) and Pr(B) are greater than 0. Pr(B) 1 1 1 B) It is possible to have Pr(A)= , Pr(B)= and Pr(A ∪ B)= . 4 3 2 C) Pr(A|B)=Pr(B|A) if and only if Pr(A)=Pr(B)>0. D) If A is an event such that Pr(A)=1, then A′=∅. E) none of these Answer: C

133) If E and F are independent and Pr(E) = 0.3 and Pr(F) = 0.6, find Pr(E ∪ F). A) 0 B) 0.18 C) 0.72 D) 0.90 E) none of these Answer: C 134) Suppose that Pr(E) = 0.85, Pr(F) = 0.4, and Pr(E ∩ F) = 0.3. Calculate Pr(F|E). 3 A) 4 B)

3 10

6 11

6 17

E) none of these Answer: D 135) Suppose that Pr(E) = 0.85, Pr(F) = 0.4, and Pr(E ∩ F) = 0.3. Calculate Pr(F|E′). 11 A) 7 B)

1 5

2 3

5 11

E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Suppose that Pr(E) = 0.3, Pr(F) = 0.5, and Pr(E ∩ F) = 0.2. 136) Calculate Pr(F|E′). Answer:

3 7

137) Calculate Pr(F′|E′). Answer:

4 7

Let E and F be events with Pr(E) = 0.4, Pr(F) = 0.6, and Pr(E ∪ F) = 0.8. 138) Find Pr(E|F). Answer:

1 3

139) Find Pr(F|E). Answer: 0.5 Let A and B be events such that Pr(A) = 0.6, Pr(B) = 0.5, and Pr(A|B) = 0.4. 140) Compute Pr(A ∪ B). Answer: 0.9 141) Compute Pr(B|A). Answer:

1 3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The table below gives crime statistics relating to the location of the crime and the type of crime. Robbery Murder Assault Residential 130 40 30 Commercial 102 28 20 142) Find the probability that a randomly-selected crime committed in a residential area is a murder. 1 A) 5 B)

4 35

10 17

34 175

E) none of these Answer: A 143) Find the probability that a randomly-selected crime was committed in a commercial area given that it was an assault. 1 A) 7 B)

2 15

2 5

1 3

E) none of these Answer: C

144) Two cards are drawn (without replacement) from an ordinary deck of 52 cards. Find the probability that the second card is black if the first card is the ace of hearts. 2 A) 51 B)

26 51

1 104

1 2

E) none of these Answer: B 145) A biased coin with Pr(H) = A)

2 3

1 3

3 2

9 16

1 3 and Pr(T) = is thrown twice, find the probability of getting two tails. 4 4

E) none of these Answer: D 146) Two cards are drawn (without replacement) from an ordinary deck of 52 cards. Find the probability that both cards are aces. 2 A) 13 B)

1 221

4 663

1 17

E) none of these Answer: B

The probability that person A will pass Finite Mathematics is

5 6 and the probability that person B will pass is . Assume 8 7

the events are independent. 147) Find the probability that neither will pass. 3 A) 56 B)

53 56

C) 1 29 D) 56 E) none of these Answer: A 148) Find the probability that both will pass. 53 A) 56 B)

30 54

15 28

D) 0 E) none of these Answer: C 149) Find the probability that at least one will pass. 53 A) 56 B) 0 30 C) 56 D)

13 28

E) none of these Answer: A A shipment of twenty radios contains six defective radios. Two radios are randomly selected from the shipment. 150) Find the probability that both radios selected are defective. 15 A) 19 B)

3 10

1 3

14 17

E) none of these Answer: E

151) Find the probability that neither radio selected is defective. 1 A) 7 B)

3 17

91 190

1 2

E) none of these Answer: C 152) A shipment contains 25 defective items and 100 nondefective items. Two items are randomly chosen in succession without replacement. Find the probability that both items are defective. 1 A) 25 B)

6 155

1 16

6 99

E) none of these Answer: B 153) Of the 1000 freshmen enrolled at a certain college, 100 have verbal SAT scores above 650. Thirty of these 100 students earned an A in freshman composition. Find the probability that a freshman has a verbal SAT score above 650 and earned an A in freshman composition. 1 A) 10 B)

3 100

13 100

D) impossible to determine. E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. A coin has Pr(T) =

1 . It is tossed six times in succession. 4

154) Find the probability of getting exactly four tails. Answer:

6 1 5 3 2 135 = 4 4 4 4096

155) Find the probability of getting at least five tails. Answer:

6 1 5 3 1 6 19 + = 5 4 4 4 4096

156) Find the probability of getting at most five tails. Answer: 1-

3 6 3367 = 4 4096

Two cards are drawn in succession (without replacement) from an ordinary deck of 52 cards. 157) Find the probability that the second card is red if the first card is the king of hearts. Answer:

25 51

158) Find the probability that both cards are black. Answer:

1 25 25 ∙ = 2 51 102

The probability that a person passes organic chemistry the first time he enrolls is 0.8. The probability that a person passes organic chemistry the second time he enrolls is 0.9. 159) Find the probability that a person fails the first time but passes the second time. Answer: 0.18 160) Find the probability that a person fails both times. Answer: 0.02 Enrollment statistics at a certain college show that 45% of all students are men, 10% of the student body consists of women majoring in business administration, and 35% of all students major in business administration. A student is selected at random. 161) What is the probability that the selected student majors in business administration if the selected student is a women? Answer:

2 11

162) What is the probability that the selected student is a male business administration major? Answer:

1 4

163) What is the probability that the selected student is a woman if the selected student is a business administration major? Answer:

2 7

Two people X and Y enter a supermarket. The probabilities that person X and Y will make a purchase are 0.3 and 0.4 respectively. Assume that whether each makes a purchase is independent of whether the other makes a purchase. 164) What is the probability that at least one of the two people described above makes a purchase? Answer: 0.58

165) What is the probability that neither of the two people described above makes a purchase? Answer: 0.42 166) What is the probability that exactly one of the two people described above makes a purchase? Answer: 0.46 Solve the problem. 167) There are three children in a family. Are the events "there are more boys than girls" and "the first child is a girl" (a) mutually exclusive? (b) independent? Answer: (a) no (b) no The table below gives the distribution of blood type by sex in a group of 1000 individuals. Blood Type Male Female Total O 80 370 450 A 150 250 400 B 50 50 100 AB 20 30 50 Total 400 600 A person is selected at random from this group. 168) What is the probability that the person selected is male? Answer: 0.4 169) What is the probability that the person selected has blood type B? Answer: 0.1 170) What is the probability that the person selected is a female with blood type O? Answer: 0.37 171) What is the probability that the person selected is female if the person's blood type is O? Answer:

37 45

172) What is the probability that the person selected has blood type AB if the person is male? Answer: 0.05 Solve the problem. 173) An urn contains an equal number of red and blue balls. One third of the red balls have a white dot on them. What is the probability that a randomly selected ball is red with a white dot? Answer:

1 6

174) Suppose that two independent events have probability p each. What is the probability that both events will occur in a single outcome? Answer: p2 = p ∙ p

175) The probabilities that two species will become extinct in five years are 0.3 and 0.2 respectively. Given that these probabilities are independent, what is the probability that at least one group will become extinct in the next five years? Answer: 0.44 176) A fair die is rolled four times. Compute the probability that a 6 appears at least once. Answer: 1 -

5 4 ≈ 0.518 6

177) If heads is five times as likely as tails when a biased coin is flipped, what values should be given to Pr(H) and Pr(T)? Answer: Pr(H)=

5 1 and Pr(T)= 6 6

178) Four radar systems work independently of each other. Each system can detect an approaching airplane with a probability of 0.95. Find the probability that an approaching airplane will escape all four systems. Answer: (0.05) 4 = 0.00000625 The figure below shows a map of the streets in part of a certain city.

A tourist starts at point A and selects at random a path to point B with no backtracking. 179) Compute the probability that the tourist passed through point C given that he passed through point D. Answer: 0.3 180) Compute the probability that the tourist passed through point D given that he passed through point C. Answer:

1 5

Solve the problem. 181) A shipment contains 25 defective and 100 nondefective items. Two items are randomly chosen in succession without replacement and the shipment is rejected if at least one of these is defective. What is the probability that the shipment will be rejected? Answer:

56 155

182) A pair of dice is tossed and the numbers on the uppermost faces are observed. If the sum of the numbers is 8, what is the probability that both of them are even? Answer: 0.6 30

A fair coin is thrown. Let A be the event "at most four heads appear" and B be the event "exactly two heads appear." 183) Find Pr(A|B). Answer: 1 184) Find Pr(B|A).

Answer:

5 2

Pr(B) 10 = = Pr(A) 31 31

Solve the problem. 185) A pair of dice is tossed twice and the numbers on the uppermost faces are observed. What is the probability of getting a sum of 9 in each of the two successive tosses? Answer:

1 81

186) The probability of a rainy day is 0.30, whereas the probability that a rainy day is followed by another rainy day is 0.45. What is the probability of two rainy days following each other? Answer: 0.135 187) A basketball player makes 60% of all foul shots that she tries. What is the probability that, in two foul shots, she makes at least one? Answer: 1 - (0.40) 2 = 0.84 A certain soccer goalkeeper catches 30% of all penalty kicks against her team. 188) What is the probability that out of five penalty kicks she catches two? Answer: C(5, 2) ∙ (0.3)2 ∙ (0.7)3 = 0.3087 189) What is the probability that out of five penalty kicks she catches at least two? Answer: 1 - (0.7)5 - [C(5, 1) ∙ (0.3) ∙ (0.7)4 ] = 0.47178 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the probability. 190) If 81% of scheduled flights actually take place and cancellations are independent events, what is the probability that 3 separate flights will all take place? A) .53 B) .01 C) .81 D) .66 Answer: A 191) A calculator requires a keystroke assembly and a logic circuit. Assume that 97% of the keystroke assemblies and 94% of the logic circuits are satisfactory. Find the probability that a finished calculator will be satisfactory. Assume that defects in keystroke assemblies are independent of defects in logic circuits. A) .8836 B) .9118 C) .9409 D) .9600 Answer: B

Answer the question. 192) A pair of fair dice are rolled. Let E be the event that the sum is less than six. Let F be the event that at least one die shows a two. Are E and F independent events? A) No B) Yes Answer: A 193) A pair of fair dice are rolled. Let E be the event that the total showing is greater than seven. Let F be the event that the total showing is odd. Are E and F independent events? A) Yes B) No Answer: B 194) A pair of fair dice are rolled. Let E be the event that a one shows on the first die. Let F be the event that the total showing is even. Are E and F independent events? A) Yes B) No Answer: A Solve the problem. 195) There are 3 balls in a hat; one with the number 3 on it, one with the number 4 on it, and one with the number 7 on it. You pick a ball from the hat at random and then you flip a coin. Using a tree diagram , obtain the sample space for the experiment. Then, find the probability that the number on the ball is greater than 6. 1 A) 4 B)

2 3

1 3

1 6

Answer: C Fifty percent of students enrolled in an astronomy class have previously taken physics. Thirty percent of these students received an A for the astronomy class, whereas twenty percent of the other students received an A for astronomy. 196) Find the probability that a student selected at random previously took a physics course and did not receive an A in the astronomy course. A) 0.10 B) 0.40 C) 0.35 D) 0.15 E) none of these Answer: C

197) Find the probability that a student selected at random received an A in the astronomy course. A) 0.45 B) 0.10 C) 0.50 D) 0.25 E) none of these Answer: D 198) Find the probability that a student selected at random previously took a physics course, given that they received an A in the astronomy course. A) 0.6 B) 0.15 C) 0.40 D) 0.25 E) none of these Answer: A Suppose that 30% of all small businesses are undercapitalized, 40% of all undercapitalized small business fail, and 20% of all small businesses that are not undercapitalized fail. 199) A small business is chosen at random. Find the probability that the small business succeeds if it is undercapitalized. A) 0.60 B) 0.18 C) 0.80 D) 0.56 E) none of these Answer: A 200) A small business is chosen at random. Find the probability that the small business is not undercapitalized and yet fails. A) 0.40 B) 0.12 C) 0.20 D) 0.14 E) none of these Answer: D 201) A small business is chosen at random. Find the probability that the small business succeeds. A) 0.60 B) 0.88 C) 0.56 D) 0.74 E) none of these Answer: D

202) A small business is chosen at random and is found to have failed. Find the probability that the business was undercapitalized. 6 A) 37 B)

3 10

7 10

6 13

E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 203) A box contains four good light bulbs and three defective ones. Bulbs are selected one at a time (without replacement). Find the probability that the second defective bulb is found on the third selection. Answer:

8 35

204) A pitcher plans to throw a ball to a particular hitter. The probability of the ball being thrown outside the plate is 0.3. The pitcher knows from experience that in this situation the probability of the hitter swinging is 0.8. Given that the hitter swings with a probability of 0.59, what is the probability that the hitter will swing if the pitch is thrown inside the plate? Answer: 0.5 205) Urn I contains three red balls and one white ball. Urn II contains two red and two white balls. An urn is selected at random, and a ball is chosen. If the ball is red, what is the probability that Urn I was chosen? Answer: 0.6 206) An election between two candidates is held in two districts. The first district, which has 60% of the voters, votes 40% for candidate I and 60% for candidate II. The second district, with 40% of the voters, votes 60% for candidate I and 40% for candidate II. Who wins? Answer: Candidate II 207) Every day Aaron, Bebe, and Cindy eat a piece of fruit for lunch. Aaron will eat an apple, banana, peach, or orange with equal likelihood. Bebe will eat an apple, banana, or mango with a 50% likelihood for an apple, 30% for a banana, and 20% for a mango. Cindy will eat either an apple or a banana with equal likelihood. What is the probability that they all eat the same kind of fruit for lunch? Answer: 10% A local store orders lightbulbs from two suppliers, AAA Electronics and ZZZ Electronics. The local store purchases 30% of the bulbs from AAA and 70% of the bulbs from ZZZ. Two percent of the bulbs from AAA are defective while 3% of the bulbs from ZZZ are defective. 208) Find the probability that a bulb was purchased from AAA and is defective. Answer: 0.006 209) Find the probability that a bulb was purchased from ZZZ and is not defective. Answer: 0.679 34

Solve the problem. 210) Of the 1000 freshman enrolled at a certain college, 100 have verbal SAT scores above 650. Thirty of these 100 students earned an A in freshman composition. Out of the remaining 900, 180 earned an A in freshman composition. (a) Draw a tree diagram for this situation and label it with the appropriate probabilities. (b) Compute the probability that a randomly chosen freshman earns an A in freshman composition. Answer: (a)

(b) 0.21 211) Urn I contains two white balls and five red balls; urn II contains six white balls and four red balls. An urn is chosen at random and then a ball is chosen from that urn. (a) Draw a tree diagram for this two-stage experiment, and label it with the appropriate probabilities. (b) What is the probability that the chosen ball is white? (c) What is the probability that the chosen ball came from urn I, if it is white? Answer: (a)

(b)

31 70

(c)

10 31

212) Seventy percent of the students enrolled in a calculus course had previously taken a precalculus course. Twenty percent of the students who took a precalculus course earned an A in the calculus course, whereas 10% of the other students earned an A in the calculus course. (a) Draw a tree diagram summarizing these data and label it with the appropriate probabilities. (b) Find the probability that a student selected at random did not earn an A in calculus if he/she did not take a precalculus course. (c) Find the probability that a student selected at random did not take a precalculus course and earned an A in calculus. (d) Find the probability that a student selected at random earned an A in calculus. (e) Find the probability that a student selected at random previously took a precalculus course, given that he/she earned an A in calculus. Answer: (a)

(b) 0.9 (c) 0.03 (d) 0.17 14 (e) 17

213) Suppose that 5% of all adults over 40 have diabetes. A certain physician correctly diagnoses 90% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 3% of all adults over 40 without diabetes as having the disease. (a) Draw a tree diagram and label it with appropriate probabilities. (b) Find the probability that a randomly-selected adult over 40 does not have diabetes but is diagnosed as having diabetes. (c) Find the probability that a randomly-selected adult over 40 is diagnosed as not having diabetes. (d) Find the probability that a randomly-selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes. Answer: (a)

(b) 0.0285 (c) 0.9265 10 ≈ 0.005397 (d) 1853 214) An auto mechanic uses a two-step procedure to determine the problem with a faulty engine. Step I finds the problem with probability 0.7. Step II (which is executed only if step I fails to locate the problem) locates the problem with probability 0.5. (a) Draw a tree diagram representing the sequence of steps and label it with the appropriate probabilities. (b) What is the probability of this procedure locating the problem in a given faulty engine? Answer: (a)

(b) 0.85

215) Two persons, A and B, are competing in a tournament which ends if three games are played or one of them wins two games in a row. The winner has to win two games. The probability that A wins any game is 0.6. (a) Draw a tree diagram showing all the possibilities for the tournament and label it with appropriate probabilities. (b) What is the probability that three games are played? (c) What is the probability that A wins the tournament? Answer: (a)

(b) 0.48 (c) 0.648 216) An urn contains a white (W), a red (R), and a green (G) ball. A ball is drawn at random from the urn and its color is noted. Without replacing the first ball, a second ball is drawn and its color noted. (a) Draw a tree diagram for this situation and label it with appropriate probabilities. (b) Find the probability that the second ball is green. (c) If the second ball is green, what is the probability that the first was white? Answer: (a)

(b)

1 3

(c)

1 2

217) An urn contains two red (R) balls and six blue (B) balls. Balls are drawn randomly, in succession, and without replacement until a blue ball is drawn. (a) Draw a tree diagram that summarizes all possible selections and label it with appropriate probabilities. (b) Find the probability that the number of balls drawn is two. (c) Find the probability that the number of balls selected is three. Answer: (a)

(b)

3 14

(c)

1 28

Data maintained by a university records the distribution of the student population by college and by the proportion of each college's population who are honors students. Proportion Proportion who College of university are honors students Arts and Sciences 0.35 0.10 Education 0.10 0.08 Engineering 0.30 0.15 Journalism 0.05 0.12 Nursing 0.20 0.16 218) A student is chosen at random from the university. If the student is an engineering student, what is the probability that he/she is an honors student? Answer: 0.15 219) A student is chosen at random from the university. If the student is an honors student, what is the probability that he/she is a nursing student? Answer:

16 ≈ 0.262 61

220) A student is chosen at random from the university. If the student is an honors student, what is the probability that he/she is not a nursing student? Answer:

45 ≈ 0.738 61

In a factory, assembly lines I, II, and III produce 60%, 30%, and 10% of the total output, respectively. One percent of line I's output is defective, 2% of line II's output is defective, and 3% of line III's output is defective. 221) An item is chosen at random. What is the probability that the selected item is defective. Answer: 0.015

222) An item is chosen at random. If the selected item is defective, what is the probability that it was produced by line III? Answer:

1 5

Balls of different colors are placed in two urns as follows: Red Green Blue Urn I 3 4 3 Urn II 5 3 4 223) Based on the data above, what is the probability that the a ball chosen from urn I is green? Answer: 0.4 224) Based on the data above, given that a ball chosen is blue, what is the probability that it came from urn I? Answer:

9 19

In the current first-year class of a community college, all the students come from three local high schools. Schools I, II, and III supply respectively 40%, 50%, and 10% of the students. The failure rate of students is 4%, 2%, and 6%, respectively. 225) What is the probability that a student chosen at random will fail? Answer: 0.032 226) Given that a student fails, what is the probability that he or she came from school I? Answer: 0.5 Three boxesI, II, and IIIcontain three red and two green chips, two red and four green chips, and four red and five green chips, respectively. A box is selected at random and a chip is drawn at random from the box . 227) What is the probability that the chip is green? Answer:

73 135

228) Given the the chip is green, what is the probability that it came from box II? Answer:

30 73

A local store orders lightbulbs from two suppliers, AAA Electronics and ZZZ Electronics. The local store purchases 30% of the bulbs from AAA and 70% of the bulbs from ZZZ. Two percent of the bulbs from AAA are defective while 3% of the bulbs from ZZZ are defective. 229) Find the probability that a bulb is defective. Answer: 0.027 230) Find the probability that a randomly selected defective light bulb was purchased from AAA Electronics. Answer: 0.222 A test for a certain drug produces a false negative 5% of the time and a false positive 8% of the time. Suppose 12% of the employees at a certain company use the drug. 231) If an employee at the company tests positive, what is the probability that he or she does not use the drug? Answer: Approximately 38% 40

232) What is the probability that a nondrug user at the company tests positive twice in a row? Answer: 0.64% 233) What is the probability that a company employee who tests positive twice in a row is not a drug user? Answer: Approximately 4.9% MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the method of natural frequencies to find the indicated probability. 234) Refer to the table which summarizes the results of testing for a certain disease. Positive Test Result Negative Test Result Subject has the disease 98 8 Subject does not have the disease 22 98 A test subject is randomly selected and tested for the disease. What is the probability the subject does not have the disease given that the test result is positive? A) 0.183 B) 0.597 C) 0.817 D) 0.14 Answer: A 235) Refer to the table which summarizes the results of testing for a certain disease. Positive Test Result Negative Test Result Subject has the disease 97 8 Subject does not have the disease 24 318 A test subject is randomly selected and tested for the disease. What is the probability the subject has the disease given that the test result is negative? A) 0.975 B) 0.235 C) 0.076 D) 0.025 Answer: D 236) Use the results summarized in the table.

Republican Democrat Independent

Approve of mayor 8 9 6

Do not approve of mayor 20 18 39

One of the 100 test subjects is selected at random. Given that the person selected approves of the mayor, what is the probability that they vote Democrat? A) 0.333 B) 0.659 C) 0.391 D) 0.22 Answer: C

237) 3.1% of a population are infected with a certain disease. There is a test for the disease, however the test is not completely accurate. 94.7% of those who have the disease test positive. However 3.9% of those who do not have the disease also test positive (false positives). A person is randomly selected and tested for the disease. What is the probability that the person has the disease given that the test result is positive? A) 0.563 B) 0.777 C) 0.031 D) 0.437 Answer: D 238) In the town of Maplewood a certain type of DVD player is sold at just two stores. 44% of the sales are from store A and 56% of the sales are from store B. 2.3% of the DVD players sold at store A are defective while 4.2% of the DVD players sold at store B are defective. If Kate receives one of these DVD players as a gift and finds that it is defective, what is the probability that it came from store A? A) 0.301 B) 0.699 C) 0.023 D) 0.430 Answer: A 239) Some employers use lie detector tests to screen job applicants. Lie detector tests are not completely reliable. Suppose that in a lie detector test, 66% of lies are identified as lies and that 14% of true statements are also identified as lies. A company gives its job applicants a polygraph test, asking "Did you tell the truth on your job application?". Suppose that 94% of the job applicants tell the truth during the polygraph test. What is the probability that a person who fails the test was actually telling the truth? A) 0.301 B) 0.769 C) 0.231 D) 0.14 Answer: B 240) Shameel has a flight to catch on Monday morning. His father will give him a ride to the airport. If it rains, the traffic will be bad and the probability that he will miss his flight is 0.06. If it doesn't rain, the probability that he will miss his flight is 0.01. The probability that it will rain on Monday is 0.16. Suppose that Shameel misses his flight. What is the probability that it was raining? A) 0.16 B) 0.533 C) 0.06 D) 0.467 Answer: B 241) In one town, 10% of 18-29 year olds own a home, as do 35% of 30-50 year olds and 65% of those over 50. According to a recent census taken in the town, 27.3% of adults in the town are 18-29 years old, 36.8% are 30-50 years old, and 35.9% are over 50. If an adult is selected at random from the town and found to be a home-owner, what is the probability that they are aged between 30 and 50? A) 0.368 B) 0.129 C) 0.35 D) 0.331 Answer: D

242) At Sally's Hair Salon there are three hair stylists. 20% of the hair cuts are done by Chris, 35% are done by Karine, and 45% are done by Amy. Chris finds that when he does hair cuts, 6% of the customers are not satisfied. Karine finds that when she does hair cuts, 3% of the customers are not satisfied. Amy finds that when she does hair cuts, 8% of the customers are not satisfied. Suppose that a customer leaving the salon is selected at random. If the customer is not satisfied, what is the probability that their hair was done by Amy? A) 0.615 B) 0.205 C) 0.08 D) 1.6 Answer: A 243) A company manufacturing electronic components for home entertainment systems buys electrical connectors from three suppliers. It buys 23% of the connectors from supplier A, 34% from supplier B, and 43% from supplier C. It finds that 2% of the connectors from supplier A are defective, 3% of the connectors from supplier B are defective, and 5% of the connectors from supplier C are defective. If a customer buys one of these components and finds that the connector is defective, what is the probability that it came from supplier B? A) 0.03 B) 0.127 C) 0.391 D) 0.281 Answer: D 244) Pierre has three coins one of which is biased so that it turns up tails 60% of the time. If he randomly selects one of the coins, tosses it three times, and obtains three tails, what is the probability that this is the biased coin? A) 0.216 B) 0.44 C) 0.464 D) 0.333 Answer: C 245) Refer to the table which summarizes the results of testing for a certain disease. Positive Test Result Negative Test Result Subject has the disease 91 7 Subject does not have the disease 27 91 A test subject is randomly selected and tested for the disease. What is the probability the subject does not have the disease given that the test result is positive? A) 0.25 B) 0.229 C) 0.524 D) 0.771 Answer: B

246) Refer to the table which summarizes the results of testing for a certain disease. Positive Test Result Negative Test Result Subject has the disease 89 7 Subject does not have the disease 24 311 A test subject is randomly selected and tested for the disease. What is the probability the subject has the disease given that the test result is negative? A) 0.073 B) 0.022 C) 0.223 D) 0.978 Answer: B 247) Use the results summarized in the table.

Republican Democrat Independent

Approve of mayor 7 18 10

Do not approve of mayor 15 19 31

One of the 100 test subjects is selected at random. Given that the person selected approves of the mayor, what is the probability that they vote Democrat? A) 0.333 B) 0.514 C) 0.685 D) 0.486 Answer: B 248) 3.7% of a population are infected with a certain disease. There is a test for the disease, however the test is not completely accurate. 90.2% of those who have the disease test positive. However 3.9% of those who do not have the disease also test positive (false positives). A person is randomly selected and tested for the disease. What is the probability that the person has the disease given that the test result is positive? A) 0.037 B) 0.889 C) 0.529 D) 0.471 Answer: D 249) In the town of Maplewood a certain type of DVD player is sold at just two stores. 27% of the sales are from store A and 73% of the sales are from store B. 2.4% of the DVD players sold at store A are defective while 4.2% of the DVD players sold at store B are defective. If Kate receives one of these DVD players as a gift and finds that it is defective, what is the probability that it came from store A? A) 0.211 B) 0.174 C) 0.024 D) 0.826 Answer: B

250) Some employers use lie detector tests to screen job applicants. Lie detector tests are not completely reliable. Suppose that in a lie detector test, 67% of lies are identified as lies and that 13% of true statements are also identified as lies. A company gives its job applicants a polygraph test, asking "Did you tell the truth on your job application?". Suppose that 89% of the job applicants tell the truth during the polygraph test. What is the probability that a person who fails the test was actually telling the truth? A) 0.13 B) 0.389 C) 0.611 D) 0.637 Answer: C 251) Shameel has a flight to catch on Monday morning. His father will give him a ride to the airport. If it rains, the traffic will be bad and the probability that he will miss his flight is 0.06. If it doesn't rain, the probability that he will miss his flight is 0.01. The probability that it will rain on Monday is 0.29. Suppose that Shameel misses his flight. What is the probability that it was raining? A) 0.710 B) 0.06 C) 0.290 D) 0.29 Answer: A 252) At Sally's Hair Salon there are three hair stylists. 22% of the hair cuts are done by Chris, 30% are done by Karine, and 48% are done by Amy. Chris finds that when he does hair cuts, 5% of the customers are not satisfied. Karine finds that when she does hair cuts, 3% of the customers are not satisfied. Amy finds that when she does hair cuts, 7% of the customers are not satisfied. Suppose that a customer leaving the salon is selected at random. If the customer is not satisfied, what is the probability that their hair was done by Amy? A) 1.68 B) 0.205 C) 0.07 D) 0.627 Answer: D 253) A company manufacturing electronic components for home entertainment systems buys electrical connectors from three suppliers. It buys 21% of the connectors from supplier A, 38% from supplier B, and 41% from supplier C. It finds that 1% of the connectors from supplier A are defective, 2% of the connectors from supplier B are defective, and 3% of the connectors from supplier C are defective. If a customer buys one of these components and finds that the connector is defective, what is the probability that it came from supplier B? A) 0.345 B) 0.528 C) 0.02 D) 0.095 Answer: A

254) Simulate 20 rolls of a pair of dice where the sum is observed. Give the relative frequency and corresponding theoretical probability of each of the outcomes. Make a table showing your results. The simulation is shown below. Relative Theoretical randInt (1,6,20) + randInt (1,6,20) Sum of Dice Frequency Probability { 12 6 8 9 7 3 5 7 3 4 2 9 8 7 7 4 5 7 2 9 7} 3 4 5 6 7 8 9 10 11 12 A) Relative Theoretical Sum of Dice Frequency Probability 2 5% 1/20 3 10% 1/10 4 10% 1/10 5 10% 3/20 6 5% 1/5 7 30% 1/4 8 10% 1/5 9 15% 3/20 10 0% 1/10 11 0% 1/10 12 5% 1/20 B) Relative Theoretical Sum of Dice Frequency Probability 2 8.333% 1/36 3 16.666% 1/18 4 16.666% 1/12 5 16.666% 1/9 6 8.333% 5/12 7 50% 1/6 8 16.666% 5/12 9 25% 1/9 10 0% 1/12 11 0% 1/18 12 8.333% 1/36

C) Relative Theoretical Sum of Dice Frequency Probability 2 5% 1/36 3 10% 1/18 4 10% 1/12 5 10% 1/9 6 5% 5/36 7 30% 1/6 8 10% 5/36 9 15% 1/9 10 0% 1/12 11 0% 1/18 12 5% 1/36 D) Relative Theoretical Sum of Dice Frequency Probability 2 8.333% 1/11 3 16.666% 1/11 4 16.666% 1/11 5 16.666% 1/11 6 8.333% 1/11 7 50% 1/11 8 16.666% 1/11 9 25% 1/11 10 0% 1/11 11 0% 1/11 12 8.333% 1/11 Answer: C 255) Simulate 10 free-throws for a basketball player whose free-throw average was 78%. How many of his shots were successful? The simulation is shown below. randInt (1,100,10)→L1 { 57 28 40 48 87 90 29 47 69 28 } SortA (L1) Done L1 { 28 28 29 40 47 48 57 69 87 90 } A) 3 shots B) 8 shots C) 2 shots D) 7 shots Answer: B

256) A student who has not studied for an 8-question multiple-choice test, with 4 choices among the answers (a = 1, b = 2, c = 3, d = 4) for each question, decides to simulate such a test and answer the questions according to a simulation in which each choice of answer has the same probability. Assume the correct answers are a, b, d, c, b, b, a, d. The simulation is shown below. What is the student's score? randInt (1,4,8) {3 2 1 3 2 4 1 4}

A) 62.5% B) 50% C) 37.5% D) 25% Answer: B

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) A study was conducted to determine sources used to find employment. Four hundred subjects were randomly selected and the results are listed below. Job Sources of Survey Respondents Frequency Newspaper want ads 72 Online services 124 Executive search firms 69 Mailings 32 Networking 103 Display the data in a bar chart that shows frequencies on the y-axis. Answer:

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Construct a bar graph of the given frequency distribution. 2) The frequency distribution indicates the height in feet of persons in a group of 108 people. Height (feet) Number of persons 4-5 12 5-6 68 6-7 24 7-8 4

A) 160

B) 160

C) 80

D) 80

Answer: C 3) The frequency distribution indicates the number of fish caught by each fisherman in a group of 250 fishermen. Number of Number of fish caught persons 1 80 2 60 3 50 4 10 5 30 6 20

A) 80

0 B) 40

0 C) 80

D) 80

0 Answer: D Display the data from the table in a bar chart showing percentages on the y-axis. 4) The table shows the number of students enrolled in the College of Business at a local university. Year Number of students Freshman 71 Sophomore 121 Junior 21 Senior 50 A)

Answer: A 5) What is the approximate measure of the angle that represents 40 out of 100 responses in a pie chart? A) 144° B) 60° C) 216° D) 40° Answer: A 6) What is the approximate measure of the angle that represents 50 out of 100 responses in a pie chart? A) 180° B) 144° C) 216° D) 50° Answer: A Construct a pie graph, with sectors given in percent, to represent the data in the given table. 7) Age Number of people 25-35 294 35-45 396 45-55 348 55-65 186 65-75 90

A) 7% 14%

22%

27%

30%

B) 7% 19%

22%

30%

C) 90% 186%

294%

348%

396%

D) 10% 14%

24%

27%

26%

Answer: A 8) Favorite Restaurant Style Number of Responses Chinese 161 Indian 105 Mexican 168 Thai 126

23%

30%

29%

19%

21%

31%

32%

20%

126%

168%

161%

105%

22.5%

28.8%

30%

18.7%

Answer: A 9) Favorite Beverage Number of responses Cola 272 Juice 168 Milk 184 Tea 248 Water 120

120% 272%

248% 168% 184% B)

12% 27%

25% 17% 19% C)

5% 32%

28% 16% 19%

8% 24%

28% 21% 19% Answer: B Use the pie chart the answer the question. 10) Students at a local college were asked to name their favorite sport to watch on television. The results are shown in the pie chart. What is the probability that a randomly selected student answered either baseball or basketball? Other, 5.2% Soccer, 19.4% Baseball, 21.8%

Golf, 6.7%

Basketball, 8.4%

Football, 38.5% A) 0.402 B) 0.603 C) 30.2 D) 0.302 Answer: D 11) A type of visual representation of data that tallies the number of times a response occurs is A) a box plot. B) a histogram. C) a bar chart. D) a frequency table. E) none of these Answer: D 12) The type of visual representation of data that best shows the relative amounts or percentages for different categories is A) a scatter plot. B) a bar chart. C) a frequency table. D) a histogram. E) none of these Answer: B 11

Construct a histogram of the given frequency distribution. 13) The frequency distribution indicates the age of 2190 students in a college chemistry course. Age (years) Number of People 18 425 19 370 20 345 21 295 22 260 23 205 24 180 25 110

A) 500

250

B) 250

125

C) 500

250

D) 500

250

Answer: C

14) The frequency distribution indicates the number of speeding tickets for each person in a group of 456 drivers. Number of Number of Speeding Tickets People 0 168 1 24 2 64 3 96 4 72 5 32

160

320

160

Answer: A 15) The frequency distribution indicates the number of fish caught by each fisherman in a group of 500 fishermen. Number of Number of Fish Caught People 1 160 2 120 3 100 4 20 5 60 6 40

A) 160

B) 160

C) 80

D) 160

Answer: D 16) A five-number summary for a box plot includes which of the following? A) Q1 , Q2 , Q3 , Q4, average B) min, Q1, Q2 , Q3 , max C) Q1 , Q2 , Q3 , Q4, total D) min, median, max, average, frequency total Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 17) Eleven college professors were asked to give the numbers of students in their classes. The sample data follows: 36, 31, 30, 31, 20, 19, 24, 34, 21, 28, 24. Find the five-number summary. Answer: 19, 21, 28, 31, 36 18) Find the five-number summary for the sample data: 10, 11, 13, 14, 16, 17, 19, 20, 21, 23, 24. Answer: 10, 13, 17, 21, 24

19) The box plot for the price (in cents) of a can of chicken noodle soup is shown.

(a) Approximately what percentage of the soups are priced below 30 cents? (b) Approximately what percentage of the soups are priced above 75 cents? (c) Approximately what percentage of the soups are priced below 50 cents? (d) Approximately what percentage of the soups are prices between 50 and 75 cents? (e) What is the median price of a can of chicken noodle soup? Answer: (a) 25% (b) 25% (c) 50% (d) 25% (e) 50 cents MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 20) Quartiles are measures used in constructing A) histograms. B) pie charts. C) frequency tables. D) box plots. E) none of these Answer: D 21) Find the first quartile for a box plot of the following numbers: 11 13 15 18 21 22 25 26 30 35 42 45 A) 39 B) 23.5 C) 32.5 D) 16.5 Answer: D 22) Given the following five-number summary, find the interquartile range. 29, 37, 50, 66, 94 A) 50 B) 65 C) 29 D) 32.5 Answer: C 23) The first quartile (Q1 ) is A) the average of the two smallest numbers in a set of data. B) the same as the frequency of the first category in a histogram. C) the smallest number in a set of data. D) the median of the numbers less than the median of a set of data. Answer: D

24) The test scores of 30 students are listed below. Find Q3 . 31 41 45 48 52 55 56 56 63 65 67 67 69 70 70 74 75 78 79 79 80 81 83 85 85 87 90 92 95 99 A) 83 B) 31 C) 78 D) 85 Answer: A 25) Find the median of the numbers: 35 38 29 42 45 A) 42 B) 29.5 C) 29 D) 38 Answer: D 26) Find the median of the numbers: 35 38 29 42 A) 36 B) 73 C) 36.5 D) 35.5 Answer: C 27) Find the median of the numbers: 8 3 14 11 6 A) 8 B) 8.5 C) 8.4 D) 6.5 E) 14 Answer: A 28) The median of the numbers greater than the median of a set of data is called A) the high median. B) the true median. C) the third quartile. D) the second quartile. E) none of these Answer: C 29) Given the numbers 1 3 6 29 35 38 42 45 48 55, find the first quartile. A) 3 B) 20.5 C) 35 D) 6 E) none of these Answer: D

30) Given the numbers 1 3 6 29 35 38 42 45 48 55, find the interquartile range. A) 38 B) 39 C) 45 D) 10 E) none of these Answer: B Obtain the five-number summary for the given data. 31) The test scores of 15 students are listed below. 43 47 50 53 59 61 65 71 75 79 85 87 90 94 95 A) 43, 53, 73.0, 87, 95 B) 43, 52.25, 71, 85.5, 95 C) 43, 53, 71, 87, 95 D) 43, 52.25, 73.0, 85.5, 95 Answer: C 32) The weekly salaries (in dollars) of sixteen government workers are listed below. 690 603 813 658 728 569 487 626 522 664 685 452 558 787 510 826 A) 452, 540.0, 642.0, 709, 826 B) 452, 531.00, 642.0, 718.5, 826 C) 452, 522, 626, 690, 826 D) 452, 531.00, 626, 718.5, 826 Answer: A

Draw the box plot corresponding to the given five-number summary. 33) min = 16, Q1 = 20, Q2 = 22, Q3 = 25, max = 26 A)

16 20

Answer: A

Find the five-number summary and interquartile range for the given set of numbers, then draw the box plot. 34) 16, 22, 27, 33, 34, 36, 41, 43, 46 A)

24.5

42 46

36 46

41 46

interquartile range = 34 B)

interquartile range = 30 C)

24.5

interquartile range = 17.5 D)

interquartile range = 14 Answer: C

Solve the problem. 35) The manager of a small retail store counted the number of sales each hour during a 60-hour week. The frequency distribution is given below. Number of sales Number of during hour occurrences 6 25 7 20 8 10 9 0 10 5 Find the relative frequency of seven sales during an hour. 1 A) 4 B)

7 40

7 60

1 3

E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 36) Eighty families having four children each were surveyed and the number of girls was recorded for each. The table shows the resulting frequency distribution. number of number of girls occurrences 0 5 1 18 2 32 3 21 4 4 Determine the corresponding relative frequency distribution. Answer: number of girls 0 1 2 3 4

relative frequency 0.0625 0.2250 0.4000 0.2625 0.0500

37) From the following frequency distribution, construct a relative frequency distribution. Number Frequency 0 4 1 6 2 2 3 5 Answer: Number 0

Rel. Frequency 4 17

6 17

2 17

5 17

38) The number of customers waiting to be served in a certain bank was counted every 15 minutes for five hours. The results are given in the frequency distribution below. Number of Number of customers occurrences 0 2 1 2 2 3 3 5 4 4 5 3 6 1 (a) Determine the corresponding relative frequency distribution. (b) Draw a histogram for the relative frequency distribution. Answer: (a) No. customers 0 1 2 3 4 5 6 (b)

Rel. freq. 0.10 0.10 0.15 0.25 0.20 0.15 0.05

39) The number of accidents per week on a certain highway was recorded over a 40-week period, and the results are summarized below. Number of Number of accidents occurrences 0 7 1 12 2 7 3 8 4 4 5 2 (a) Determine the relative frequency distribution for the number of accidents per week. (b) What is the highest number of accidents per week. (c) What is the number of accidents per week with the highest frequency? Answer: (a) No. accidents 0 1 2 3 4 5 (b) 5 (c) 1

Rel. freq 0.175 0.300 0.175 0.200 0.100 0.050

40) Fifty customers at a restaurant were asked to rate the quality of service they received, and the results are summarized in the following frequency distribution. (very good = 5, good = 4, fair = 3, poor = 2, very poor = 1) Number of Quality occurrences 1 4 2 6 3 8 4 20 5 12 Determine the corresponding relative frequency distribution and draw a histogram. Answer: Quality 1 2 3 4 5

Rel. freq. 0.08 0.12 0.16 0.40 0.24

41) An urn contains three white balls and two red balls. The balls are drawn from the urn, one at a time without replacement, until a white ball is drawn. Let X be the number of the draw on which a white ball is drawn for the first time. Determine the probability distribution for X. Answer: k 1

Pr(X = k) 3 5

3 10

1 10

42) On a new version of the Price is Right, a bag contains two white balls and four red balls. A contestant receives $5000 if they draw two white balls, $1000 if they draw one white ball and lose if they draw two red balls. Determine the probability distribution for playing the game. Answer: Pr(X = k) 1 $5000 15 k

$1000

8 15

6 15

43) A jury of 6 people is to be selected from an available pool of eight women and four men. Let X denote the number of men in the selected jury. Determine the probability distribution of X. Answer: k 0

Pr(X = k) 1 33

8 33

5 11

8 33

1 33

44) Let X denote the number of girls in a family with four children. Determine the probability distribution of X. Answer: k 0

Pr(X = k) 1 16

1 4

3 8

1 4

1 16

45) A coin is flipped three times; give the probability distribution of the number of heads. Answer: Number of heads 0

Probability 1 8

3 8

1 8

A pair of dice is tossed and the number of dots on the uppermost faces are observed. Let X denote the random variable defined as the larger of the two numbers. (If the two numbers are equal, then the value of X is either of them.) 46) Determine the probability distribution of X. Answer: k 1

Pr(X = k) 1 36

3 36

5 36

7 36

1 4

11 36

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 47) Which of the following can be a probability distribution for the random variable X? A) k Pr(X = k) 1 0 6

5 2

2 3

k -2

5 12

1 4

k -3

Pr(X = k) 1 3

Pr(X = k) 1 12

5 12

1 3

k 1 2 3

Pr(X = k) 1 3 -

1 6 5 6

Answer: B Make a probability distribution for the given set of events. 48) The following table displays a frequency distribution of the number of siblings for students at one middle school. Make a probability distribution for number of siblings. Number of siblings 0 1 2 3 4 5 6 7 Frequency 191 259 122 53 25 10 6 3 A) Siblings Probability 1 0.542 2 0.255 3 0.111 4 0.052 5 0.021 6 0.013 7 0.006

B) Siblings Probability 0 0.286 1 0.387 2 0.182 3 0.079 4 0.037 5 0.015 6 0.009 7 0.004 C) Siblings Probability 0 0.125 1 0.125 2 0.125 3 0.125 4 0.125 5 0.125 6 0.125 7 0.125 D) Siblings Probability 0 0.300 1 0.372 2 0.194 3 0.067 4 0.037 5 0.015 6 0.009 7 0.004 Answer: B The histogram for a probability distribution is given. Use the histogram to answer the question. 49) What is the probability that the outcome is between 33 and 35, inclusive?

A) 0.8 B) 0.05 C) 0.7 D) 0.65 Answer: C

50) To what event do the shaded rectangles correspond?

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 A) The outcome is greater than 45. B) The outcome is between 45 and 49, inclusive. C) The probability that the outcome is between 45 and 49 is 0.33. D) The outcome is less than 49. Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. A pair of dice is tossed and the number of dots on the uppermost faces are observed. Let X denote the random variable defined as the larger of the two numbers. (If the two numbers are equal, then the value of X is either of them.) 51) Compute Pr(X = 4). Answer:

7 36

52) Compute Pr(X ≤ 3). Answer:

1 4

53) Compute Pr(X + 2 ≤ 3). Answer:

1 36

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 54) An urn contains four white balls and three red balls. The balls are drawn from the urn, one at a time without replacement, until two white balls have been drawn. Let X be the number of the draw on which the second white ball is drawn. Find the values that X may be. A) 2, 3, 4, 5 B) 1, 2, 3, 4, 5 C) 1, 2, 3, 4, 5, 6 D) 2, 3, 4, 5, 6, 7 E) none of these Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 55) Let X be a random variable. What value of a is necessary in the chart below to make it a probability distribution? k 0 1 2

Pr(X = k) 0.5 a 0.35

Answer: a = 0.15 56) Let X have the following probability distribution: k Pr(X = k) 0 0.2 1 0.1 2 0.4 3 0.3 Determine the probability distribution of (X - 1)2. Answer: k 0 1 4

Pr((X - 1)2 = k) 0.1 0.6 0.3

57) Let X have the following probability distribution: k Pr(X = k) 0.2 -1 0 0.3 1 0.1 2 0.4 Determine the probability distribution of X 2 . Answer: k 0 1 4

Pr(X 2 ) = k) 0.3 0.3 0.4

58) Assume that A is a random variable with the given probability distribution. k Pr(A=k) 3 0 15 5

2 15

4 15

5 15

(a) Determine the probability that A = 20 (b) Determine the probability that A ≥ 12.. (c) Find that probability that A+10 is less than 20. Answer: (a)

1 15

(b)

2 5

(c)

9 15

59) The following table gives the probability distribution of the variable X. k Pr(X = k) 1 -2 16 -1

3 16

4 16

5 16

? 1 16

(a) Determine the probability that X = 2. (b) Find Pr(X ≥ 1). (c) Find the probability that X2 is at least 1. (d) Find the probability that X + 2 is positive. Answer: (a)

2 1 or 16 8

(b)

8 1 or 16 2

(c)

12 3 or 16 4

(d)

15 16

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 60) The probability that exactly k successes occur when a binomial trial is repeated independently n times and the probability of success on any given trial p is A) n pn-k (1 - p)k. k B) pk (1 - p) n-k. C) n . k D) 1 - p. E) none of these Answer: E 61) Let X denote the number of boys in a family with four children. Calculate Pr(X ≥ 3). 2 A) 3 B)

1 4

11 16

5 16

E) none of these Answer: D 62) Let X denote the number of boys in a family with four children. Calculate Pr(X < 0). A) 1 1 B) 5 C)

1 4

D) 0 E) none of these Answer: D 63) A single die is tossed six times. Find the probability that a two appears exactly four times. 25 A) 66 B)

375 66

2 3

15 64

E) none of these Answer: B

64) Suppose that the probability that a federal income tax return contains an arithmetic error is 0.2. If 10 federal income tax returns are selected at random, find the probability that fewer than two of them contain an arithmetic error. A) (0.8)10 + (0.2)10 B) 0.20 C) 1 - [(0.8)10 + 10(0.2)(0.8)9] D) (0.8)10 + 10(0.2)(0.8)9 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 65) A single die is tossed five times. Find the probability that either a one or a two appears exactly four times. Answer:

10 243

66) A true false test has four questions. What is the probability of getting at least three questions correct by guessing? Answer:

5 16

67) A baseball player hits a home run with probability .02. What is the probability that he will hit a home run in a given game with four times at bat? Round to the nearest five decimal places. Answer: ≈ 0.07763 68) At a particular school 15% of the students are mathematics majors. Given a sample of 10 students, what is the probability that no more than one of them is a mathematics major? Round to the nearest three decimal places. Answer: ≈ 0.544 69) A coin is flipped 10 times, and the number of heads is observed. What is the probability that heads appears nine times, given that it appears at least nine times? Answer:

10 11

70) A car rental agency finds that 20% of all reservations made in advance are not used. If 10 independent reservations are selected, what is the probability that more than two are not used? Round to the nearest four decimal places. Answer: 1 - 10 ∙ (0.8)10 - 10 ∙ (0.2) ∙ (0.8)9 - 10 ∙ (0.2)2 ∙ (0.8)8 ≈ 0.3222 0 1 2 71) The probability for receiving an A in Calculus is given to be 20% in a class of 1000. If 20 students are chosen (with replacement) from the class, what is the probability that five of them received A's? Round to the nearest four decimal places. Answer:

20 (0.2)5 (0.8)15 ≈ 0.1746 5

72) Statistics show that 60% of adult breakfasts are eaten at home. Five adults are randomly selected. What is the probability that at least one of them has eaten breakfast at home? Answer: 1- 5 0

∙ (0.60) 0 (0.40) 5 ≈ 0.98976

A box contains nine balls labeled 1 through 9. A game consists of randomly drawing five balls from the box in succession, with replacement, and recording the number of each ball drawn. If two even numbers are recorded, the player wins the game. 73) What is the probability of winning this game? Round to the nearest four decimal places. Answer: 5 2

∙

4 2 5 3 ∙ ≈ 0.3387 9 9

74) If the game is repeated 10 times, what is the probability of winning four times? Round to the nearest four decimal places. Answer: 10 (0.3387) 4 (0.6613)6 ≈ 0.2311 4 A box contains 26 balls each of which is labeled with a different letter of the alphabet. An experiment consists of randomly drawing 10 of these balls and recording the letter of each ball drawn. 75) Compute the probability that two of the 10 letters are vowels. Round to the nearest four decimal places. Answer: 5 21 2 8 1260 ≈ 0.3831 = 26 3289 10 76) If the experiment is repeated nine times, find the probability that two vowels are recorded in four different runs of the experiment. Round to the nearest four decimal places. Answer: 9 (0.3831)4 (0.6169)5 ≈ 0.2425 4 Solve the problem. 77) A test for a certain drug produces a false positive 22% of the time. If a person who does not use the drug takes the test six times, what is the probability the person tests positive at least four times? Round to the nearest three decimal places. Answer: ≈ 0.024

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 78) The binomial coefficient n is equal to p n A) . n-p B) n - p . p n +1 + n+1 . C) p p+1 D) all of these E) none of these Answer: A 79) Which of the following is NOT true A) n = 1 whenever n ≥ 1 . 0 B) n = 1 whenever n is a natural number n C) n = n whenever n is a natural number. n n D) = n whenever p ≤ n.. n-p p E) none of these Answer: C Find the requested probability. 80) What is the probability that 15 rolls of a fair die will show three fives? A) 0.1182 B) 0.2363 C) 0.0473 D) 0.4726 Answer: B 81) What is the probability that 19 rolls of a fair die will show 6 sixes? A) 0.0272 B) 0.0544 C) 0.0109 D) 0.1088 Answer: B 82) A fair coin is tossed 5 times. What is the probability of exactly 4 heads? A) 0.1250 B) 0.0313 C) 0.0625 D) 0.1563 Answer: D

83) What is the probability that 16 tosses of a fair coin will show 7 tails? A) 0.0873 B) 0.1746 C) 0.3492 D) 0.0349 Answer: B 84) A coin is biased to show 40% heads and 60% tails. The coin is tossed twice. What is the probability that the coin turns up tails on both tosses? A) 40% B) 60% C) 20% D) 36% Answer: D 85) A coin is biased to show 43% heads and 57% tails. The coin is tossed twice. What is the probability that the coin turns up heads once and tails once? A) 24.51% B) 49.02% C) 57% D) 43% Answer: B 86) A coin is biased to show 42% heads and 58% tails. The coin is tossed twice. What is the probability that the coin turns up heads on the second toss? A) 17.64% B) 24.36% C) 58% D) 42% Answer: D Use the histogram for the given binomial random variable to answer the question. 87) The following is a histogram for the binomial random variable with n = 6 and p = 0.35. What outcome is most likely to occur?

A) 6 B) 0 C) 2 D) All are equally likely Answer: C

88) The following is a histogram for the binomial random variable with n = 6 and p =0.33. The histogram shows that the values of Pr(X = 0) and Pr(X = 4) are nearly the same. Calculate the two probabilities to determine which is larger.

A) Pr(X = 4) > Pr(X = 0) B) Pr(X = 0) > Pr(X = 4) C) Pr(X = 0) = Pr(X = 4) Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 89) Write the probability distribution for a binomial random variable with parameters n = 5, p = 0.3. Answer: k 0 1 2 3 4 5

Pr(X = k) 0.16807 0.36015 0.30870 0.13230 0.02835 0.00243

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate

n k n-k p q for the given values of n, k, and p. Round to four decimal places. k

90) n = 15, k = 7, p =

1 4

A) 0.0511 B) 0.0472 C) 0.0393 D) 0.0433 Answer: C 91) n = 12, k = 9, p = 0.3 A) 0.0019 B) 0.0014 C) 0.0016 D) 0.0015 Answer: D

Find the expected value for the probability distribution. 92)

Value Probability A) 4.5 B) 3.4 C) 4 D) 3.6

0.1 0.3

0.5

0.1

Answer: D

93)

Value Probability A) 9 B) 7.26 C) 7.66 D) 7.94

0.4 0.4 0.17 0.03

Answer: C

94)

Value

Probability 0.14 0.03 0.36 0.37 0.10 A) 7.11 B) 9.78 C) 7.74 D) 7.32 Answer: B

95)

Value

Probability 0.33 0.12 0.46 0.07 0.08 A) 21.12 B) 28.58 C) 30.14 D) 28 Answer: B

Find the expected value for the random variable x having the specified histogram. 96) p

a = 18 c = 20 A) 15.9 B) 19.5 C) 15.6 D) 20

b = 19 d = 21

Answer: D 97) p

a=9 d = 15 A) 13.4 B) 13 C) 16 D) 14.7

b = 11 e = 17

c = 13

Answer: A 98) p

a = 10 d = 13 A) 9.2 B) 9.8 C) 12 D) 10

b = 11 e = 14

c = 12

Answer: C

99) p

a = 25 c = 35 A) 30 B) 30.5 C) 27 D) 33

b = 30

Answer: B 100) Consider the probability distribution below. k -2 0 1 2 3

Pr(X = k) 0.1 0.2 0.1 0.2 0.4

Find the mean. A) 0.8 B) 1.5 C) 1.0 D) 3.0 E) none of these Answer: B 101) Consider the probability distribution below: k -10 20 25

Pr(X = k) 0.2 0.6 0.2

Find the mean. A) 15 B) 20 C) 35 D) 25 E) none of these Answer: A 43

102) Consider the probability distribution below of the number of tails in 4 tosses of a fair coin. k 0

Pr(X = k) 1 16

4 16

6 16

4 16

1 16

Find the mean. A) 1.0 B) 3.0 C) 1.75 D) 1.5 E) none of these Answer: C 103) The manager of a small retail store counted the number of sales each hour during a 60-hour week. The frequency distribution is given below. Number of sales during hour 6 7 8 9 10

Number of occurrences 25 20 10 0 5

Find the average number of sales during an hour. A) 8 1 B) 8 2 C) 6 D) 7 E) none of these Answer: D

104) Two people play a game. A single die is thrown. If the outcome is a 2 or a 3, then player A pays player B $6. How much should B pay A when a 1, 4, 5, or 6 is thrown so that A and B break even, on average, over many repetitions of the game? A) $4 B) $8 C) $2 D) $6 E) none of these Answer: E 105) A luncheon at the cafeteria sold x adult tickets at $6 each and y children's tickets at $3 each. The average (arithmetic mean) revenue per ticket was 6x+3y A) 9 B)

6x+3y xy

9xy x+y

6x+3y . x+y

E) none of these Answer: D 106) If three distinct positive integers have an average (arithmetic mean) of 80, and if the smallest of the three integers is 60, then the largest of the three integers can be at most A) 80 B) 100 C) 120 D) 119 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 107) Two people play a game. A single die is thrown. If the outcome is a 3, then player A pays player B $4. How much should B pay A when a 1, 2, 4, 5, or 6 is thrown so that A and B break even, on average, over many repetitions of the game? Answer: $0.80 108) Your grades on the four exams this term are 100, 75, 80, and 60. If the final test grade is worth 1.5 times a regular test, and the grade is a percentage, can you still get an average of 90? Answer: no 109) Suppose a basketball team averages 82 points per game for 20 games and then averages 86 points per game for 10 games. If the team scores 96 and 88 points in the next two games, how many points per game did the team average over the 32 games? Answer: 83.875 points per game

110) You pay $d to play a game. The game is played as follows: a coin is flipped twice, and you are paid according to the outcome; HH pays $2, HT pays $0.75, TH pays nothing, TT pays $0.25. How much can you pay to play and expect to stay even? Answer: $0.75 per game 111) Your friend wants to play a game with a pair of dice. You win if the sum of the top faces is 5, 6, 7, or 8. The winner pays the loser $1 each game. Should you play? Answer: yes 112) According to life insurance tables, the probability that a 35-year-old man will live an additional five years is 0.99. How much should a 35-year-old man be willing to pay for a policy which pays $10,000 in the event of his death any time within the next five years? Answer: $100 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 113) The mean of a binomial random variable based on n independent repetitions of a Bernoulli trial with constant probability of success p is A) n p B) np C) np(1 - p) D) np(1 - p) E) none of these Answer: B Solve the problem. 114) To get a C in history, Nandan must average 71 on four tests. Scores on the first three tests were 65, 79, and 60. What is the lowest score that Nandan can get on the last test and still receive a C? A) 80 B) 68 C) 69 D) 9 Answer: A 115) Bengisu was pregnant 271 days and 264 days for her first two pregnancies. In order for Bengisu's average pregnancy to equal a national average of 266 days, how long must her third pregnancy last? A) 257 days B) 263 days C) 268 days D) 267 days Answer: B 116) Samuel consumed 2133 calories of food on Monday, 2305 calories on Tuesday, and 1826 calories on Wednesday. In order for Samuel's average calorie intake to equal a daily average of 2100 calories, how many calories of food must he consume on Thursday? A) 2136 calories B) 2091 calories C) 2088 calories D) 2237 calories Answer: A 46

117) Jackie's sisters weigh 111 lb, 142 lb, 124 lb, and 128 lb. The average female in her city weighs 131.5 lb. How much does Jackie weigh if she and her sisters have an average weight of 131.5 lb? A) 127.3 lb B) 152.5 lb C) 263.5 lb D) 126.3lb Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 118) In a new The Price is Right game, two chips are drawn without replacement from a bag that contains two white chips and four red chips. The game proceeds until a white chip is drawn. The player bets $1000 to play and receives $500 for each chip drawn. Find the expected value of the game. Answer: $433.33 119) A box contains three red balls and two green balls. A player pays $5 and draws two balls in succession, without replacement. If the two balls are the same color, the player gets $10; otherwise, he loses the $5. Compute the expected value of the player's net gain. Answer: -1 (that is, $1 loss) 120) A shop supervisor counted the number of accidents each week during a year. The frequency distribution is given below. Number of accidents during week 0 1 2 3 4

Number of occurrences 20 15 10 4 3

(a) Determine the probability distribution for the possible number of accidents during the week. (b) Find the expected number of accidents during the week. Round to the nearest three decimal places. Answer: (a) Number of accidents during a week 0

(b)

Probability 5 13

15 52

5 26

1 13

3 52

59 ≈ 1.135 52

121) A multiple-choice test consists of five questions, each of which has a choice of four answers (only one of which is correct). A student guesses answers at random. Let X denote the number of questions the student answers correctly. (a) Find the probability distribution for X. (b) What is the expected value of X? Answer: (a) k 0

(b)

Pr(X = k) 243 1024

405 1024

135 512

45 512

15 1024

1 1024

5 = 1.25 4

122) A church sells 3000 lottery tickets on a new car worth $7000. Each ticket costs $4. If you buy one ticket, what is your expected winning? Answer: $ -

5 ≈ - $1.67 3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 123) A church sells 2000 lottery tickets on a new car worth $7000. Each ticket costs $5. If you buy one ticket, what is your expected winning? 7 A) 2 B) -

1999 400

C) -

599 400

D) -

3 2

E) none of these Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 124) Consider the probability distribution below of the number of tails in 4 tosses of a fair coin. k 0

Pr(X = k) 1 16

4 16

6 16

4 16

1 16

(a) Find the mean. (b) Find the expected number of tails during the experiment. Answer: (a) 2 (b) 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 125) In a game, you have a 1/49 probability of winning $140 and a 48/49 probability of losing $2. What is your expected winning? A) -$1.96 B) $0.90 C) $2.86 D) $4.82 Answer: B 126) Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 3 or a 1, nothing otherwise. What is your expected net winnings? A) $4.00 B) -$2.00 C) -$0.67 D) $2.00 Answer: C 127) Suppose you pay $3.00 to roll a fair die with the understanding that you will get back $9.00 for rolling a 1 or a 3, nothing otherwise. What are your expected winnings? A) $9.00 B) -$3.00 C) $3.00 D) $0.00 Answer: D

128) Bob and Fred play the following game. Bob rolls a single die. If the result is the number 1, 2, 3, 4, or 5 , Bob must pay Fred the number of dollars indicated by the number rolled. If a 6 is rolled, Fred must pay Bob $33. Find Fred's expectation. A) $1.00 B) $0.00 C) -$10.00 D) -$3.00 Answer: D 129) A contractor is considering a sale that promises a profit of $24,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $6000 with a probability of 0.3. What is the expected profit? A) $18,000 B) $16,800 C) $21,000 D) $15,000 Answer: D 130) Experience shows that a ski lodge will be full (178 guests) if there is a heavy snow fall in December, while only partially full (88 guests) with a light snow fall. What is the expected number of guests if the probability for a heavy snow fall is 0.40? A) 106.8 B) 124 C) 142 D) 71.2 Answer: B 131) An insurance company has written 61 policies of $50,000, 470 of $25,000, and 927 of $10,000 on people of age 20. If the probability that a person will die at age 20 is 0.001, how much can the company expect to pay during the year the policies were written? A) $2407 B) $0 C) $240,700 D) $24,070 Answer: D 132) An insurance company says that at age 50 one must choose to take $10,000 at age 60, $30,000 at 70, or $50,000 at 80 ($0 death benefit). The probability of living from 50 to 60 is 0.84, from 50 to 70, 0.63, and from 50 to 80, 0.43. Find the expected value at each age. A) 60: $8400 70: $25,200 80: $42,000 B) 60: $6300 70: $18,900 80: $21,500 C) 60: $8400 70: $12,900 80: $21,500 D) 60: $8400 70: $18,900 80: $21,500 Answer: D

133) At age 50, Ann must choose between taking $18,000 at age 60 if she is alive then, or $28,000 at age 70 if she is alive then. The probability for a person aged 50 living to be 60 and 70 is 0.82 and 0.58, respectively. Using expected value, what is Ann's best option? A) $18,000 at age 60 B) $28,000 at age 70 Answer: B Find the variance for the given data. Round your answer to one more decimal place than the original data. 134) Jeanne is currently taking college zoology. The instructor often gives quizzes. On the past five quizzes, Jeanne got the following scores: 1 15 10 1 4 A) 37.7 B) 45.7 C) 30.2 D) 37.6 Answer: A 135) The owner of a small manufacturing plant employs six people. As part of their personnel file, she asked each one to record to the nearest one-tenth of a mile the distance they travel one way from home to work. The six distances are listed below: 50 51 29 45 14 27 A) 26.9 mi2 B) 223.2 mi2 C) 8658.7 mi2 D) 22.5 mi2 Answer: B 136) To get the best deal on a microwave oven, Jeremy called six appliance stores and asked the cost of a specific model. The prices he was quoted are listed below: $361 $657 $617 $151 $294 $179 A) 38,903.9 dollars2 B) 46,684.6 dollars2 C) 1,058,421.6 dollars2 D) 46,684.7 dollars2 Answer: D 137) A class of sixth grade students kept accurate records on the amount of time they spent playing video games during a one-week period. The times (in hours) are listed below: 19.8 25.0 20.5 22.8 14.7 17.2 13.2 12.5 17.4 21.6 A) 17.50 hr2 B) 116.90 hr2 C) 15.75 hr2 D) 17.40 hr2 Answer: A

Solve the problem. 138) The manager of an electrical supply store measured the diameters of the rolls of wire in the inventory. The diameters of the rolls (in m) are listed below. Find the standard deviation. Round your result to four decimal places. 0.403 0.287 0.464 0.116 0.356 0.112 0.4 A) 0.6530 B) 0.1160 C) 0.1413 D) 0.7728 Answer: C 139) Christine is currently taking college astronomy. The instructor often gives quizzes. On the past seven quizzes, Christine got the following scores. Find the standard deviation. Round your result to one decimal place. 45 10 33 25 18 55 64 A) 33 B) 19.8 C) 11,284 D) 8928.6 Answer: B 140) The manager of a small dry cleaner employs six people. As part of their personnel file, she asked each one to record to the nearest one-tenth of a mile the distance they travel one way from home to work. The six distances are listed below. Find the standard deviation. Round your result to two decimal places. 23.3 17.0 26.6 41.6 20.6 23.0 A) 34.10 miles B) 8.57 miles C) 3855.74 miles D) 4223.37 miles Answer: B 141) To get the best deal on a CD player, Tom called eight appliance stores and asked the cost of a specific model. The prices he was quoted are listed below. Find the standard deviation. Round your result to the nearest ten cents. $307 $340 $405 $267 $366 $202 $260 $438 A) $835,278.10 B) $879,367.00 C) $79.40 D) $402.00 Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 142) Find the mean, variance, and standard deviation for the probability distribution below. k 0 1 2 3

Pr(X = k) 0.4 0.2 0.3 0.1

Answer: mean = 1.1, variance = 1.09, standard deviation =

1.09

143) Find the mean, variance, and standard deviation of the following probability distribution. k -2

Pr(X = k) 1 12

1 3

7 12

Answer: μ = 2.5, σ2 =

15 , σ= 4

15 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 144) Consider the probability distribution below: k -10 20 25

Pr(X = k) 0.2 0.6 0.2

Find the variance. A) 200 B) 160 C) 35 D) 140 E) none of these Answer: B 145) Consider the probability distribution below: k -2 0 1 2 3

Pr(X = k) 0.1 0.2 0.1 0.2 0.4

Find the variance. A) 3.05 B) 2.65 C) 0.0 D) 1.18 E) none of these Answer: B

146) A binomial random variable with n = 1000 and p =

3 has a variance of 4

A) 187.5. B) 187.5. C) 750. D) 750. E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 147) Find the variance of k -1 0 1 2

Pr(X = k) 0.2 0.3 0.1 0.4

Answer: 1.41 148) Find the variance of k 0 1 2 3

Pr(X = k) .1 .3 .5 .1

Answer: .64 149) Student A earned the following course grades during his freshman year: 4, 4, 4, 3, 3, 3, 3, 3, 2, 2. Student B earned the following course grades during his freshman year: 4, 4, 3, 3, 3, 2, 1, 1, 0, 0. (a) Compute the means and variances for each student. (b) Which student has the better grade point average? (c) Which student was more consistent? Explain. Answer: (a) Student A: μ = 3.1, σ2 = 0.49 Student B: μ = 2.1, σ2 = 2.09 (b) A (c) A; the variance of his score is lower. 150) A sample of six papers is chosen from a large statistics class. The grades are: 87, 93, 96, 82, 95, 87. For the grades on these papers, determine: (a) the sample mean. (b) the sample variance. Answer: (a) x = 90 (b) s2 = 30.4

151) Eighteen workers were sampled, and their days absent from work last year were recorded. The data are: 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 5, 6. (a) Determine the frequency distribution for these data. (b) Find the sample mean. (c) Find the sample variance. Answer: (a) xi 1 2 3 4 5 6

fi 4 5 1 4 3 1

(b) x = 3 46 ≈ 2.71 (c) s2 = 17 152) It is estimated that 1% of all items coming off an assembly line are defective. Let X be the number of defective items in a random sample of 1,000 items from the assembly line. Compute (a) the mean of X. (b) the variance of X. Answer: (a) 10 (b) 9.9 153) If X is a random variable such that E X2 = Answer:

15 5 and μ = , calculate σ2. 2 2

5 4

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 154) The variance of a probability distribution A) is small if the mean is large. B) has little practical value. C) is always smaller than the mean. D) is large if the mean is large. E) none of these Answer: E 155) A certain probability distribution has mean 100 and variance 5. The standard deviation is A) 500. B) 25. C) 5. D) 20. E) none of these Answer: C

Use the given probability distribution to answer the question. 156) Determine by inspection which probability distribution, A or B, has the lesser variance. A

A) B B) A Answer: A 157) According to the Chebychev inequality, if a probability distribution with numerical outcomes has mean μ and standard deviation σ, then the probability that a randomly chosen outcome lies between μ - c and μ + c is σ2 A) no more than 1 . c2 B) at least 1 -

σ2 . c2

C) no more than

D) at least

σ2 - 1. c2

E) none of these Answer: B 158) Suppose that a probability distribution has mean 20 and standard deviation 3. The Chebychev inequality states that the probability that an outcome lies between 16 and 24 is 1 A) at most . 4 B) at least

1 . 4

C) at least

7 . 16

D) less than

7 . 16

E) none of these Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 159) Suppose that a probability distribution has expected value 50 and standard deviation 2. Use the Chebychev inequality to estimate the probability that an outcome lies between 44 and 56. Answer: The probability is at least

8 . 9

160) Suppose that a probability distribution has mean 40 and standard deviation 2. Use the Chebychev inequality to estimate the probability that an outcome lies between 30 and 50. Answer: p ≥ 0.96 161) Suppose that a probability distribution has mean 20 and standard deviation 0.5. Use the Chebychev inequality 16 to find the value of C for which the probability that the outcome lies between 20 - C and 20 + C is at least . 25 Answer: C =

5 6

162) Let X be the number of heads obtained when a fair coin is tossed 1000 times. Using the Chebychev inequality, estimate the probability that X lies between 450 and 550. Answer: p ≥ 0.9 163) State Chebychev's Inequality. Answer: Suppose that a probability distribution with numerical outcomes has expected value μ and standard deviation σ. Then the probability that a randomly chosen outcome lies between μ - c and μ + c is at least 1 σ2 . c2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 164) The heights of the adults in one town have a mean of 67.1 inches and a standard deviation of 3.4 inches. What can you conclude from Chebyshev's theorem about the percentage of adults in the town whose heights are between 60.3 and 73.9 inches? A) The percentage is at most 75% B) The percentage is at least 95% C) The percentage is at least 75% D) The percentage is at most 95% Answer: C 165) The ages of the members of a gym have a mean of 47 years and a standard deviation of 12 years. What can you conclude from Chebyshev's theorem about the percentage of gym members aged between 20.6 and 73.4? A) The percentage is at most 79.3% B) The percentage is approximately 54.5% C) The percentage is at least 79.3% D) The percentage is at least 54.5% Answer: C

166) If Z is the standard normal random variable, find Pr(Z≤ 0.5). A) 0.6915 B) 0.2277 C) 0.3085 D) 0.7723 E) none of these Answer: A 167) If Z is the standard normal random variable, find Pr(Z ≥ 0.6). A) 0.7257 B) 0.5987 C) 0.2254 D) 0.2743 E) none of these Answer: D 168) If X is the standard normal random variable, find Pr(-1.5 ≤ X ≤ 0). A) 0.5000 B) 0.4332 C) 0.9332 D) 0.0668 E) none of these Answer: B 169) If X is the standard normal random variable, find Pr(-1.5 ≤ X ≤ 1.5). A) 0.5000 B) 0.9332 C) 0.0668 D) 0.8664 E) none of these Answer: D 170) If Z is the standard normal random variable and Pr(-z ≤ Z ≤ z) = 0.6578, find z. A) 1.00 B) 0.40 C) 0.45 D) 0.95 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use a table to find the area of the shaded regions under the standard normal curve. 171)

Answer: 0.3085 58

172)

Answer: 0.0919 173)

Answer: 0.7745 174)

Answer: 0.4672 Find the value of z for which the area of the shaded region under the standard normal curve is given. 175)

Answer: z = 1.4 176)

Answer: -z = -0.45

177)

Answer: -z = -1.55 and z = 1.55 178)

Answer: -z = -0.70 and z = 0.70 Solve the problem. 179) Find the 96th percentile of the standard normal distribution. Answer: ≈ 1.75 180) Find the 99th percentile of the standard normal distribution. Answer: ≈ 2.35 181) Find the 90th percentile of the normal distribution with μ = 10, σ = 1.5. Answer: 11.92 182) In a large group we find that men's weights are normally distributed with μ = 130 lb. and σ = 15 lb. Find (a) the 92nd percentile of men's weights. (b) the 21st percentile of men's weights. Answer: (a) 151 lb. (b) 118 lb. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 183) Suppose X is the random variable with a normal distribution with mean μ = 100 and the 95th percentile equal to 116.5.The value of σ is A) 0.95. B) -10. C) 16.5. D) 10. E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 184) Suppose X is a random variable having a normal distribution with σ = 10 and 96th percentile of 22.5. Find μ. Answer: 5 60

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine μ and σ by inspection. 185)

A) μ = 2 σ = 23 B) μ = 23 σ =2 C) μ = 23 σ=4 D) μ = 23 σ = 25 Answer: B Use a table to find the area of the shaded region under the given normal curve. 186)

μ = 34 σ=6

38.5

A) 0.8402 B) 0.7066 C) 0.1598 D) 0.2934 Answer: D 187) If X has a normal distribution with μ = 5 and σ = 2, find Pr(2 ≤ X ≤6). A) 0.6247 B) 0.7583 C) 0.2417 D) 0.9772 E) none of these Answer: A

188) The IQ of adults in a certain large population is normally distributed with mean 100 and standard deviation 10. If a person is chosen at random from this group, find the probability that the person's IQ is less than 85 or greater than 110. A) 0.0919 B) 0.2255 C) 0.9081 D) 0.7745 E) none of these Answer: B 189) Newborn infants' birth weights in a certain community are normally distributed with μ = 7 lb 8 oz and σ = 1 lb (8 oz = 0.5 lb). Infants whose birth weights fall below the 5th percentile are kept in a special nursery. What range of birth weights qualifies for the special care? A) infants under 7.5 lb B) infants under 5 lb C) infants under 9.5 lb D) infants under 5.85 lb E) none of these Answer: D 190) Suppose X is the random variable with a normal distribution with mean μ = 3.3 and σ = 0.2. What is the probability of selecting a number greater than 4? A) 0.002 B) 0.0004 C) 0.004 D) 0.0002 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. X has a normal distribution with μ = 100 and σ = 10. 191) Find the probability that X is smaller than 96 or larger than 106. Answer: 0.6186 192) Find the probability that X is between 95 and 115. Answer: 0.6247 Solve the problem. 193) Let X be a random variable having a normal distribution with mean 25 and standard deviation 2. Find Pr(21 ≤ X ≤ 27). Answer: 0.8185 The weights of men in a certain large group is normally distributed with μ = 160 lb. and σ = 15 lb. If X represents the weight of a man selected at random from the group, determine the following probabilities. 194) The man weighs less than 160 lb. Answer: 0.5 195) The man weighs more than 190 lb. Answer: 0.0228

196) The man weighs less between 160 and 190 lb. Answer: 0.4772 197) The man weighs less than 130 lb. Answer: 0.0228 Solve the problem. 198) Suppose that the grade point average of graduating seniors at Noswot University is normally distributed with μ = 2.7 and σ = 0.5. To graduate with honors, one must rank in the top 8% of the class. What is the minimum grade point average to graduate with honors? Answer: 3.4 199) The lifetimes of a certain model of washing machine are normally distributed with μ = 36 months and σ = 4 months. The manufacturer wants to issue a warranty that will be written so that about 96% of the machines will outlast the warranty. For how many months should the machines be guaranteed? Answer: 29 months 200) In filling 10 oz. jars with instant coffee, the machine dispenses a normally distributed amount of coffee with μ = 10 oz. and σ = 0.1 oz. How many ounces of coffee should a jar hold so that no more than 5% of the jars overflow? Answer: 10.165 oz. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 201) The lifetimes of a certain model of television's picture tubes are normally distributed with μ = 48 months and σ = 8 months. The manufacturer wants to issue a warranty that will be written so that about 92% of the picture tubes will outlast the warranty. For how many months should the picture tubes be guaranteed? A) 44.16 B) 40.64 C) 36.80 D) 59.20 E) none of these Answer: C 202) Tax returns are monitored by a state if the reported yearly income is in the top 15% of incomes in that state. Assume income is normally distributed with mean μ = $28,000 and standard deviation σ = $5000. Find the minimum income in the monitored group. A) $22,750 B) $23,000 C) $33,200 D) $33,000 E) none of these Answer: C 203) Which of the following statements is true? A) Some normal curves are not symmetric. B) There are many different normal curves with the same mean. C) A normal curve is completely described by its standard deviation. D) A normal curve is flatter when its standard deviation is small. E) none of these Answer: B 63

204) Consider the normal distributions drawn below(with different scales). (a) (b)

They both have μ = 20 and the area of the shaded region of each is 0.90. Which of the following holds? A) There is not enough information. B) x1 = x2 C) x1 < x2 D) x1 > x2 E) x1 ≥ x2 Answer: C 205) If X is a normal random variable with mean 12 and standard deviation

5 , an x-value of 14 corresponds to a 4

standard value of A) 5.5. B) 0.8. C) 2.5. D) 1.6. E) none of these Answer: D 206) If X is the normal random variable with mean 12 and standard deviation

5 , a standard value of 2 corresponds 4

to an x-value of A) 9.5. B) 10.4. C) 14.5. D) 13.6. E) none of these Answer: C Provide an appropriate response. 207) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 470 seconds and a standard deviation of 40 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 378 seconds. A) 0.5107 B) 0.4893 C) 0.0107 D) 0.9893 Answer: C

208) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 470 seconds and a standard deviation of 60 seconds. Find the probability that a randomly selected boy in secondary school will take longer than 332 seconds to run the mile. A) 0.9893 B) 0.4893 C) 0.5107 D) 0.0107 Answer: A 209) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.31 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains fewer than 12.21 ounces of beer. A) 0.0062 B) 0.9938 C) 0.5062 D) 0.4938 Answer: A 210) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.14 onces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains more than 12.14 ounces of beer. A) 0.4 B) 0.5 C) 1 D) 0 Answer: B 211) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.19 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains between 12.09 and 12.15 ounces. A) 0.1525 B) 0.1649 C) 0.8475 D) 0.8351 Answer: A 212) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 4.5 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 4.0 minutes. A) 0.2674 B) 0.3085 C) 0.1915 D) 0.3551 Answer: B

213) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 4.0 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will take between 2.5 and 5.0 minutes to find a parking spot in the library lot. A) 0.0919 B) 0.4938 C) 0.7745 D) 0.2255 Answer: C 214) The amount of soda a dispensing machine pours into a 12 ounce can of soda follows a normal distribution with a mean of 12.09 ounces and a standard deviation of 0.06 ounce. The cans only hold 12.15 ounces of soda. Every can that has more than 12.15 ounces of soda poured into it causes a spill and the can needs to go through a special cleaning process before it can be sold. What is the probability a randomly selected can will need to go through this process? A) 0.6587 B) 0.8413 C) 0.3413 D) 0.1587 Answer: D 215) A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls followed a normal distribution with an average of $400 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Using the distribution above, what is the probability that a randomly selected month had a PCE of between $275.00 and $490.00? A) 0.9999 B) 0.9579 C) 0.0421 D) 0.0001 Answer: B 216) A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $400 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the probability that a randomly selected month had a PCE that falls below $250. A) 0.9987 B) 0.6250 C) 0.0013 D) 0.3750 Answer: C 217) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 2500 miles. What is the probability a particular tire of this brand will last longer than 57,500 miles? A) 0.1587 B) 0.7266 C) 0.2266 D) 0.8413 Answer: D

218) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 2100 miles. What is the probability a certain tire of this brand will last between 55,590 miles and 56,220 miles? A) 0.4920 B) 0.0180 C) 0.9813 D) 0.4649 Answer: B 219) Assume that 80% of all children who are exposed to chicken pox contract the disease. If 1225 children are exposed to chicken pox, the probability that more than 945 of them will contract the disease is approximately the area under the standard normal curve A) to the right -2.46. B) to the right of -2.50. C) to the left 2.46. D) to the right -0.18. E) none of these Answer: B 220) A fair coin is tossed 150 times. Find the probability of observing exactly 75 heads. A) 0.0651 B) 0.0067 C) 0.796 D) 0.067 E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. The staff of a clinic knows from literature distributed by a pharmaceutical company that

1 of the patients given a certain 7

drug will have bad side effects. Suppose that the clinic gives 294 patients this drug. Use an approximating normal curve to estimate the probability. 221) Find the probability that none will have any bad side effects. Answer: 0 222) Find the probability that exactly 40 will have bad side effects. Answer: approximately ≈ 0.0641 223) Find the probability that at least 30 will have bad side effects. Answer: approximately ≈ 0.9812

Solve the problem. 224) A person claims to have E.S.P. (extrasensory perception). A card is drawn from the following cards 192 times.

The person identified the chosen card correctly 56 times. Does the result of this experiment support the person's claim that he has E.S.P.? Explain. Answer: If the person is guessing, the probability that he correctly identifies at least 56 cards correctly is approximately 0.1056. It is unlikely that the person can guess correctly at least 56 times. The result of the experiment supports the person's claim. 225) Suppose that a fair coin is tossed 100 times. Find the probability of observing at least 60 heads. Answer: 0.0287 226) A true-false exam consists of 75 questions. What is the probability that someone guessing will get no more than 50 correct answers? Answer: 0.9987 227) A TV company finds that 2% of its TV's break down in the first year. Find the probability that 30 out of 2000 break down within the first year. Answer: 0.0178 228) The recovery rate for a certain cattle disease is 25%. If 200 are affected, what is the probability that at least 60 will recover? Answer: 0.0606 229) A die is thrown 60 times. What is the probability that a 6 shows up exactly 10 times? Answer: 0.0531 230) Two college students challenge themselves to a contest. A coin is flipped 100 times. The first student thinks (with a better than 50-50 chance) that he can guess 65 or more of the flips before they are flipped. The second student says this is almost impossible. Who is right and why? Answer: The second student. The probability of this occurring is approximately 0.006. 231) On a 100 question multiple-choice test, each question has four possible answers. What is the probability of randomly guessing at least 40 of the questions right? (Use the normal curve to approximate the probability.) Answer: 0.04% 232) The manager of a store finds that, for a certain brand, 15% of all VCRs sold are returned within a month. Determine the probability that, out of the 500 VCRs sold, more than 90 will be returned within a month. Answer: 0.0256

233) Let X be the number of red balls drawn in three draws (with replacement) from an urn with six red and four white balls. (a) Write the probability distribution of X. (b) In 20 draws (with replacement) from the same urn, approximate the probability that one draws at least 17 reds. Answer: (a) k 0

Pr(X = k) 3 ∙ 0.6 0 ∙ 0.4 3 = 0.064 0

3 1

∙

0.6 1 ∙ 0.4 2 = 0.288

3 2

∙

0.6 2 ∙ 0.4 1 = 0.432

3 3

∙

0.6 3 ∙ 0.4 0 = 0.216

(b) 0.0202

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Consider the matrices: 1 5 6 3 I. 5 II. 1 0.7 2 0 0.3 6 3

III. 0.3 0.7 0.1 0.9

0.4 0 1 IV. 0.6 0.2 0 0 0.8 0

0.2 0.5 V. 0.1 0.1 0.7 0.4

Which of them are stochastic matrices? A) all except V B) II and IV only C) II, III, and IV only D) all except I E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the matrix is a stochastic matrix. Give a reason for your conclusion. 2) 1 1 3 3 2 3

4 3

Answer: No; an entry is negative. 3) 0.5 0.3 0.1 0.4 0.2 0.1 0.1 0.5 0.8 Answer: Yes; this is a square matrix with nonnegative entries, and each column sums to 1. 4) 0.2 0 0 0 0.3 0 0.8 0.4 1 Answer: No; the second column does not sum to 1. 5) 0.3 0.1 0.3 0.2 0.4 0.7 Answer: No; the matrix is not square. 6) 0.2 0.8 0.8 0.2 Answer: Yes; it is a square matrix with non-negative entries whose columns sum to 1.

7) 0.2 0.8 0.3 0.7 Answer: No; neither column sums to 1. 8) 0.2 0 0.8 1 Answer: Yes; this is a square matrix with nonnegative entries and each column sums to 1. 9) 0.3 0.3 0.4 0.2 0.4 0.4 0.5 0.2 0.3 Answer: No; neither the second nor the third column sums to 1. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the matrix is stochastic. 10) 7 0 0 0.3 A) Yes B) No Answer: B 11) 5 11

4 13

6 11

9 13

A) Yes B) No Answer: A 12) 0.1 0.5 0.9 0.5 A) Yes B) No Answer: A 13) 0.3 0.2 0.4 0.2 0.4 0.1 0.5 0.5 0.5 A) Yes B) No Answer: B

14) 1 1 1 4 4 1 0 0 4 1 3 0 2 4 A) Yes B) No Answer: A Write a stochastic matrix corresponding to the transition diagram. 15) 0.9 0.3 A)

S1 0.1 0.7 S2 0.9 0.3 B)

S1 0.3 0.9 S2 0.7 0.1 Answer: A 16)

A = 0.5 C = 0.3 A) S1 S2 S3

B = 0.6 D = 0.6 S 1 S2 S 3 0 0.3 0.6 0.5 0.6 0.4 0.5 0.1 0

S1 S1 S2 S3

0.5 0.1 0 0.5 0.6 0.4 0 0.3 0.6

Answer: B

17)

A = 0.4 D = 0.5 A) S1 S2 S3 B)

B = 0.4 E = 0.2 S 1 S2 S 3 0 0.4 0.1 0.4 0.5 0.7 0.6 0.1 0.2 S1

S1 S2 S3

C = 0.1

0.6 0.1 0.2 0.4 0.5 0 0 0.4 0.1

Answer: A Sketch a transition diagram for the given transition matrix. 18)

S1 S2 S1 0.7 0.4 S2 0.3 0.6 A)

A = 0.7 C = 0.6

B = 0.3 D = 0.4

A = 0.3 C = 0.4

B = 0.7 D = 0.6

Answer: A

19)

S1 S2 S1 0.6 1 S2 0.4 0 A)

A = 0.4 C=1

B = 0.6

A = 0.6 C = 0.1

B = 0.4

Answer: A 20) Suppose that the people in a certain city are catching cold. It is observed that after one week, 40% of the people who were sick are still sick. Of the people who were well, 30% are sick after one week. Set up the 2 × 2 stochastic matrix with columns and rows labeled S and W that describes this situation. A) S W S 0.4 0.6 W 0.3 0.7 B) S W S 0.4 0.3 W 0 0 C) S W S 0.4 0.3 W 0.6 0.7 D) S W S 0.3 0.4 W 0.7 0.6 Answer: C

21) Suppose in Florida it was observed that 70% of parents who were vegetarian were succeeded by children who were vegetarian and 30% by non-vegetarian. Also, 40% of non-vegetarians were succeeded by vegetarians and 60% by non-vegetarians. Set up the 2 × 2 stochastic matrix with columns and rows labeled V and N-V that describes this situation. A) V N-V V 0.7 0.4 N-V 0.3 0.6 B) V N-V V 0.7 0.4 N-V 0.6 0.3 C) V N-V V 0.7 0.6 N-V 0.3 0.4 D) V N-V V 0.4 0.7 N-V 0.3 0.6 Answer: A 22) Experience with the Finite Mathematics course at a certain college suggests that of those students who get a grade of C or better on exam I, 90% get a C or better on exam II. Of those who get D or F on exam I, 70% get a D or F on exam II. This year 80% got C or better on exam I. Set up the 2 × 2 stochastic matrix with columns and rows labeled ABC and DF that describes this situation. A) ABC DF ABC 0.9 0.7 DF 0.1 0.3 B) ABC DF ABC 0.9 0.3 DF 0.1 0.7 C) ABC DF ABC 0.8 0.2 DF 0.2 0.8 D) ABC DF ABC 0.9 0.1 DF 0.7 0.3 Answer: B

23) Assume that college graduates and noncollege graduates have children in the same numbers. Suppose also that 70% of the children of college graduates also graduate from college. Of the children of noncollege graduates, 55% will graduate from college. Set up the 2 × 2 stochastic matrix with columns and rows labeled G and NG that describes this situation. A) G NG G 0.3 0.55 NG 0.7 0.45 B) G NG G 0.7 0.3 NG 0.5 0.45 C) G NG G 0.7 0.55 NG 0.3 0.45 D) G NG G 0.3 0.45 NG 0.7 0.55 Answer: C Solve the problem. 24) If it snows today, there is a 90 percent chance of snow tomorrow; however if it does not snow today, there is a 70 percent chance that it will not snow tomorrow. Construct the stochastic matrix that represents this situation with columns and rows labeled S and NS. A) S NS S 0.3 0.7 NS 0.9 0.1 B) S NS S 0.9 0.1 NS 0.3 0.7 C) S NS S 0.9 0.1 NS 0.7 0.3 D) S NS S 0.7 0.3 NS 0.9 0.1 Answer: B

25) 10 percent of the people in one generation who have a certain physical characteristic will pass that characteristic on to the next generation. 70 percent of the people in one generation who do not have this characteristic will pass it on to the next generation. Construct the stochastic matrix that represents this situation with columns and rows labeled H and NH. A) H NH H 0.7 0.3 NH 0.1 0.9 B) H NH H 0.9 0.1 NH 0.3 0.7 C) H NH H 0.3 0.7 NH 0.9 0.1 D) H NH H 0.1 0.9 NH 0.7 0.3 Answer: D 26) Suppose that the people in a certain city are catching cold. It is observed that after one week, 40% of the people who were sick are still sick. Of the people who were well, 30% are sick after one week. If, at some particular time, 20% of the people are ill, find the proportion of people who are sick after two weeks. A) 0.668 B) 0.332 C) 0.320 D) 0.104 Answer: B 27) Experience with the Finite Mathematics course at a certain college suggests that of those students who get a grade of C or better on exam I, 90% get a C or better on exam II. Of those who get D or F on exam I, 70% get a D or F on exam II. This year 80% got C or better on exam I. Find the proportion of Finite Mathematics students who are likely to get D or F on exam II. A) 0.78 B) 0.70 C) 0.14 D) 0.22 Answer: D 28) Suppose that of the sons of right-handed fathers, 90% are right-handed and 10% are left-handed. Of the sons of left-handed fathers, 30% are right-handed and 70% are left-handed. (Assume that right-handed and left-handed fathers have sons in equal numbers.) Suppose that in the initial generation all of the fathers are right-handed. What is the percentage of their grandsons that are left-handed? A) 10% B) 16% C) 1% D) 84% E) none of these Answer: B

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 29) Assume 50% of women currently work. Of those who work, 70% of their daughters work the next generation. Of those who don't work, only 40% of their daughters work the next generation. Find the percentage of women in the next generation who work. Answer: 55% 30) Assume that a city is divided into three areas and that during a given year nobody moves in or out of the city. If movement within the city is given by the matrix Old Home I II III New I 0.5 0.3 0.4 Home II 0.4 0.4 0.2 III 0.1 0.3 0.4 and 1000 people currently live in area I, 1500 in area II, and 750 in area III, determine the population of each area after one year. Answer: Area I: 1250, Area II: 1150, Area III: 850 31) In a certain company it has been determined that in one year 10% of the skilled workers become unskilled and 40% of the unskilled workers become skilled. (a) Write a stochastic matrix, labeling the rows and columns with S for skilled and U for unskilled, which describes the transitions. (b) Assume that at the beginning of a year 70% of the workers are skilled and 30% of the workers are unskilled. What percentage of the workers will be unskilled at the beginning of the next year? Answer: (a)

S S 0.9 U 0.1

U 0.4 0.6

(b) 25% 32) In a certain city, it has been determined that 70% of the small businesses (employing 25 or fewer employees) grow to be medium-sized businesses (26-75 employees) each year and 40% of the medium-sized businesses shrink to small businesses. (a) Write a stochastic matrix, labeling the rows and columns with S for small and M for medium, which describes the transitions. (b) Assume that at the beginning of a year, 20% of the businesses are small and 80% are medium-sized. What percentage of the businesses will be medium-sized at the beginning of the next year? Answer: (a)

S M S 0.3 0.40 M 0.70 0.60 (b) 62%

33) Assume that the transition matrix of a Markov process is given by the following. 1 0 0 0 1 0 0 0 1 What does this say about the change from year to year of a given initial distribution? Answer: There is no change.

34) Determine the values of a and b such that the following matrix is a transition matrix of a Markov process. a 0.31 0.4 b Answer: a = 0.6 b = 0.69 35) A Markov process has the transition matrix A = 0.9 0.4 0.1 0.6 and the initial distribution matrix 0.5 . 0.5 Find the first and second distribution matrices. Answer: 0.65 and 0.725 0.35 0.275 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A small town has only two dry cleaners, Fast and Speedy. Fast hopes to increase its market share by conducting an extensive advertising campaign. The initial market share for Fast was 40% and 60% for Speedy. Solve the problem. 36) Find the probability that a customer using Fast initially will use Fast for his second batch of clothes. Use the following transition matrix. Fast Speedy Fast

0.57 Speedy 0.43

0.36 0.64

A) 0.64 B) 0.57 C) 0.36 D) 0.43 Answer: B 37) Find the probability that a customer using Fast initially will use Fast for his third batch of clothes. Use the following transition matrix. Round your answer to the nearest hundredth. Fast Speedy Fast

0.23 Speedy 0.77

0.85 0.15

A) 0.71 B) 0.29 C) 0.68 D) 0.32 Answer: A

38) Find the probability that a customer using Fast initially will use Fast for his fourth batch of clothes. Use the following transition matrix. Round your answer to the nearest hundredth. Fast Speedy Fast

0.16 Speedy 0.84

0.34 0.66

A) 0.71 B) 0.72 C) 0.29 D) 0.28 Answer: D 39) Use the initial market share as an initial distribution matrix, along with the following stochastic matrix, to find the market share for each firm for the first batch of clothing after the campaign. Round your answer to the nearest percent. Fast Speedy Fast

0.44 Speedy 0.56

0.74 0.26

A) 67% for Fast and 33% for Speedy B) 62% for Fast and 38% for Speedy C) 33% for Fast and 67% for Speedy D) 38% for Fast and 62% for Speedy Answer: B 40) Use the initial market share as an initial distribution matrix, along with the following stochastic matrix, to find the market share for each firm for the second batch of clothing after the campaign. Round your answer to the nearest percent. Fast Speedy Fast

0.65

0.75

Speedy 0.35

0.25

A) 33% for Fast and 68% for Speedy B) 53% for Fast and 47% for Speedy C) 47% for Fast and 53% for Speedy D) 68% for Fast and 33% for Speedy Answer: D

41) Suppose A = 0.2 0.4 . Find A2 . 0 0.6 0.04 0.16 2 A) A = 0 0.36 B) A2 = 0.008 0.064 0 0.216 C) A2 = 0.6 1.2 0 1.8 D) A2 = 0.04 0.32 0 0.36 Answer: D 42) Suppose A = 0.3 0.1 . Find A2 . 0 0.6 A) A2 = 0.27 .0064 0 0.216 B) A2 = 0.09 0.81 0 0.36 C) A2 = 0.027 0.64 0 0.216 D) A2 = 0.09 0.09 0 0.36 Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 43) Suppose A = 0.2 0.8 . Find A2 and A3 . 0.8 0.2 Answer: A2 = 0.68 0.32 , 0.32 0.68

A3 = 0.392 0.608 0.608 0.392

44) Suppose A = 0.3 0.4 . Find A2 and A3 . 0.7 0.6 Answer: A2 = 0.37 0.36 , 0.63 0.64

A3 = 0.363 0.364 0.637 0.636

45) 0.4 0.8 0.1 If A = 0.3 0.1 0.4 . Find A2 . 0.3 0.1 0.5 Answer: 0.43 0.41 0.41 A2 = 0.27 0.29 0.27 0.3 0.3 0.32 46) 0.5 0.7 0.1 If A = 0.2 0.2 0.1 . Find A2 . 0.3 0.1 0.8 Answer: 0.42 0.5 0.2 A2 = 0.17 0.19 0.12 0.41 0.31 0.68 Use a graphing calculator to answer the question. 47) Successive powers of the matrix A = 0.30 0.40 get closer and closer to a certain 2 × 2 matrix. What is that 0.70 0.60 matrix? (Round the entries to two decimal places.) Answer: 0.36 0.36 0.64 0.64 48) Let A = 0.4 0.2 and B = 1 be the transition and the initial distribution matrices of a certain Markov process. 0.6 0.8 0 2 8 Compute AB, A B, ..., A B. Answer: 0.4 , 0.28 , 0.256 , 0.2512 , 0.25024 , 0.250048 , 0.2500096 , 0.25000192 0.6 0.72 0.744 0.7488 0.74976 0.749952 0.7499904 0.74999808 49) Let A = 0.1 0.6 and B = .5 be the transition and the initial distribution matrices of a certain Markov process. 0.9 0.4 .5 2 4 Compute AB, A B, ..., A B. Answer: 0.35 , 0.425 , 0.3875 , 0.4063 0.65 0.575 0.6125 0.5938 50) 0.4 0.2 0.8 0.3 Let A = 0.3 0.5 0.1 and B = 0.4 be the transition and the initial distribution matrices of a certain Markov 0.3 0.3 0.1 0.1 2 process. Compute AB, A B, ..., A7 B. Answer: 0.28 0.348 0.3472 0.34992 0.349888 0.3499968 0.34999552 0.3 , 0.256 , 0.252 , 0.25024 , 0.25008 , 0.2500096 , 0.2500032 0.22 0.196 0.2008 0.19984 0.200032 0.1999936 0.20000128

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 51) Which of the following are regular stochastic matrices? 0.2 0 0.3 I. 0.4 0.5 II. 0 1 III. 0.1 1 0.4 IV. 0.1 0.8 0.6 0.2 1 0 0.9 0.2 0.7 0 0.3 A) II and III only B) I and IV only C) IV only D) IV and V only E) II, III, and V only

V. 0 0.6 1 0.4

Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the matrix is a regular stochastic matrix. Give a reason for your conclusion. 52) 0.3 0.9 0.7 0.1 Answer: Yes; the matrix is a stochastic all of whose entries are positive. 53) -0.1 0.2 1.1 0.8 Answer: No; the matrix has a negative entry, so it is not stochastic. 54) 0.3 1 0.7 0 Answer: Yes; squaring the matrix yields 0.79 0.3 , which has all positive entries. 0.21 0.7 55) 0 .5 .5 .5 Answer: No. The first column does not add to 1. 56) 1 0.3 0 0 0.4 0 0 0.3 0 1 Answer: No; the first column in every power is 0 . 0 57) 0 1 1 0 Answer: No; the powers of the matrix alternate between 0 1 and the identity matrix. 1 0

58) 1 0.5 1 0 0.5 0 0 0 0 Answer: No; the third row in every power is [0 0 0]. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide whether or not the stochastic matrix is regular. 59) 0.5 0.5 0.5 0.5 A) Yes B) No Answer: A 60) 1 0.1 0 0.9 A) Yes B) No Answer: B 61) 0.44 0.57 0 0 0.43 0.75 0.56 0 0.25 A) Yes B) No Answer: A 62) 0.69 0 0 0 1 0 0.31 0 1 A) Yes B) No Answer: B 63) 0.73 1 0 0 0 0.19 0.27 0 0.81 A) Yes B) No Answer: A

Find the stable distribution for the regular stochastic matrix. 64) 0.6 0.3 0.4 0.7 Round the numbers in your answer to the nearest thousandth. A) 0.667 0.333 B) 0.571 0.429 C) 0.429 0.571 D) 0.333 0.667 Answer: C 65) 0.66 0.74 0.34 0.26 Round the numbers in your answer to the nearest thousandth. A) 0.685 0.315 B) 0.315 0.685 C) 0.529 0.471 D) 0.471 0.529 Answer: A

66) 0.3 0.2 0.8 Consider the regular stochastic matrix 0.3 0.6 0.1 . 0.4 0.2 0.1 In order to find its stable distribution, it is necessary to solve which system of equations? A) x+ y+ z=0 0.3x + 0.3y + 0.4z = 1 0.2x + 0.6y + 0.2z = 1 0.8x + 0.1y + 0.1z = 1 B) x + y+ z=1 0.3x + 0.2y + 0.8z = x 0.3x + 0.6y + 0.1z = y 0.4x + 0.2y + 0.1z = z C) x+ y+ z=1 0.3x + 0.2y + 0.8z = 0 0.3x + 0.6y + 0.1z = 0 0.4x + 0.2y + 0.1z = 0 D) x + y+ z=1 0.3x + 0.3y + 0.4z = x 0.2x + 0.6y + 0.2z = y 0.8x + 0.1y + 0.1z = z Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 67) 0.1 0.6 0.2 Consider the stochastic matrix 0.4 0.3 0.6 . 0.5 0.1 0.2 Write the system of equations that must be solved in order to find the stable distribution associated with it. Answer:

x+ y+ z=1 0.1x + 0.6y + 0.2z = x 0.4x + 0.3y + 0.6z = y 0.5x + 0.1y + 0.2z = z

68) List simultaneous equations whose solutions will give the stable distribution for the stochastic matrix 0.2 0.9 0.3 0.1 0 0.3 . 0.7 0.1 0.4 Answer: x+ y+ z=1 0.2x + 0.9y + 0.3z = x 0.1x + 0.3z = y 0.7x + 0.1y + 0.4z = z Find the stable distribution of the regular stochastic matrix. 69) 0.2 0.3 0.8 0.7 Answer: 3 11 8 11 70) 0.6 0.5 0.4 0.5 Answer: 5 9 4 9 71) 0.8 0.3 0.2 0.7 Answer: .6 .4 72) 0.4 0.5 0.6 0.5 Answer: 5 11 6 11

73) 0.6 0.1 0.1 0.3 0.8 0.2 0.1 0.1 0.7 Answer: 0.2 0.55 0.25 74) 0 0.3 0.2 0.5 0.4 0.3 0.5 0.3 0.5 Answer: 21 106 20 53 45 106

≈

0.20 0.38 0.42

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 75) Find the stable distribution for the regular stochastic matrix 0.6 0.2 . 0.4 0.8 A) 2 3 1 3 B) 1 3 2 3 C) 1 4 3 4 D) 3 4

3 4

1 4

Answer: B

76) Find the stable distribution for the regular stochastic matrix 0.4 0.2 . 0.6 0.8 A) 1 3 2 3 B) 2 3 1 3 C) 3 4

3 4

1 4

D) 1 4 3 4 Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 77) Find the matrix whose stable distribution is 0.4 . 0.6 Answer: 0.4 0.4 0.6 0.6 Use a graphing calculator to answer the question. 78) Find the stable distribution of the regular stochastic matrix A = 0.4 1 0.6 0 by calculating A2 , A3 , A4 , ..., until successive matrices do not change. Answer: 0.625 0.375

79) Consider the stochastic matrix A and distribution matrix B below. A = 0.4 1 B = 0.5 0.6 0 0.5 Determine the stable distribution of A by calculating AB, A2B, A3 B, A4 B, ..., until successive matrices do not change. Answer: 0.625 0.375 80) Find the stable distribution of the regular stochastic matrix 0.7 0.4 0.5 A = 0.2 0.2 0.2 0.1 0.4 0.3 by calculating A2 , A3 , A4 , ..., until successive matrices do not change. Answer: 0.6 0.2 0.2 81) Find the stable distribution of the regular stochastic matrix 0.1 0.4 0.1 A = 0.3 0.2 0.8 0.6 0.4 0.1 by calculating A2 , A3 , A4 , ..., until successive matrices do not change. Answer: 0.2286 0.4286 0.3429 82) Consider the stochastic matrix A and the distribution matrix B below. 0.7 0.4 0.5 0.4 A = 0.2 0.2 0.2 B = 0.3 0.1 0.4 0.3 0.2 Determine the stable distribution of A by calculating AB, A2B, A3 B, A4 B, ... , until successive matrices do not change. Answer: 0.54 0.18 0.18 83) Consider the stochastic matrix A and the distribution matrix B below. 0.1 0.4 0.1 0.4 A = 0.3 0.2 0.8 B = 0.3 0.6 0.4 0.1 0.2 Determine the stable distribution of A by calculating AB, A2B, A3 B, A4 B, ... , until successive matrices do not change. Answer: 0.2057 0.3857 0.3086 21

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 84) Assume that college graduates and noncollege graduates have children in the same numbers. Suppose also that 70% of the children of college graduates also graduate from college. Of the children of noncollege graduates, 20% will graduate from college. If this trend continues, then in the long run, what is the proportion of the population that will be college graduates? 2 A) 5 B)

2 7

3 5

D) 0 E) none of these Answer: A 85) Suppose that of the sons of right-handed fathers, 90% are right-handed and 10% are left-handed. Of the sons of left-handed fathers, 30% are right-handed and 70% are left-handed. (Assume that right-handed and left-handed fathers have sons in equal numbers.) If these trends continue, then eventually, what will be the the percentage of men who are left-handed? A) 75% B) 25% C) 50% D) 10% E) none of these Answer: B Solve the problem. 86) The probability that an assembly line works correctly depends on whether the line worked correctly the last time. Find the probability that the line will work in the long run. Round your answer to the nearest thousandth. Works Does not 0.7

0.4

Does not 0.3

0.6

Works

A) 0.665 B) 0.571 C) 0.700 D) 0.429 Answer: B

87) Weather is classified as sunny or cloudy in a certain place. What are the long-term predictions for sunny and cloudy days? Round numbers to the nearest thousandth. Sunny Cloudy Sunny Cloudy

0.8

0.1

0.2

0.9

A) 0.800 0.200 B) 0.333 0.667 C) 0.760 0.240 D) 0.667 0.333 Answer: B 88) Rats are kept in a cage with two compartments (A and B). Rats in A move to B with probability 0.2. Rats in B move to A with probability 0.3. Find the long-term trend for rats in each compartment. Round numbers to the nearest thousandth. A) 0.760 0.240 B) 0.800 0.200 C) 0.400 0.600 D) 0.600 0.400 Answer: D 89) At a liberal arts college, students are classified as humanities or science majors. There is a probability of 0.2 that a humanities major will change to science and of 0.5 that a science major will change to humanities. What are the long-term predictions for each major? Round numbers to the nearest thousandth. A) 0.286 0.714 B) 0.760 0.240 C) 0.800 0.200 D) 0.714 0.286 Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 90) Suppose that 100 cows graze in pasture I and 10 cows graze in pasture II. Assume that the location of cows from one day to the next can be modeled by a Markov process whose transition matrix is given by I II I 0.4 0.5 II 0.6 0.5 How many cows will be in pasture I after four days? Answer: 50 91) A survey in a small mining town indicates that 40% of miners' sons become miners, whereas 60% do not. Also 10% of nonminers' sons become miners. At the present time, 70% of the town's men are miners and 30% are nonminers. Considering this as a Markov process: (a) Write the initial distribution matrix. (b) Write the stochastic (transition) matrix with rows and column labeled M (for miners ) and NM (for nonminers) that describes the transitions. (c) Give the percentage of the town's men who will be miners in the first generation, ...in the second generation. (d) Write the system of linear equations needed to determine the stable distribution of the town's male population. (e) Give the proportion of the town's men who will become miners in the long run. Answer: (a) 0.7 0.3 (b)

M NM M 0.4 0.1 NM 0.6 0.9 (c) 31%; 19.3% (d) x+ y =1 0.4x + 0.1y = x 0.6x + 0.9y = y 1 (e) 7

92) A national study of collegiate beer drinkers revealed the following facts: (I) Among current beer drinkers, 30% prefer one of the "light" beers and 70% prefer "regular" beer. (II) In any given year, 20% of the "light" drinkers switch to "regular," and 25% of the "regular" drinkers switch to "light." (a) Write the initial distribution matrix. (Label the matrix using L for "light" and R for "regular.") (b) Write the transition matrix for the study. (Label columns and rows using L for "light" and R for "regular.") (c) In the long run what proportion of collegiate beer drinkers drink "light"? Answer: (a) L 0.3 R 0.7 (b) L R L 0.8 0.25 R 0.2 0.75 5 (c) 9 93) An experiment was conducted by having a group of children solve a puzzle. It was found that of those who had solved the puzzle on a given trial, 40% could solve it on the next trial. And of those who couldn't solve the puzzle on a given trial, 10% could solve it on the next trail. At the present time, 80% of the children solved the puzzle on the current trial, but 20% did not. Assume that this process can be modeled as a Markov process. (a) Write the transition matrix. Label the columns and rows, using S for "able to solve" and NS for "not able to solve." (b) What percent of the children will be able to solve the puzzle on the next trial? (c) After many trials, what proportion of the children will fail to solve the puzzle on each trial? Answer: (a)

S S 0.4 NS 0.6 (b) 34% 6 (c) 7

NS 0.1 0.9

94) Suppose that companies I, II, and III compete for the entire market of a certain kind of washing machine. During any year, company I keeps 60% of its customers and loses 25% to II and 15% to III. On the other hand, company II keeps 50% of its customers and loses 30% to I and 20% to III. Also, company III keeps 40% of its customers and loses 35% to I and 25% to II. Currently, customers are equally divided among the three companies. Considering that this process can be modeled as a Markov process: (a) Determine the transition matrix. (b) Determine the long-run market share among I, II, and III. Answer: (a)

I II III I 0.6 0.3 0.35 II 0.25 0.5 0.25 III 0.15 0.2 0.40

(b) 4 9 1 3 2 9 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the transition diagram corresponds to an absorbing stochastic matrix. 95) 0.3 0.4 0.5 0.3

0.8 0.2 0.4

0.1

A) No B) Yes Answer: B

96) 0.2

0.4 0.3

0.2

0.3 0.1

0.6

1 0.9 A) No B) Yes Answer: A Determine whether the matrix is an absorbing stochastic matrix. 97) 0.7 0 0.1 0 1 0 0.3 0 0.9 A) Yes B) No Answer: B 98) 0.5 0.2 0 0.1 0.2 0 0.4 0.6 1 A) Yes B) No Answer: A 99) 0.5 0 0.1 0 1 0.3 0.2 0 0.3 0.3 0 0.3 A) Yes B) No

0 0 0 1

Answer: A 100) 0.3 0.1 0.2 0.4 0 1 0 0 0 0 1 0 0 0 0 1

0.3 0.1 0.2 0.4

0 0 1 0 0 1 0 0

0 0 0 1

A) No B) Yes Answer: B

101) 1

1 0 7

4 0 7

2 1 7

A) No B) Yes Answer: B 102) 1 0 0 0 0 0.9 0.2 0 0 0.1 0.8 0 0 0 0 1 A) No B) Yes Answer: A 103) Consider the matrices: 0.3 III. 0.2 0.1 0.4 Which are absorbing stochastic matrices? A) II and V only B) I and II only C) III and V only D) I and III only E) all of these 0 0.2 0 I. 1 0.7 0 0 0.1 1

0 0 3 II. 1 0 0.2 0 1 0.5

0 0.5 0 0.3 0.5 0 0.7 0 0 0 0 1

IV. 0.3 1 0.4 0

1 V. 0 0 0

0 0.4 0 0.6

0 0 1 0

0 0.3 0 0.7

Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the matrix is an absorbing stochastic matrix. Give a reason for your conclusion. 104) 0.2 0 0.5 0.3 1 0.4 0.5 0 0.2 Answer: No; the matrix is not stochastic. 105) 0 0 0.3 1 0 0.4 0 1 0.3 Answer: No; there is no absorbing state.

106) 1 0 0.3 0 1 0.4 0 0 0.3 Answer: Yes. 107) 1 0 0 0

0 0.3 0 0.7

0 0 1 0

0 0.5 0 0.5

Answer: No; it is impossible to get from states 2 or 4 to an absorbing state. 108) 0 0.3 0.1 0.6

0 1 0 0

0 0 1 0

0.2 0.3 0.5 0

Answer: Yes; states 2 and 3 are absorbing, and it is possible to get from the nonabsorbing states to the absorbing states. 109) 0.1 0 0 0 1 0.4 0.9 0 0.6 Answer: Yes; state 2 is absorbing, and it is possible to get from states 1 and 3 to state 2. Solve the problem. 110) Consider the following stochastic matrix. I II III IV V I 0 1 0 0.2 0 II 1 0 0.2 0.3 0 III 0 0 0.3 0.1 0 IV 0 0 0.2 0.2 0 V 0 0 0.1 0.2 1 (a) List all the absorbing states (b) Is the matrix an absorbing stochastic matrix? Why or why not? Answer: (a) V (b) No; it is impossible to get from state I or II to state V. For the given absorbing stochastic matrix, determine the absorbing and nonabsorbing states. 111) I II III IV V I 1 0.3 0 0 0.8 II 0 0.2 0 0.2 0.1 III 0 0 1 0.4 0.1 IV 0 0.4 0 0.1 0 V 0 0.1 0 0.3 0 Answer: Absorbing states: I and III; nonabsorbing states: II, IV, and V.

112)

I I 0.4 II 0.2 III 0.3 IV 0.1

II III IV 0.5 0 0 0 0 0 0 0 0 0.5 1 1

Answer: Absorbing states: IV; nonabsorbing states: I, II, and III Determine whether or not the given matrix is stochastic. If it is stochastic, determine if it is regular, absorbing, or neither. 113) 0.2 0.3 0 0 0.5 0.7 0 0 0.1 0 1 0 0.2 0 0 1 Answer: stochastic, absorbing 114) 0.3 0.2 0.1 0.4

0 1 0 0

0 0 0.2 0.8

0 0 0.3 0.7

Answer: stochastic, neither 115) 1

1 0 4

3 0 4

0 1

Answer: stochastic, absorbing 116) 1 0

1 3

0 1

1 3

0 0

1 3

Answer: stochastic, absorbing 117) 0.1 0.6 0.5 0.2 0.3 0.4 0.7 0.1 0.1 Answer: stochastic, regular 118) 0.6 0.8 0.4 0.1 Answer: not stochastic

Identify the matrix as a regular stochastic matrix, absorbing stochastic matrix, or neither. 119) 0.2 0 0.9 0.3 0.6 0.1 0.5 0.4 0 Answer: regular stochastic 120) 1 0.2 0 0.8 Answer: absorbing stochastic 121) 0 0.2 0.2 0.6

0 0.6 0.4 0.0

0.3 0.5 0.1 0.1

0.1 0.9 0 0

Answer: regular stochastic 122) 1 0.2 0 0.9 Answer: neither MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Put the absorbing stochastic matrix in standard form and then find the fundamental matrix. 123) 1 0.4 0 0 0.4 0 0 0.2 1 5 A) 3 3 B) 5 C)

11 9

D) 3 Answer: A

124) 0.25 0 0 0.36 1 0 0.39 0 1 3 A) 4 4 B) 3 C) 3 11 D) 9 Answer: B 125) Find the stable matrix for the absorbing stochastic matrix I S . 0 R -1 A) I S(I - R) 0 R -1 B) I S(I - R ) 0 0 -1 C) I S(R - I) 0 0 -1 D) I S(I - R) 0 0 Answer: D

1 0

1 2

1 126) Find the stable matrix for the absorbing matrix 0 1 8 . 0 0 A) 1 0

1 2

0 1

1 8

0 0

1 0

1 2

0 1

1 8

0 0

3 8

1 0

4 5

0 1

1 5

0 0

3 8

1 0

4 5

0 1

1 5

0 0

Answer: D

3 8

1 0

7 10

1 127) Find the stable matrix for the absorbing matrix 0 1 10 . 0 0 A) 1 0

5 8

0 1

3 8

0 0

1 0

1 2

0 1

1 8

0 0

1 0

4 5

0 1

1 5

0 0

3 8

1 0

7 8

0 1

1 8

0 0

Answer: D

1 5

1 0.4 0.2 128) Find the stable matrix for the absorbing stochastic matrix 0 0.1 0.2 . 0 0.5 0.6 A) 100 001 010 B) 111 000 000 C) 100 011 000 D) 100 010 001 Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. For the given absorbing stochastic matrix, compute the stable matrix. 129) 17 1 0 24 0 1

1 24

0 0

1 4

Answer: 0 0

17 18

1 0

1 18

0 0

130) 1 0

1 4

0 1

1 8

0 0

5 8

Answer: 1 0

2 3

0 1

1 3

0 0 0 131) 1 0.2 0 0 0.3 0.7 0 0.5 0.3 Answer: 1 1 1 0 0 0 0 0 0 132) 1 2

1 4

0 1 0

1 2

0 0

1 6

0 0

1 3

1 4

1 0

Answer: 1 0

11 15

1 3

0 1

4 15

2 3

0 0 0 0

0 0

133) 1 0 0 0

0 1 0 0

0.3 0.5 0.2 0

0 0.1 0.1 0.8

Answer: 1 0 0 0

0 1 0 0

1 0

0.375 0.625 0 0

3 8

0.1875 0.8125 = 5 0 1 0 8 0 0 0 0 0 0 0

3 16 13 16 0 0

Use a graphing calculator to answer the question. 134) Consider the absorbing stochastic matrix 1 0 0 0.3 0.1 0 1 0 0.1 0.1 A= 0 0 1 0 0.1 0 0 0 0.6 0.1 0 0 0 0 0.6 Find the stable matrix of A by computing A2 , A3 , A4 , ..., until successive matrices do not change. Round numbers to four decimal places, if necessary. Answer: 1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0.75 0.25 0 0 0

0.4375 0.3125 0.25 0 0

135) Consider the absorbing stochastic matrix 1 0 0 0 0.3 0 0 1 0 0 0.1 0 A = 0 0 1 0 0.1 0 0 0 0 1 0 0.2 0 0 0 0 0.5 0 0 0 0 0 0 0.6 Find the stable matrix of A by computing A2 , A3 , A4 , ..., until successive matrices do not change. Round all numbers to four decimal places. Answer: 1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0.6 0.2 0.2 0 0 0

0 0 0 0.5 0 0

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the absorbing stochastic matrix to standard form. 136) A BC A 0.3 0 0 B 0.4 1 0 C 0.3 0 1 A)

ABC A 1 0 0.4 B 0 1 0.3 C 0 0 0.3 B) ABC A 1 0 0.3 B 0 1 0.4 C 0 0 0.3 C) BCA B 1 0 0.3 C 0 1 0.4 A 0 0 0.3 D) BCA B 1 0 0.4 C 0 1 0.3 A 0 0 0.3 Answer: D

137)

A B C D A 0.1 0 0.3 1 B 0.1 1 0.1 0 C 0.2 0 0.2 0 D 0.6 0 0.4 0 A)

D B A C A 1 0 0.1 0.3 B 0 1 0.1 0.1 C 0 0 0.2 0.2 D 0 0 0.4 0.4 B) B A C D B 1 0.1 0.1 0 A 0 0.1 0.3 1 C 0 0.2 0.2 0 D 0 0.6 0.4 0 C) B A C D B 1 0.6 0.4 0 A 0 0.1 0.3 1 C 0 0.2 0.2 0 D 0 0.1 0.2 0 D) B D A C B 1 0 0.1 0.1 D 0 1 0.4 0.4 A 0 0 0.1 0.3 C 0 0 0.2 0.2 Answer: B Solve the problem. 138) There are five brands of a product on the market, three of which have very loyal customers who never change brands. The transition matrix for the five brands is shown below (where the brands are labeled A, B, C, D, and E). About how many times, on average, will a person currently using brand C use brand D before trying one of the brands that have loyal followings? Round your answer to the nearest hundredth. A A 1 B 0 C 0 D 0 E 0

B 0 1 0 0 0

C 0.15 0.14 0.21 0.20 0.30

D 0.19 0.24 0.15 0.13 0.29

E 0 0 0 0 1

A) 0.23 B) 1.20 C) 1.32 D) 0.30 Answer: D

139) There are five brands of a product on the market, three of which have very loyal customers who never change brands. The transition matrix for the five brands is shown below (where the brands are labeled A, B, C, D, and E). About how many times, on average, will a person currently using brand C or brand D use his or her current brand before trying one of the brands that have loyal followings? Round your answer to the nearest hundredth. A A 1 B 0 C 0 D 0 E 0

B 0 1 0 0 0

C 0.14 0.11 0.21 0.16 0.38

D 0.17 0.21 0.14 0.14 0.34

E 0 0 0 0 1

A) 1.20 B) 0.21 C) 1.31 D) 2.51 Answer: D 140) A rat is placed at random in one of five compartments in a maze. Compartments 1, 2, and 5 have food; so the rat will remain at any of these once reached. If the rat is placed in compartment 3, what is the expected number of times it will be in compartment 3 or 4 before it finds food? Round your answer to the nearest hundredth.

1 2 3 4 5

1 1 0 0 0 0

2 0 1 0 0 0

3 0.11 0.21 0.20 0.19 0.29

4 0.17 0.21 0.13 0.14 0.35

5 0 0 0 0 1

A) 21 B) 20 C) 11 D) 1.59 Answer: D 141) A rat is placed at random in one of five compartments in a maze. Compartments 1, 2, and 5 have food; so the rat will remain at any of these once reached. If the rat is placed in compartment 3, what is the expected number of times it will be in compartment 4 before it finds food? Round your answer to the nearest hundredth.

1 2 3 4 5

1 1 0 0 0 0

2 0 1 0 0 0

3 0.15 0.13 0.25 0.15 0.32

4 0.18 0.17 0.14 0.12 0.39

5 0 0 0 0 1

A) 0.23 B) 1.17 C) 1.37 D) 0.22 Answer: A

142) A rat is placed at random in one of five compartments in a maze. Compartments 1, 2, and 5 have food; so the rat will remain at any of these once reached. If the rat is placed in compartment 4, what is the expected number of times it will be in compartment 4 before it finds food? Round your answer to the nearest hundredth.

1 2 3 4 5

1 1 0 0 0 0

2 0 1 0 0 0

3 0.11 0.15 0.15 0.15 0.44

4 0.16 0.24 0.14 0.13 0.33

5 0 0 0 0 1

A) 0.19 B) 0.21 C) 1.18 D) 1.21 Answer: C 143) Among gamblers playing the slot machines and winning at a certain time (at a certain casino), the following was determined to be true about one hour later: 50% were still winning; 21% were losing; 10% had switched from the slots to table games (blackjack, roulette, etc.); and 19% had left the casino. Similarly, among gamblers playing the slots and losing at a certain time, the following was determined to be true about one hour later: 29% were winning; 49% were still losing; 15% had switched from the slots to table games; and 7% had left the casino. Among gamblers playing the slots and winning (at that certain time), what percentage eventually switch to table games? Round your answer to the nearest whole number. A) 34% B) 43% C) 39% D) 45% Answer: B 144) Among gamblers playing the slot machines and winning at a certain time (at a certain casino), the following was determined to be true about one hour later: 45% were still winning; 27% were losing; 10% had switched from the slots to table games (blackjack, roulette, etc.); and 18% had left the casino. Similarly, among gamblers playing the slots and losing at a certain time, the following was determined to be true about one hour later: 34% were winning; 47% were still losing; 15% had switched from the slots to table games; and 4% had left the casino. Among gamblers playing the slots and losing (at that certain time), what percentage eventually leave the casino without first trying the table games? Round your answer to the nearest whole number. A) 56% B) 40% C) 46% D) 42% Answer: D

145) Among first-year students enrolled in a two-year program at a trade school, the following was determined to be true: 66% would successfully complete the year and go on to the second year; 17% would repeat the year; and 17% would drop out of the program. Among the second-year students, 76% would successfully complete the year and graduate; 14% would repeat the year; and 10% would drop out of the program. In the long run, what percentage of the second-year students will drop out of the program? Round your answer to the nearest whole number. A) 21% B) 6% C) 16% D) 12% Answer: D 146) Among first-year students enrolled in a two-year program at a trade school, the following was determined to be true: 74% would successfully complete the year and go on to the second year; 19% would repeat the year; and 7% would drop out of the program. Among the second-year students, 78% would successfully complete the year and graduate; 7% would repeat the year; and 15% would drop out of the program. What is the expected number of years that the average first-year student will spend in the program? Round your answer to the nearest hundredth. A) 2.42 years B) 2.22 years C) 2.35 years D) 2.28 years Answer: B 147) Among patients who were improving on a certain day (in the critical care unit of a certain hospital), the following was determined to be true on the next day: 63% were still improving; 10% were stable; 7% were deteriorating; and 20% had been discharged. Among the patients who were stable on a certain day, the following was determined to be true on the next day: 34% were improving; 42% were still stable; 20% were deteriorating; 3% had been discharged; and 1% had died. Among the patients who were deteriorating on a certain day, the following was determined to be true on the next day: 13% were improving; 42% were stable; 41% were still deteriorating; none had been discharged; and 4% had died. What is the expected number of additional days that a patient, who is improving on that certain day, will spend in the critical care unit? Round your answer to the nearest hundredth. A) 4.00 days B) 5.74 days C) 9.21 days D) 7.07 days Answer: D

148) Among patients who were improving on a certain day (in the critical care unit of a certain hospital), the following was determined to be true on the next day: 55% were still improving; 10% were stable; 7% were deteriorating; and 28% had been discharged. Among the patients who were stable on a certain day, the following was determined to be true on the next day: 34% were improving; 48% were still stable; 14% were deteriorating; 3% had been discharged; and 1% had died. Among the patients who were deteriorating on a certain day, the following was determined to be true on the next day: 13% were improving; 42% were stable; 41% were still deteriorating; none had been discharged; and 4% had died. What is the expected number of additional days that a patient, who is stable on that certain day, will spend in the critical care unit? Round your answer to the nearest hundredth. A) 4.05 days B) 5.38 days C) 6.72 days D) 7.49 days Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 149) Consider the absorbing Markov process whose stochastic matrix is shown below. 1 2 3 4 1 1 0 0.2 0 A = 2 0 1 0.5 0.5 3 0 0 0.2 0 4 0 0 0.1 0.5 (a) Identify the matrices S and R. (b) Find the fundamental matrix F. (c) What is the expected number of times the process will be in state 3 if it starts in state 3? (d) What is the expected number of times the process will be in state 4 if it starts in state 3? (e) Compute the expected number of steps before absorption if the process starts in state 3. Answer: (a) S = 0.2 0 , R = 0.2 0 0.5 0.5 0.1 0.5 (b) F = 1.25 0 0.25 2 (c) 1.25

(d) 0.25

(e) 1.5

150) Consider the absorbing Markov process whose stochastic matrix is shown below. 1 2 3 4 1 1 0 0.2 0 A = 2 0 1 0.5 0.5 3 0 0 0.2 0 4 0 0 0.1 0.5 (a) Identify the matrices S and R. (b) Find the stable matrix. (c) What is the probability that the process will be absorbed in state 1 if it starts in state 3? (d) What is the probability that the process will be absorbed in state 2 if it starts in state 3? Answer: (a) S = 0.2 0 , R = 0.2 0 0.5 0.5 0.1 0.5 (b) 1 0 0 0

0 1 0 0

0.25 0.75 0 0

0 1 0 0

(d) 0.75

151) A particle moves every second on the x-axis in one-unit jumps such that its coordinate at any time is 0, 1, 2, or 3. If the particle is at 0 or 3, it does not move. On the other hand, if it is at 1 or 2, it will move to the right with 2 1 probability and to the left with probability . 3 3 (a) Give the stochastic matrix of transitions for this process in standard form. (b) Determine the fundamental matrix F. (c) What is the expected number of times the particle will be at 1 if it starts at 2? (d) What is the expected time before absorption of the particle if it starts at 2? Answer: (a)

0 3 0 1 0

1 1 3

3 0 1

2 3

1 0 0

1 3

2 0 0

2 3

(b) 9 7

F= 6 7

3 7 9 7

(c)

3 7

(d)

12 7

152) A particle moves every second on the x-axis in one-unit jumps such that its coordinate at any time is 0, 1, 2, or 3. If the particle is at 0 or 3, it does not move. On the other hand, if it is at 1 or 2, it will move to the right with 2 1 probability and to the left with probability . 3 3 (a) Give the stochastic matrix of transitions for this process in standard form. (b) Determine the stable matrix. (c) What is the probability that the particle will end up at 0 if it starts at 2? (d) What is the probability that the particle will end up at 0 if it starts at 1? Answer: (a)

0 3 1

1 0 1 0 3 0 3 0 1

2 3

1 0 0

1 3

2 0 0 2 0 3

(c)

1 7

(d)

3 7

(b) 0 3

1 0

3 7

1 7

0 1

4 7

6 7

1 2

0 0 0 0

0 0

153) For the absorbing stochastic matrix given below, determine the matrices S and R. 1 0 0.6 0 0 1 0.2 0.7 0 0 0 0.3 0 0 0.4 0 Answer: S = 0.6 0 0.2 0.7

0.3 R= 0 0.4 0

154) Consider the following absorbing stochastic matrix. I II III IV I 1 0 0.2 0.1 II 0 1 0.5 0.2 III 0 0 0.1 0.4 IV 0 0 0.2 0.3 After a long period of time, what is the probability of an element moving from state IV to state II? Answer:

38 ≈ 0.69 55

155) Consider the following absorbing stochastic matrix. I II III I 1 0 0.2 II 0 1 0.6 III 0 0 0.2 After a long period of time, what is the probability of an element moving from state III to state I? Answer:

1 4

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Suppose that a game has payoff matrix 1 -2 5 . Determine the optimal pure strategies for R and C. 3 4 3 A) R - row 1, C - column 1 B) R - row 2, C - column 1 C) R - row 2, C - column 2 D) R - row 1, C - column 2 E) none of these Answer: B 2) Suppose that a game has payoff matrix

1 2 . Determine the optimal pure strategies for R and C. -1 3

A) R- row 1, C - column 1 B) R - row 2, C - column 2 C) R - row 2, C - column 1 D) R - row 1, C - column 2 E) none, because the game is not strictly determined Answer: A 3) Suppose that a game has payoff matrix -1 -2 . Determine the optimal pure strategies for R and C. 0 3 A) R - row 1, C - column 2 B) R - row 1, C - column 1 C) R- row 2, C - column 1 D) R - row 2, C - column 2 E) none, because the game is not strictly determined Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. State whether or not the game having the given payoff matrix is strictly determined. If so, give the optimal pure strategies and the value of the game. 4) 1 -3 2 4 Answer: strictly determined; R - row 2, C - column 1; 2 5) 2 -1 -1 -4 Answer: strictly determined; R - row 1, C - column 2; -1 6) 4 1 0 5 4 3 -1 0 1 Answer: strictly determined; R - row 2, C - column 3; 3

7) 1 2 -1 5 -3 0 Answer: not strictly determined 8) 0 -2 -1 1 0 3 -3 -2 -2 Answer: strictly determined; R - row 2, C - column 2; 0 Solve the problem. 9) For the following strictly determined game, determine the optimal strategies and the value. 7 0 1 -2 -1 2 3 1 4 Answer: R - row 3; C - column 2; 1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 10) 1 0 is the payoff matrix of a strictly determined game. What is the value of the game? 0 -1 A) -1 B) 0.5 C) 1 D) 0 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 11) For what values of k is the following game strictly determined such that (a) the value of the game is 1? (b) the value of the game is 2? (c) the value of the game is k? -2 1 -5 k 2 3 1 -3 4 Answer: (a) k = 1 (b) 2 ≤ k (c) 1 ≤ k ≤ 2 12) For what value of k is the following game strictly determined such that (a) the value of the game is -1? (b) the value of the game is negative? (c) the value of the game is positive? 3 0 -1 3 2 k -1 -3 -2 Answer: (a) k ≤ -1 (b) k < 0 (c) 0 < k

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2 3 13) 9 8 is the payoff matrix of a strictly determined game. Find a saddle point. 4 7 A) 3 B) 8 C) 9 D) 2 E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. The given matrix is the payoff matrix for a strictly determined game. Find a saddle point. Determine optimum pure strategies for R and C. 14) 6 4 2 3 Answer: 4; R - row 1; C - column 2 15) 2 1 2 3 6 4 0 -1 -2 Answer: 3; R - row 2; C - column 1 16) 1 2 1 1 5 1 0 -8 -1 Answer: 1; multiple entries are saddles, i.e. R - row 1 or 2; C - column 1 or 3 17) 1 -1 -2 0 4 1 Answer: 1; R - row 3; C - column 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) Consider the payoff matrices below, which are the payoff matrices of a strictly determined game? I. 5 3 II. 0 1 III. 4 1 IV. 4 0 V. 1 0 24 23 23 01 -2 -1 A) I and III only B) IV only C) I, II, and III only D) II and IV only E) all of these Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 19) For what values of a is the following game strictly determined? 1 2 3 a Answer: 2 < a < 3 or a > 3 20) In the following game, each player chooses an integer. If the sum is even, R pays C $1; if the sum is odd, C pays R $1. (a) Write the matrix for this game. (b) Is this game strictly determined? Answer:

O E O -1 1 (a) E 1 -1

(b) no

21) (a) Is the following game strictly determined? A = -1 3 1 2 (b) Is the game given by A2 strictly determined? Answer: (a) yes (b) no 22) Suppose that R and C play a game by matching coins. If R shows "heads" and C shows "tails", C pays R $2. If R shows "tails" and C shows "heads", R pays C $1. If they both show "heads", C pays R $1. If they both show "tails", R pays C $2. (a) Give the payoff matrix for this game. (b) Decide if the game is strictly determined. If so, determine the optimal strategies for R and C. Answer: (a)

C H T RH 1 2 T -1 -2 (b) strictly determined; R shows "heads", C shows "heads"

23) Player R has two cards: a red 5 and a black 9. Player C has three cards: a red 7, a black 7, and a black 8. They each place one of their cards on the table. If the cards are the same color, R receives the difference of the two numbers. If the cards are of different colors, the player with the higher number receives the difference of the two numbers. (a) Give the payoff matrix for this game. (b) Decide if the game is strictly determined. If so, determine the optimal strategies for R and C. Answer: (a)

C R7 B7 B8 R R5 2 -2 -3 B9 2 2 1

(b) strictly determined; R plays black 9, C plays black 8

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Set up the payoff matrix and decide on the best strategy. 24) An investor has $22,000 to invest in stocks, either conservative blue-chip or speculative stocks. If the stock market goes up, the speculative stocks could be worth $52,000 while the blue chips would be worth $39,000. If the market goes down, the blue chips would drop to $14,000, while the speculative stocks could drop to $12,000. Where should the investor put his money? A) up down Blue-chip 39,000 12,000 Speculative 52,000 14,000 ; blue-chip stocks B) up down Blue-chip 39,000 12,000 Speculative 52,000 14,000 ; speculative stocks C) up down Blue-chip 39,000 14,000 Speculative 52,000 12,000 ; speculative stocks D) up down Blue-chip 39,000 14,000 Speculative 52,000 12,000 ; blue-chip stocks Answer: D 25) A state is considering conducting an anti-smoking campaign for adults, youth, or both. Its costs, if there is a cause-effect relationship between smoking and cancer, are $751,000 for adults, $815,000 for youth and $1,096,000 for both. If there is no cause-effect relationship, then the cost is $758,000 for youth, $827,000 for adults, and $1,142,000 for both. Whom should the state target in the campaign? A) Cause-effect No Cause-effect Adult $758,000 $751,000 Youth $815,000 $827,000 Both $1,096,000 $1,142,000 ; adults B) Cause-effect No Cause-effect Adult $751,000 $758,000 Youth $827,000 $815,000 Both $1,096,000 $1,142,000 ; youth C) Cause-effect No Cause-effect Adult $751,000 $758,000 Youth $815,000 $827,000 Both $1,096,000 $1,142,000 ; youth D) Cause-effect No Cause-effect Adult $751,000 $758,000 Youth $815,000 $827,000 Both $1,096,000 $1,142,000 ; both Answer: D

26) A student can bring either a calculator or a reference book to an exam. The exam might stress calculations or definitions. If the exam stresses calculations then the student could gain 30 points with a calculator or 18 points with a reference book. If the exam stresses definitions, then a calculator would give the student 39 points but a reference book would give 15 points. Should the student bring a reference book or a calculator to the exam? A) Calculations Definitions Calculator 30 39 Reference book 18 15 ; calculator B) Calculations Definitions Calculator 30 39 Reference book 15 18 ; calculator C) Calculations Definitions Calculator 30 39 Reference book 15 18 ; reference book D) Calculations Definitions Calculator 30 39 Reference book 18 15 ; reference book Answer: A 27) A candidate for office can stress the deficit or nuclear power. The voters can also stress one or the other. If both stress the deficit the candidate will gain 26 points in the polls. If the candidate stresses nuclear power and the voters stress the deficit, the candidate gains 18 points. If both stress nuclear power, the candidate gains 16 points, but if the candidate stresses the deficit while the voters stress nuclear power, the candidate gains 30 points. What should the candidate stress? A) Budget Deficit Nuclear Power Budget Deficit 30 26 Nuclear Power 18 16 ; nuclear power B) Budget Deficit Nuclear Power Budget Deficit 30 26 Nuclear Power 18 16 ; budget deficit C) Budget Deficit Nuclear Power Budget Deficit 26 30 Nuclear Power 18 16 ; budget deficit D) Budget Deficit Nuclear Power Budget Deficit 26 30 Nuclear Power 18 16 ; nuclear power Answer: C

1 -2 . R uses the strategy [ 0.4 0.6 ] and C uses the strategy 0.3 . What is the 0.7 -3 0 expected value of the game? A) -0.90 B) 0.14 C) -0.98 D) 0.54 E) none of these

28) A game has payoff matrix

Answer: C 29) A game has payoff matrix A = 4 -1 . Player R uses strategy [ 0.3 0.7 ] and player C uses strategy 0.5 . What 0.5 -3 2 is the expected value of the game? A) 1.745 B) 0.255 C) 0.10 D) 0.745 E) none of these Answer: C 30) A game has payoff matrix 1 2 . Player C plays column one 10% of the time, and player R plays row one 20% 4 -5 of the time. What is the expected value of the game? A) 0.8 B) -2.9 C) -2.7 D) 0.74 E) none of these Answer: C 31) A game has payoff matrix A = -2 22 . Player R uses strategy [ 0.3 0.7 ] and player C uses strategy 0.6 . 10 -38 0.4 What is the expected value of the game? A) [0.3 0.7] 0.6 0.4 0.6 B) [ 0.3 0.7] 0.4 C) [ 0.3 0.7] A 0.6 0.4 0.6 D) A [0.3 0.7] 0.4 E) none of these Answer: C

32) Game I has payoff matrix

7 -3 . Game II has payoff matrix 13 3 . Suppose the value of game II is 49 , what 13 1 4 -5 -2

is the value of game I ? 127 A) 13 B) -

29 13

C) -

49 13

D) -

43 13

E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 33) Suppose a game has a payoff matrix 1 -1 , R plays [0.7 0.3], and C plays 0.4 . 0 2 0.6 What is the expected value? Answer: 0.22 34) A game has payoff matrix A = 3 -1 . Player R uses strategy [ 0.3 0.7 ] and player C uses strategy 0.5 . The 0.5 -1 4 expected value of the game is Answer: 1.35 2 1 35) Suppose that the payoff matrix of a game is -2 2 and C plays 0.5 . 0.5 0 3 (a) Calculate the expected values if R plays [0.2 0.3 0.5] and if R plays [ 0.6 0 0.4]. (b) Which is better for R? Answer: (a) 1.05 and 1.5 (b) the second play: [0.6 0 0.4] 36) Suppose a game has payoff matrix 3 -1 2 and R plays [ 0.9 0.1]. 2 1 0 0.2 0.7 Which of 0.3 and 0.2 are better for C? 0.5 0.1 Answer: 0.2 0.3 0.5

37) Suppose that a game has payoff matrix 5 -2 . 2 7 Calculate the expected values for the following strategies and determine which of the following situations is most advantageous to R. (a) R plays [0.5 0.5], C plays 0.8 . 0.2 (b) R plays [0.1 0.9], C plays 0.5 . 0.5 (c) R plays [0.6 0.4], C plays 1 . 0 Answer: (a) 3.3 (b) 4.2 (c) 3.8 Situation (b) is most advantageous to R. 38) A game has payoff matrix A = 3 -1 . -1 4 Calculate the expected values for the following strategies and determine which of the following situations is most advantageous to R. (a) R plays [0.5 0.5], C plays 0.8 . 0.2 (b) R plays [0.3 0.7], C plays 0.5 . 0.5 (c) R plays [0.3 0.7], C plays 1 . 0 Answer: (a) 1.1 (b) 1.35 (c) .2 Situation (b) is most advantageous to R. 0 -2 -1 39) Suppose that a game has a payoff matrix -1 1 1 . 1 -1 0 Calculate the expected values for the following strategies and determine which of the following situations is most advantageous to C. 0 (a) R plays [ 0 1 0], C plays 0 . 1 0.5 (b) R plays [0.1 0.2 0.7], C plays 0 . 0.5 0.4 (c) R plays [0.1 0.1 0.8], C plays 0.4 . 0.2 Answer: (a) 1 (b) 0.3 (c) -0.08 Situation (c) is most advantageous to C.

40) Suppose that a game has payoff matrix

1 0 and R plays row one 40% of the time. If C plays column two -2 3

30% of the time, what is expected value? Answer: 0.58 41) R and C play a game by each showing none, one, or two fingers. If each player shows the same number of fingers, R pays C a number equal to the total number of fingers shown. If the two players each show a different number of fingers, C pays R a number equal to the total number of fingers shown. (a) Give the payoff matrix. (b) Suppose R shows one finger 50% of the time and two fingers 50% of the time, and C always shows two fingers. Calculate the expected value for this strategy. Answer:

0 1 2 0 0 1 2 (a) 1 1 -2 3 2 2 3 -4

(b) -0.5

42) R chooses one, two, or three. C chooses one, two, or three. If the sum of the numbers is odd, R receives the amount equal to the sum. If the numbers match, C receives $3. Otherwise, C receives $1. (a) Give the payoff matrix. (b) Suppose R chooses one 30% of the time, two 40% of the time, and three 30% of the time, and C chooses one 40% of the time, two 20% of the time, and three 40% of the time. Calculate the expected value for this strategy. Answer:

1 2 3 1 -3 3 -1 (a) 2 3 -3 5 3 -1 5 -3

(b) 0.56

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 43) A game has payoff matrix 1 3 . To find the optimal strategy for R, one may 4 2 y1 + 3y2 ≤ 1 A) minimize y1 + y2 subject to 4y1 + 2y2 ≥ 1 y1 ≥ 0, y2 ≥ 0 y1 + 3y2 ≥ 1 B) minimize y1 + y2 subject to 4y1 + 2y2 ≥ 1 y1 ≥ 0, y2 ≥ 0 y1 + 4y2 ≤ 1 C) minimize y1 + y2 subject to 3y1 + 2y2 ≤ 1 y1 ≥ 0, y2 ≥ 0 y1 + 4y2 ≥ 1 D) minimize y1 + y2 subject to 3y1 + 2y2 ≥ 1 y1 ≥ 0, y2 ≥ 0 E) none of these Answer: D

44) A game has payoff matrix 2 4 . To find the optimal strategy for C, one may 5 3 2z1 + 4z 2 ≤ 1 A) minimize z 1 + z 2 subject to 5z1 + 3z 2 ≤ 1 z 1 ≥ 0, z2 ≥ 0 2z 1 + 4z2 ≥ 1 B) maximize z 1 + z2 subject to 5z 1 + 3z2 ≥ 1 z 1 ≥ 0, z 2 ≥ 0 2z 1 + 4z 2 ≤ 1 C) maximize z 1 + z2 subject to 5z 1 + 3z 2 ≤ 1 z 1 ≥ 0, z 2 ≥ 0 2z1 + 4z 2 ≥ 1 D) minimize z 1 + z 2 subject to 5z1 + 3z 2 ≥ 1 z 1 ≥ 0, z2 ≥ 0 E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 45) Set up the linear programming problem to find the optimal strategy for R, given the payoff matrix 2 3 7 . 5 1 1 Answer:

Minimize y1 + y2 subject to

y1 ≥ 0, y2 ≥ 0 2y1 + 5y2 ≥ 1 3y1 + y2 ≥ 1 7y1 + y2 ≥ 1

46) Set up the linear programming problem to find the optimal strategy for C, given the payoff matrix -1 2 . 3 1 Answer: Answers may vary. Choosing 2 as the constant to yield a positive-value payoff matrix, the resulting programming problem is: z1 + 4z 2 ≤ 1 Maximize z1 + z 2 subject to 5z1 + 3z 2 ≤ 1 z 1 ≥ 0, z 2 ≥ 0

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the optimal strategies for R and for C for the game whose payoff matrix is given. 47) 3 3 6 1 A) 0 1 ; 1 0 B) 0.4 0.6 ; 1 0 Answer: B 48) -8 8 2 -4 0.27 0.73 B) 0.55 0.45 ; 0.27 0.73

A) 0.45 0.55

Answer: B

49) -10 -2 -3 -1 -6 -4 A) 2 1 , ; 3 3

8 17 9 17 0

B) 2 2 , ; 3 3

8 17 9 17 0

C) 4 13

5 8 , ; 13 13

9 13 0

D) 1 1 , ; 2 2

8 17 9 17 0

Answer: C

50) -7 1 0 -10 -8 -7 2 -3 -1 A) 1 1 , 0, ; 2 2

8 17 9 17 0

B) 5 8 , 0, ; 13 13

4 13 9 13 0

C) 1 2 , 0, ; 3 13

8 17 9 17 0

D) 2 1 , 0, ; 3 13

8 17 9 17 0

Answer: B 51) Consider the game 6 2 . Determine the optimal strategy for R. 1 4 A) 2 5 7 7 B) 3 7

4 7

5 7

2 7

4 7

3 7

Answer: B

52) Consider the game 5 3 . Determine the optimal strategy for R. 1 4 A) 3 2 5 5 B) 4 5

1 5

4 5

2 5

3 5

Answer: A 53) Consider the game 6 2 . Determine the optimal strategy for C. 1 4 A) 5 7 2 7 B) 4 7 3 7 C) 2 7 5 7 D) 3 7 4 7 Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 54) A game has payoff matrix 10 30 . Suppose R plays strategy [0.2 0.8]. 40 20 (a) Determine C's best counterstrategy. (b) If C uses the best counterstrategy, what is the expected value of the game? Answer: (a)

0 1

(b) 22

3 -6 . -5 4 (a) Determine the optimal strategy for R. (b) Determine the optimal strategy for C.

55) A game has payoff matrix

Answer:

(a)

1 2

1 4 (b)

3 4

56) The payoff matrix for a game is 0 -1 . -4 5 (a) Determine the optimal strategy for R. (b) Determine the optimal strategy for C. (c) Whom does the game favor? Answer: 9 (a) R = 10

1 10

3 5

(b) C = 2 5

57) The payoff matrix for a game is 3 5 . 5 2 (a) Determine the optimal strategy for R. (b) Determine the optimal strategy for C. (c) Determine the value of the game. Answer:

(a)

3 5

2 5

3 5 (b)

2 5

(c)

19 5

4 1 . -3 2 (a) Determine the optimal strategy for R. (b) Determine the optimal strategy for C. (c) Determine the value of the game.

58) A game has payoff matrix

Answer:

(a)

5 8

3 8

1 8 (b)

7 8

(c)

11 8

59) Find the optimal strategy for R given the game 0 3 . 2 1 Answer: 1 4

3 4

60) Find the optimal strategy C given the game

7 -2 . -1 8

Answer: 5 9 4 9 61) Find the optimal strategy for R and C for the game with the following payoff matrix. -1 2 3 -2 2 -1 Answer:

strategy for R:

5 8

3 0 ; strategy for C: 8

1 2 1 2

62) Find the optimal strategy for R and C for the game with the following payoff matrix. -1 1 2 2 -1 -2 Answer:

strategy for R:

3 5

2 5

2 ; strategy for C: 3 5 5 0

63) A game is played by two people. R conceals either a $1 bill or a $2 bill in his hand. C guesses 1 or 2 and wins the bill if he guesses its number. (a) Write the payoff matrix for the game. (b) Determine R's optimal strategy. (c) Determine C's optimal strategy. (d) Determine the value of the game. Answer:

(a)

-1 0 0 -2

(b)

2 3

1 3

2 3 (c)

1 3

(d) -

2 3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 64) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. The experts cannot predict inflation for the next year. The company has three plans of action and will implement these at varying percentages of its total operation. The payoff matrix for these three plans is given below, with values given in hundred thousands. What is the market strategy for the company that will yield the best expected value? Above 6% 3-6% Below 3% Plan 1 10 18 17 Plan 2 19 14 16 Plan 3 10 11 12 A) The company should use Plan 1 with probability 4/13, Plan 2 with probability 9/13, and Plan 3 with probability 0. B) The company should use Plan 1 with probability 1, Plan 2 with probability 0, and Plan 3 with probability 0. C) The company should use Plan 1 with probability 0, Plan 2 with probability 1, and Plan 3 with probability 0. D) The company should use Plan 1 with probability 5/13, Plan 2 with probability 8/13, and Plan 3 with probability 0. Answer: D

65) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. Three different experts predict inflation for the next year, but cannot agree. Each predicts one of the three levels. The company's board of directors has three plans of action and will implement one of these. Since the situation has repeated itself for several years, they decided to use game theory to assign probabilities and use these probabilities in making their decision. What assignment of probabilities will maximize their expected value? The payoff matrix below is in hundreds of thousands of dollars. Above 6% 3-6% Below 3% Plan 1 10 18 17 Plan 2 19 14 16 Plan 3 10 11 12 A) The company should use Plan 1 with probability 5/13, Plan 2 with probability 8/13, and Plan 3 with probability 0. B) The company should use Plan 1 with probability 1, Plan 2 with probability 0, and Plan 3 with probability 0. C) The company should use Plan 1 with probability 0, Plan 2 with probability 1, and Plan 3 with probability 0. D) The company should use Plan 1 with probability 4/13, Plan 2 with probability 9/13, and Plan 3 with probability 0. Answer: A 66) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. The experts cannot predict inflation for the next year. The company has three plans of action and will implement these at varying percentages of its total operation. The payoff matrix for these three plans is given below, with values given in hundred thousands. What is the marketing strategy for the company that will yield the best expected value? Above 6% 3-6% Below 3% Plan 1 6 14 13 Plan 2 6 7 8 Plan 3 15 10 12 A) The company should use Plan 1 with probability 1, Plan 2 with probability 0, and Plan 3 with probability 0. B) The company should use Plan 1 with probability 5/13, Plan 2 with probability 0, and Plan 3 with probability 8/13. C) The company should use Plan 1 with probability 0, Plan 2 with probability 1, and Plan 3 with probability 0. D) The company should use Plan 1 with probability 1/3, Plan 2 with probability 1/3, and Plan 3 with probability 1/3. Answer: B

67) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. The experts cannot predict inflation for the next year. The company has three plans of action and will implement these at varying percentages of its total operation. The payoff matrix for these three plans is given below, with values given in hundred thousands. What is the market strategy for the company that will yield the best expected value? Above 6% 3-6% Below 3% Plan 1 9 17 16 Plan 2 9 10 11 Plan 3 18 13 15 A) The company should use Plan 1 with probability 1/3, Plan 2 with probability 1/3, and Plan 3 with probability 1/3. B) The company should use Plan 1 with probability 1, Plan 2 with probability 0, and Plan 3 with probability 0. C) The company should use Plan 1 with probability 5/13, Plan 2 with probability 0, and Plan 3 with probability 8/13. D) The company should use Plan 1 with probability 0, Plan 2 with probability 1, and Plan 3 with probability 0. Answer: C 68) A company has three products that fluctuate in favorability. The payoff matrix for the expected profit in tens of thousands of dollars is given below. What are the percentages of each product that the company should make that would maximize the expected value of the profit? Round to the nearest tenth of a percent, if necessary. Preference Product 1 Product 2 Product 3 Product 1 7 15 14 Product 2 16 11 13 Product 3 7 8 9 A) Product #1: 100%, Product #2: 0%, Product #3: 0% B) Product #1: 0%, Product #2: 100%, Product #3: 0% C) Product #1: 30.8%, Product #2: 69.2%, Product #3: 0% D) Product #1: 38.5%, Product #2: 61.5%, Product #3: 0% Answer: D 69) A company has three products that fluctuate in favorability. The payoff matrix for the expected profit in tens of thousands of dollars is given below. What are the percentages of each product that the company should make that would maximize the expected value of the profit? Round to the nearest tenth of a percent, if necessary. Preference Product 1 Product 2 Product 3 Product 1 10 18 17 Product 2 10 11 12 Product 3 19 14 16 A) Product #1: 0%, Product #2: 100%, Product #3: 0% B) Product #1: 33.3%, Product #2: 33.3%, Product #3: 33.3% C) Product #1: 38.5%, Product #2: 0%, Product #3: 61.5% D) Product #1: 100%, Product #2: 0%, Product #3: 0% Answer: C

70) A person is considering three different stocks, and each is sensitive to a certain economic indicator. The indicator will be positive, neutral, or negative, and fluctuate randomly. The payoffs are given in the table below, in thousands of dollars. What investment strategy should the person make to obtain the best expected value of profit? Payoff (in thousands of dollars) Positive Neutral Negative Stock A 8 16 15 Stock B 17 12 14 Stock C 8 9 10 A) Invest in Stock A with probability 0, invest in Stock B with probability 1, and invest in Stock C with probability 0. B) Invest in Stock A with probability 1, invest in Stock B with probability 0, and invest in Stock C with probability 0. C) Invest in Stock A with probability 4/13, invest in Stock B with probability 9/13, and invest in Stock C with probability 0. D) Invest in Stock A with probability 5/13, invest in Stock B with probability 8/13, and invest in Stock C with probability 0. Answer: D 71) A person has hired an investment broker to buy stock. The broker has three different stock funds that are of interest, but each is sensitive to a certain economic indicator that is impossible to predict. The indicator will be positive, neutral, or negative. The table below shows the payoffs in thousands of dollars. Find the strategy that the broker should recommend to maximize the expected value of the investment. Payoff (in thousands of dollars) Positive Neutral Negative Stock A 7 15 14 Stock B 16 11 13 Stock C 7 8 9 A) Invest in Stock A with probability 1, invest in Stock B with probability 0, and invest in Stock C with probability 0. B) Invest in Stock A with probability 0, invest in Stock B with probability 1, and invest in Stock C with probability 0. C) Invest in Stock A with probability 4/13, invest in Stock B with probability 9/13, and invest in Stock C with probability 0. D) Invest in Stock A with probability 5/13, invest in Stock B with probability 8/13, and invest in Stock C with probability 0. Answer: D

72) A person is considering three different stocks, and each is sensitive to a certain economic indicator. The indicator will be positive, neutral, or negative, and fluctuate randomly. The payoffs are given in the table below, in thousands of dollars. What investment strategy should the person make to obtain the best expected value of profit? Payoff (in thousands of dollars) Positive Neutral Negative Stock A 10 18 17 Stock B 10 11 12 Stock C 19 14 16 A) Invest in Stock A with probability 0, invest in Stock B with probability 1, and invest in Stock C with probability 0. B) Invest in Stock A with probability 1, invest in Stock B with probability 0, and invest in Stock C with probability 0. C) Invest in Stock A with probability 1/3, invest in Stock B with probability 1/3, and invest in Stock C with probability 1/3. D) Invest in Stock A with probability 5/13, invest in Stock B with probability 0, and invest in Stock C with probability 8/13. Answer: D 73) A person has hired an investment broker to buy stock. The broker has three different stock funds that are of interest, but each is sensitive to a certain economic indicator that is impossible to predict. The indicator will be positive, neutral, or negative. The table below shows the payoffs in thousands of dollars. Find the strategy that the broker should recommend to maximize the expected value of the investment. Payoff (in thousands of dollars) Positive Neutral Negative Stock A 10 18 17 Stock B 10 11 12 Stock C 19 14 16 A) Invest in Stock A with probability 1, invest in Stock B with probability 0, and invest in Stock C with probability 0. B) Invest in Stock A with probability 1/3, invest in Stock B with probability 1/3, and invest in Stock C with probability 1/3. C) Invest in Stock A with probability 5/13, invest in Stock B with probability 0, and invest in Stock C with probability 8/13. D) Invest in Stock A with probability 0, invest in Stock B with probability 1, and invest in Stock C with probability 0. Answer: C

Solve the problem. Assume that probabilities are representative of the best strategies. Round to two decimal places, if necessary. 74) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. The experts cannot predict inflation for the next year. The company has three plans of action and will implement these at varying percentages of its total operation. The payoff matrix for these three plans is given below, with values given in hundred thousands. What is the expected value?

Plan 1 Plan 2 Plan 3 A) 42 B) 46 C) 9.78 D) 14.54

Above 6% 9 18 9

3-6% 17 13 10

Below 3% 16 15 11

Answer: D 75) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. Three different experts predict inflation for the next year, but cannot agree. The company's board of directors has three plans of action and will implement one of these. Since the situation has repeated itself for several years, they decide to use game theory to assign probabilities and use these probabilities in making their decision. What is the expected value? The payoff matrix below is in hundreds of thousands of dollars.

Plan 1 Plan 2 Plan 3 A) 8.44 B) 40 C) 12.54 D) 36

Above 6% 7 16 7

3-6% 15 11 8

Below 3% 14 13 9

Answer: C 76) A company has three products that fluctuate in favorability. The payoff matrix for the expected profit in tens of thousands of dollars is given below. What is the expected value of the profit? Preference Product 1 Product 2 Product 3 Product 1 7 15 14 Product 2 16 11 13 Product 3 7 8 9 A) 12.54 B) 36 C) 8.44 D) 40 Answer: A

77) A person is considering three different stocks, and each is sensitive to a certain economic indicator. The indicator will be positive, neutral, or negative, and fluctuate randomly. The payoffs are given in the table below, in thousands of dollars. What is the expected value of profit? Payoff (in thousands of dollars) Positive Stock A 5 Stock B 14 Stock C 5 A) 34 B) 5.18 C) 10.54 D) 30

Neutral Negative 13 12 9 11 6 7

Answer: C 78) A person has hired an investment broker to buy stock. The broker has three different stock funds that are of interest, but each is sensitive to a certain economic indicator that is impossible to predict. The indicator will be positive, neutral, or negative. The table below shows the payoffs in thousands of dollars. Find the expected value of the investment. Payoff (in thousands of dollars) Positive Stock A 9 Stock B 18 Stock C 9 A) 9.18 B) 42 C) 46 D) 14.54

Neutral Negative 17 16 13 15 10 11

Answer: D 79) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. The experts cannot predict inflation for the next year. The company has three plans of action and will implement these at varying percentages of its total operation. The payoff matrix for these three plans is given below, with values given in hundred thousands. What is the expected value?

Plan 1 Plan 2 Plan 3 A) 36 B) 12.54 C) 7.18 D) 40

Above 6% 7 7 16

3-6% 15 8 11

Below 3% 14 9 13

Answer: B

80) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. Three different experts predict inflation for the next year, but cannot agree. The company's board of directors has three plans of action and will implement one of these. Since the situation has repeated itself for several years, they decide to use game theory to assign probabilities and use these probabilities in making their decision. What is the expected value? The payoff matrix below is in hundred of thousands of dollars.

Plan 1 Plan 2 Plan 3 A) 10.18 B) 15.54 C) 45 D) 49

Above 6% 10 10 19

3-6% 18 11 14

Below 3% 17 12 16

Answer: B 81) A company has three products that fluctuate in favorability. The payoff matrix for the expected profit in tens of thousands of dollars is given below. What is the expected value of the profit? Preference Product 1 Product 2 Product 3 A) 6.18 B) 37 C) 33 D) 11.54

Product 1 Product 2 Product 3 6 14 13 6 7 8 15 10 12

Answer: D 82) A person is considering three different stocks, and each is sensitive to a certain economic indicator. The indicator will be positive, neutral, or negative, and fluctuate randomly. The payoffs are given in the table below, in thousands of dollars. What is the expected value of profit? Payoff (in thousands of dollars) Positive Stock A 5 Stock B 5 Stock C 14 A) 5.18 B) 30 C) 10.54 D) 34

Neutral Negative 13 12 6 7 9 11

Answer: C

83) A person has hired an investment broker to buy stock. The broker has three different stock funds that are of interest, but each is sensitive to a certain economic indicator that is impossible to predict. The indicator will be positive, neutral, or negative. The table below shows the payoffs in thousands of dollars. Find the expected value of the investment. Payoff (in thousands of dollars) Positive Stock A 9 Stock B 9 Stock C 18 A) 42 B) 14.54 C) 46 D) 9.18

Neutral Negative 17 16 10 11 13 15

Answer: B

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) $4000 deposited at 6% interest compounded monthly will grow to $13,240.82 in 20 years. What is the i, n, P, and F for this situation? A) 0.05; 240; $13,240.82; $4000 B) 0.01; 20; $4000; $13,240.82 C) 0.005; 240; $4000; $13,240.82 D) 6%; 20; $4000; $13,240.82 E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Express percents as decimals. Round dollar amounts to the nearest cent. 2) Determine i, n, and P for $500 invested at 6% compounded quarterly for three years. Answer: i = 0.015, n = 12, P = 500 3) $5000 deposited at 3% interest compounded monthly will grow to $12,284.21 in 30 years. Determine the i, n, P, and F for this situation. Answer: 0.0025; 360; $5000; $12,284.21 Solve the problem. 4) Consider the following savings account passbook: Date 4/1/12

1/1/12 7/1/12 Deposit $1000.00 Withdrawal Interest $15.00 $15.23 Balance $1000.00 $1015.00 $1030.23 (a) What annual interest rate is this bank paying? (b) Give the interest and balance on 10/1/12. (c) Give the balance on 1/1/15. Answer: (a) 6% (b) interest: $15.45; balance: $1045.68 (c) $1195.62 Express percents as decimals. Round dollar amounts to the nearest cent. 5) Which would be the better investment: 6% interest compounded annually or 5.9% compounded daily? Answer: 5.9% compounded daily has an effective rate of approximately 6.08% and is the better investment 6) Which would be the better investment: 4.1% interest compounded annually or 3.89% compounded daily? Answer: 3.89% compounded daily has an effective rate of approximately 3.97% so 4.1% compounded annually is the better investment MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) Is it more profitable to receive $9000 now or $23,000 in 12 years? Assume that money can earn 8% interest compounded quarterly. A) $9000 now B) $23,000 in 12 years Answer: A 1

8) Is it more profitable to receive $5000 now or $8400 is 13 years? Assume that money can earn 4% interest compounded quarterly. A) $5000 now B) $8400 in 13 years Answer: B 9) Is it more profitable to receive $2000 now or $3400 in 9 years? Assume that money can earn 6% interest compounded semiannually. A) $3400 in 9 years B) $2000 now Answer: B 10) Is it more profitable to receive $20,000 now or $30,000 in 10 years? Assume that money can earn 5% interest compounded semiannually. A) $30,00 in 10 years B) $20,000 now Answer: B Find the compound amount for the deposit. Round to the nearest cent. 11) $7000 at 6% compounded annually for 15 years A) $12,880.00 B) $13,300.00 C) $15,826.33 D) $16,775.91 Answer: D 12) $19,000 at 7% compounded semiannually for 12 years A) $28,710.30 B) $42,791.64 C) $34,960.00 D) $43,383.24 Answer: D 13) $1100 at 5% compounded quarterly for 4 years A) $1341.88 B) $1337.06 C) $1320.00 D) $1156.04 Answer: A 14) $1880 at 7.9% compounded annually for 15 years A) $4107.80 B) $5450.77 C) $3959.28 D) $5881.38 Answer: D

15) $2520 at 7.3% compounded semiannually for 10 years A) $4359.60 B) $5097.98 C) $3606.56 D) $5161.63 Answer: D Find the amount of compound interest earned. 16) $7000 at 5% compounded annually for 3 years A) $1,505,000.00 B) $1103.38 C) $1126.38 D) $15,050.00 Answer: B 17) $9000 at 4% compounded semiannually for 5 years A) $936.73 B) $4322.20 C) $1970.95 D) $1949.88 Answer: C 18) $6500 at 6% compounded quarterly for 4 years A) $3860.01 B) $10,012.29 C) $398.86 D) $1748.41 Answer: D 19) $9000 at 4.5% compounded monthly for 4 years A) $413.46 B) $135.76 C) $1732.67 D) $1771.33 Answer: D Find the interest rate for each deposit and compound amount. 20) $6000 accumulating to $8414.58, compounded quarterly for 8 years. A) 3.75% B) 4.5% C) 4.25% D) 4.75% Answer: C 21) $9800 accumulating to $13,055.32, compounded monthly for 5 years. A) 5.9% B) 5.5% C) 5% D) 5.75% Answer: D

Find the amount that should be invested now to accumulate the following amount, if the money is compounded as indicated. 22) $8500 at 6% compounded annually for 9 years. A) $14,360.57 B) $5333.01 C) $3468.86 D) $5031.14 Answer: D 23) $23,400 at 7% compounded annually for 15 years A) $64,561.34 B) $9074.92 C) $8481.24 D) $14,918.76 Answer: C 24) $7000 at 5.7% compounded semiannually for 11 years A) $12,989.50 B) $3804.29 C) $3227.72 D) $3772.28 Answer: D 25) $49,000 at 6.2% compounded semiannually for 3 years A) $8201.51 B) $40,909.35 C) $40,798.49 D) $58,850.22 Answer: C 26) $4900 at 5.7% compounded quarterly for 9 years A) $1955.74 B) $2944.26 C) $8154.86 D) $2975.23 Answer: B Solve the problem. 27) If inflation is 4% a year compounded annually, what will it cost in 8 years to buy a house currently valued at $60,000? A) $78,955.91 B) $85,398.71 C) $82,114.14 D) $70,299.56 Answer: C

28) June made an initial deposit of $6100 in an account for her son. Assuming an interest rate of 2% compounded quarterly, how much will the account be worth in 13 years? A) $7866.82 B) $7906.15 C) $6979.33 D) $7901.06 Answer: B 29) A small company borrows $20,000 at 4% compounded monthly. The loan is due in 4 years. How much interest will the company pay? A) $23,463.97 B) $3386.02 C) $3451.57 D) $3463.97 Answer: D 30) John Lee's savings account has a balance of $1717. After 2 years, what will the amount of interest be at 2% compounded semiannually? A) $69.37 B) $68.68 C) $69.72 D) $34.51 Answer: C 31) Andrea Gilford's savings account has a balance of $94. After 2 years, what will the amount of interest be at 3% compounded quarterly? A) $1.41 B) $-3.21 C) $10.79 D) $5.79 Answer: D 32) Barry Newman's savings account has a balance of $769. After 5 years, what will the amount of interest be at 5% compounded annually? A) $217.46 B) $212.46 C) $203.46 D) $384.50 Answer: B 33) Nora Oretega's savings account has a balance of $4237. After 10 years, what will the amount of interest be at 3% compounded semiannually? A) $1271.10 B) $1474.62 C) $1469.62 D) $1460.62 Answer: C

34) Felipe Rivera's savings account has a balance of $4969. After 2 years what will the amount of interest be at 5% compounded quarterly? A) $124.23 B) $519.19 C) $524.19 D) $510.19 Answer: B 35) Barbara knows that she will need to buy a new car in 4 years. The car will cost $15,000 by then. How much should she invest now at 3%, compounded quarterly, so that she will have enough to buy a new car? A) $14,132.76 B) $13,309.76 C) $12,939.13 D) $13,727.12 Answer: B 36) Southwest Dry Cleaners believes that it will need new equipment in 8 years. The equipment will cost $26,000. What lump sum should be invested today at 4% compounded semiannually, to yield $26,000? A) $22,275.40 B) $23,886.38 C) $22,164.57 D) $18,939.59 Answer: D 37) How many months are required for $1500 to grow to $1650 at 5% simple interest? A) 36 B) 20 C) 18 D) 24 E) none of these Answer: D 38) How many months are required for $2500 to grow to $3300 at 4% simple interest? A) 72 B) 8 C) 100 D) 96 E) none of these Answer: D 39) How many years are required for $2500 to grow to $5,000 at 2% simple interest? A) 45.5 B) 55 C) 50 D) 36.5 E) none of these Answer: C

40) Find the present value of $5000 payable in three years at 3.5% simple interest. A) $3203.29 B) $5525 C) $4310.34 D) $4524.89 E) none of these Answer: D 41) Find the present value of $10,000 payable in five years at 6% simple interest. A) $7,378.38 B) $7,692.31 C) $4,310.34 D) $3,203.29 E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Express percents as decimals. Round dollar amounts to the nearest cent. 42) Find the present value of $4000 in two years at 4% simple interest. Answer: $3703.70 43) Compute the compound amount after 1 year for $100 invested at 5% interest compounded quarterly. What simple interest rate will yield the same amount in 1 year? Answer: 5.09% MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the simple interest. Assume a 360-day year. Round results to the nearest cent. 44) $7126 at 6% for 11 months A) $391.93 B) $356.30 C) $427.56 D) $395.22 Answer: A 45) $10,896 at 3.5% for 10 months A) $286.02 B) $317.80 C) $349.58 D) $320.47 Answer: B 46) $52,988 at 5.5% for 15 months A) $3642.93 B) $3673.54 C) $3885.79 D) $3400.06 Answer: A

47) $16,000 at 5% for 96 days A) $800.00 B) $211.11 C) $213.33 D) $16,213.33 Answer: C 48) $2720 at 5% for 343 days A) $129.58 B) $136.00 C) $129.20 D) $2849.58 Answer: A 49) $24,000 at 1.25% for 77 days A) $64.17 B) $63.33 C) $24,064.17 D) $300.00 Answer: A 50) $28,120 at 3.75% for 104 days A) $28,424.63 B) $1054.50 C) $304.63 D) $301.70 Answer: C Find the future value and the amount of simple interest earned. Round to the nearest cent. 51) $3549 at 3.9% for 5 months A) $3618.21; $69.21 B) $3606.67; $57.67 C) $3607.16; $58.16 D) $3595.14; $46.14 Answer: B 52) $11,025 at 3.4% for 4 months A) $11,118.71; $93.71 B) $11,151.00; $126.00 C) $156.19; $11,181.19 D) $11,149.95; $124.95 Answer: D Solve the problem. 53) If $4800 earned simple interest of $190.40 in 7 months, what was the simple interest rate? A) 6.8% B) 8.8% C) 5.8% D) 7.8% Answer: A

54) If $23,200 earned simple interest of $812.00 in 6 months, what was the simple interest rate? A) 7.6% B) 6.2% C) 6% D) 7% Answer: D Find the effective rate corresponding to the given nominal rate. Round to the nearest hundredth. 55) 7% compounded semiannually A) 7.00% B) 7.19% C) 7.12% D) 7.23% Answer: C 56) 3% compounded quarterly A) 3.00% B) 3.03% C) 3.02% D) 3.04% Answer: B 57) Calculate the effective rate for 4.4% interest compounded monthly. Round to the nearest hundredth of a percent. A) 4.6% B) 4.30% C) 4.1% D) 4.49% E) none of these Answer: D Calculate the nominal rate of interest corresponding to the given effective rate. 58) 3.6% with interest compounded quarterly. A) ≈ 3.57% B) ≈ 3.55% C) ≈ 3.75% D) ≈ 3.48% Answer: B 59) 4.89% with interest compounded monthly. A) ≈ 4.88% B) ≈ 4.78% C) ≈ 4.85% D) ≈ 4.76% Answer: B

60) Bill needs $8000 to buy a new car in five years. How much should he deposit at the end of every quarter into an account that earns 8% interest compounded quarterly? A) $489.25 B) $329.25 C) $345.97 D) $666.33 E) none of these Answer: B 61) Jane needs $9000 to buy a new car in four years. How much should she deposit at the end of each month into an account that earns 6% interest compounded monthly? A) $166.37 B) $129.38 C) $211.37 D) $147.00 E) none of these Answer: A 62) Jane needs $30,000 to buy another new car in eight years. How much should she deposit at the end of each half year into an account that earns 4% interest compounded semiannually? A) $1263.36 B) $1663.37 C) $160.95 D) $1609.50 E) none of these Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 63) Explain why each of the following annuities is not an ordinary annuity. (a) $100 is deposited at the beginning of each month for eight years into an account paying 5% interest compounded quarterly. (b) $100 is deposited at the beginning of each month for eight years into an account paying 6% interest compounded monthly. Answer: (a) The payment period is different from the interest period. (b) Payments are made at the beginning of the interest period. 64) Specify i, n, R, and F for the following annuity. At the end of each quarter, $100 is deposited into an account paying 4% interest compounded quarterly. The balance after five years will be $2201.90. Answer: i = 0.01, n= 20, R = 100, F = 2201.90 65) Calculate the future value of an annuity of $200 per year for 10 years at 6% interest compounded annually. Answer: $2636.16 66) Calculate the rent of an annuity at 4% compounded semiannually if payments are made every half year and the future value after 12 years is $30,000. Answer: $986.13

67) What should the quarterly payments be for an annuity valued at $10,000 4 years from now if interest is 8% compounded quarterly? Answer: $536.50 68) How many monthly payments of $100 must be planned to have a future value of $1878.58 when the interest is 6% compounded monthly? Answer: 18 69) On her eighth birthday, a girl inherits $15,000 that is to be used for her college education. The money is deposited into a trust fund that will pay her R dollars on her 17th, 18th, 19th, and 20th birthdays. Find R if the money earns 5% compounded annually. Answer: $6249.90 70) Fifty dollars is deposited in the bank at the end of each year for 15 years. During the first eight years, the bank paid 5% compounded annually and after that paid 6% interest compounded annually. Find the amount in the account after 15 years. Answer: $1137.60 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the interest rate needed to accumulate the following amounts in a sinking fund, with monthly payments as given. 71) Accumulate $148,000, monthly payments of $900 over 11 years. A) 2.35% B) 3.05% C) 3.85% D) 4.55% Answer: C Find the amount of each payment to be made into a sinking fund so that enough will be present to accumulate the following amount. Payments are made at the end of each period. The interest rate given is per period. 72) $7000; money earns 7% compounded annually; 10 annual payments A) $506.64 B) $443.50 C) $249.09 D) $584.41 Answer: A 73) $79,000; money earns 5% compounded semiannually for 15 years A) $1799.43 B) $1116.44 C) $913.94 D) $1267.60 Answer: A 74) $71,000; money earns 5% compounded quarterly for 4 years A) $4036.12 B) $1548.85 C) $942.91 D) $1003.38 Answer: A 11

75) $62,000; money earns 4.8% compounded monthly for 1

5 years 12

A) $804.74 B) $1279.76 C) $758.49 D) $3531.75 Answer: D Solve the problem. Round to the nearest cent. 76) If Bob deposits $5000 at the end of each year for 7 years in an account paying 6% interest compounded annually, find the final amount he will have on deposit. A) $36,969.19 B) $49,487.34 C) $41,969.19 D) $34,876.59 Answer: C 77) Sandra deposits $3000 at the end of each semiannual period for 24 years at 5% interest compounded semiannually. Find the final amount she will have on deposit. A) $269,578.75 B) $260,003.66 C) $272,578.75 D) $140,181.30 Answer: C 78) $765.13 is deposited at the end of each month for 4 years in an account paying 2% interest compounded monthly. Find the final amount of the account. A) $36,609.59 B) $37,437.01 C) $38,202.14 D) $38,967.27 Answer: C 79) Lou has an account with $10,000 which pays 7% interest compounded annually. If to that account, Lou deposits $5000 at the end of each year for 3 years, find out the amount in the account after the last deposit. A) $28,324.93 B) $16,074.50 C) $26,074.50 D) $28,174.50 Answer: A 80) At the end of every 3 months, Teresa deposits $100 into an account that pays 4% compounded quarterly. After 4 years, she puts the accumulated amount into a certificate of deposit paying 8.5% compounded semiannually for 1 year. When this certificate matures, how much will Teresa have accumulated? A) $1805.44 B) $1875.60 C) $1866.61 D) $2031.64 Answer: B

81) If $900,000 is to be saved over 20 years, how much should be deposited monthly if the investment earns 9.25% interest compounded monthly? A) $1090.27 B) $1305.30 C) $914.59 D) $769.94 Answer: B 82) If $100,000 is to be saved over 30 years, how much should be deposited annually if the investment earns 5.25% interest compounded annually? A) $2174.34 B) $1190.03 C) $1761.13 D) $1441.69 Answer: D 83) Mary needs $9000 in 7 years. What amount can she deposit at the end of each quarter at 5% interest compounded quarterly so she will have her $9000? A) $282.30 B) $1105.38 C) $270.44 D) $259.40 Answer: C 84) If $700,000 is to be saved over 14 years, how much should be deposited monthly if the investment earns 5% interest compounded monthly? A) $2091.52 B) $2885.43 C) $2316.45 D) $2578.03 Answer: B 85) If $100,000 is to be saved over 25 years, how much should be deposited annually if the investment earns 7.5% interest compounded annually? A) $1833.88 B) $1471.07 C) $1189.18 D) $2309.22 Answer: B 86) Find the amount of each payment into a sinking fund if $8000 must be accumulated. Payments are made at the end of each quarter for 3 years, with interest of 8% compounded quarterly. A) $421.60 B) $596.48 C) $474.24 D) $787.44 Answer: B

87) Green Thumb Landscaping wants to build a $116,000 greenhouse in 2 years. The company sets up a sinking fund with payments made quarterly. Find the payment into this fund if the money earns 4% compounded quarterly. A) $8173.36 B) $6113.20 C) $11,417.88 D) $14,000.04 Answer: D 88) How much should be deposited semiannually into a sinking fund over 5 years to accumulate $107,000 if the money earns 5% compounded semiannually? A) $9550.82 B) $9333.61 C) $6423.21 D) $18,239.22 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 89) What is the present value of an annuity of $150 per month for three years if the interest is 3% compounded monthly? Answer: $5157.97 90) Calculate the present value of an annuity of $300 per quarter for nine years at 6% interest compounded quarterly. Answer: $8298.21 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 91) How much money must you deposit now at 6% interest compounded quarterly in order to be able to withdraw $6000 at the end of each quarter for five years? A) $154,517.75 B) $68,819.53 C) $138,742.00 D) $103,011.83 E) none of these Answer: D 92) If you deposit $2000 into a fund paying 4% interest compounded monthly, how much can you withdraw at the end of each month for one year? A) $177.48 B) $189.12 C) $170.30 D) $153.36 E) none of these Answer: C

93) If you deposit $5000 into a fund paying 6% interest compounded quarterly, how much can you withdraw at the end of each quarter for five years? A) $970.45 B) $291.23 C) $216.23 D) $215.17 E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 94) Specify i, n, R, and P for the following annuity. $36,029.38 is deposited into an account paying 6% interest compounded monthly. At the end of each month, $400 is withdrawn from the account for 10 years. Answer: i =

1 % = 0.005, n = 120, R = 400, P = 36,029.38 2

95) How much money must you deposit now at 6% interest compounded monthly in order to be able to withdraw $200 at the end of each month for 10 years? Answer: $18,014.69 96) Calculate the rent of a decreasing annuity at 8% compounded quarterly if payments are made every quarter-year for 7 years and the present value is $100,000. Answer: $4698.97 97) Calculate the rent of a decreasing annuity at 4% compounded semiannually if payments are made every half year for 12 years and the present value is $30,000. Answer: $1586.13 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 98) A loan of $8000 is to be repaid with monthly payments for three years at 3.5% interest compounded monthly. .Find the unpaid balance after one year. A) $1555.56 B) $1656.56 C) $5156.32 D) $5425.44 E) none of these Answer: D 99) You buy a car with a down payment of $300 and payments of $200 per month for 3 years. If the interest rate is 4% compounded monthly, what is the total cost of the car? A) $8615.38 B) $5532.14 C) $6574.20 D) $7074.15 E) none of these Answer: D

100) A mortgage at 2.8% interest compounded monthly with a monthly payment of $500 has an unpaid balance of $5035.56 after 109 months. Find the unpaid balance after 110 months. A) $4547.31 B) $4550 C) $4575.82 D) $425 E) none of these Answer: A 101) How much money can you borrow at 4.8% interest compounded monthly if you agree to pay $200 at the end of each month for three years and, in addition, a balloon payment of $2000 at the end of the third year? A) $9692.18 B) $8021.50 C) $8693.18 D) $9002.28 E) none of these Answer: D 102) A mortgage at 1.2% interest compounded monthly with a monthly payment of $600 has an unpaid balance of $6753 after 100 months. Find the unpaid balance after 101 months. A) $6159.75 B) $535 C) $6085.47 D) $502.50 E) none of these Answer: A 103) How much money can you borrow at 6% interest compounded monthly if you agree to pay $100 at the end of each month for 2 years and, in addition, a balloon payment of $3000 at the end of the second year? A) $4917.85 B) $2261.56 C) $2256.29 D) $5356.29 E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 104) A loan of $10,000 is to be repaid with monthly payments for five years at 6% interest compounded monthly. Calculate the monthly payment Answer: $193.33 105) A loan has a term of six months at 2.4% interest compounded monthly. If the payments are $115.61, what is the principal? Answer: $688.83 106) A loan is to be amortized over a six year term at 3.6% interest compounded semiannually with payments of $500 every 6 months, and a balloon payment of $2000 at the end of the term. What is the principal amount of this loan? Answer: $6967.77 16

107) Suppose a loan has an interest rate of 2.4% compounded monthly for eight months. Given that the monthly payment is $220, calculate the unpaid balance after five months. Answer: $657.37 108) A car loan of $8500 is to be amortized over two years at 2.9% interest compounded monthly. How much is the monthly payment? Answer: $364.96 109) On December 31, 2005, a house was purchased with the buyer taking out a 30-year $90,000 mortgage at 3% interest compounded monthly. The mortgage payments are made at the end of each month. Calculate: (a) the unpaid balance of the loan on December 31, 2015, just after the 120th payment. (b) the interest that will be paid during January 2016. Answer: (a) $68,417.83 (b) $171.04 110) On December 31, 2005, a house was purchased with the buyer taking out a 30-year $90,000 mortgage at 3% interest compounded monthly. The mortgage payments are made at the end of each month. (a) How much of the principal will be paid off during the year 2041? (b) How much interest will be paid during the year 2041? Answer: (a) $3994.06 (b) $559.22 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the payment necessary to amortize the loan. 111) $7500; 6% compounded annually; 7 annual payments A) $1356.63 B) $1525.23 C) $1343.51 D) $1207.77 Answer: C 112) $40,000; 8% compounded annually; 10 annual payments A) $5603.03 B) $5960.27 C) $6403.18 D) $5961.16 Answer: D 113) $7000; 7% compounded semiannually; 10 semiannual payments A) $846.56 B) $1090.73 C) $841.69 D) $784.60 Answer: C Find the monthly house payment necessary to amortize the following loan. 114) $40,000 at 5.31% for 30 years A) $5.90 B) $241.12 C) $177.00 D) $222.37 Answer: D 17

115) $102,000 at 6.6% for 15 years A) $894.15 B) $1047.47 C) $561.01 D) $7103.03 Answer: A 116) $465,000 at 6.7% for 15 years A) $32,402.65 B) $4101.95 C) $4800.99 D) $2596.27 Answer: B Suppose that in the loan described, the borrower made a larger payment, as indicated. Calculate (a) the time needed to pay off the loan, (b) the total amount of the payments, and (c) the amount of interest saved, compared with the original loan and payments. 117) $90,000; 6% compounded annually; 12 annual payments; with larger payment of $14,000. A) (a) 9 years (b) $117,174.69 (c) $11,644.47 B) (a) 8 years (b) $105,530.22 (c) $10,966.47 C) (a) 8 years (b) $112,000 (c) $20,409.47 D) (a) 9 years (b) $126,000 (c) $10,410.47 Answer: A 118) $140,000; 8% compounded quarterly; 15 quarterly payments; with larger payment of $20,000. A) (a) 8 quarters (b) $160,000 (c) $16,715.32 B) (a) 8 quarters (b) $146,841.23 (c) $16,592.32 C) (a) 7 quarters (b) $130,248.91 (c) $16,560.32 D) (a) 7 quarters (b) $140,000 (c) $16,657.32 Answer: B

119) $7400; 6.2% compounded semiannually; 18 semiannual payments; with larger payment of $910. A) (a) 9 semiannual periods (b) $8190 (c) $1150.15 B) (a) 10 semiannual periods (b) $9100 (c) $1176.15 C) (a) 9 semiannual periods (b) $7556.50 (c) $1097.15 D) (a) 10 semiannual periods (b) $8661.65 (c) $1105.15 Answer: D Solve the problem. 120) You want to take out a loan to buy a new car for which you need to finance $20,020. Your bank will give you a loan at 4% compounded monthly. You look at your budget and decide that you can afford a payment of $229 a month. How many years, to the nearest tenth of a year, must the loan be taken out to meet these conditions? A) 12.1 years B) 8.6 years C) 18.1 years D) 5.2 years Answer: B 121) In order to purchase a home, a family borrows $75,000 at an annual interest rate of 6.35%, to be paid back over a 20 year period in equal monthly payments. What is their monthly payment? A) $608.71 B) $552.58 C) $396.88 D) $19.84 Answer: B 122) In order to purchase a home, a family borrows $589,000 at 8.3% for 15 yr. What is their monthly payment? Round the answer to the nearest cent. A) $41,474.36 B) $4073.92 C) $5731.27 D) $6614.71 Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 123) Ken is 60 years old, is in the 26% marginal tax bracket, and has $333,000 in his traditional IRA. How much money will he have after taxes if he withdraws all the money from the account? Answer: $246,420 124) Ken is 60 years old, is in the 45% marginal tax bracket, and has $300,000 in his traditional IRA. How much money will he have after taxes if he withdraws all the money from the account? Answer: $165,000; With the Roth IRA, all taxes were paid before investing.

125) If you are 18 years old, deposit $3000 each year into a traditional IRA for 47 years at 5% interest compounded annually, and retire at age 65, how much money will be in the account upon retirement? Answer: $534,358.27 126) Ken is 60 years old, is in the 26% marginal tax bracket, and has $333,000 in his Roth IRA. How much money will he have after taxes if he withdraws all the money from the account? Answer: $333,000; With the Roth IRA, all taxes were paid before investing. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. Round to the nearest cent. 127) Larry wants to start an IRA that will have $740,000 in it when he retires in 24 years. How much should he invest semiannually in his IRA to do this if the interest is 5% compounded semiannually? A) $22,875.49 B) $8144.44 C) $8131.44 D) $5364.05 Answer: B 128) Joan wants to start an IRA that will have $250,000 in it when she retires in 26 years. How much should she invest annually in her IRA to do this if the interest is 4% compounded annually? A) $5641.85 B) $20,339.22 C) $15,332.82 D) $4089.77 Answer: A 129) Mark wants to start an IRA that will have $250,000 in it when he retires in 25 years. How much should he invest quarterly in his IRA to do this if the interest is 7% compounded quarterly? A) $7181.79 B) $1748.94 C) $1514.79 D) $937.20 Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 130) Use the add-on method to determine the monthly payment of a $5000 loan at 8% interest for one year. Answer: $450 131) Use the add-on method to determine the monthly payment of a $15,000 loan at 9% interest for three years. Answer: $529.17 132) Give the add-on interest rate for a $4000 loan for 1 year at 6% APR with a monthly payment of $344.27 Answer: 3.28% 133) Give the add-on interest rate for a $40,000 loan for 10 years at 9% APR with a monthly payment of $506.70. Answer: 5.2%

134) For any mortgage with discount points, will the APR be greater than or less than the stated interest rate? Answer: greater than 135) Find the APR on an add-on loan of $15,000 at 8% interest compounded monthly for three years. Answer: 14.5486% MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 136) Find the APR for a 30-year, $250,000 mortgage at 6% interest compounded monthly and two discount points. A) 5.87% B) 6.25% C) 6.12% D) 6.19% Answer: D 137) Find the APR for a 30-year, $250,000 mortgage at 9% interest compounded monthly and two discount points. A) 9.25% B) 9.05% C) 9.10% D) 12% Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 138) True or False: The longer a loan with up-front fees is held, the greater the effective mortgage rate. Answer: False. Although the effective rate is higher than the APR, the longer time the loan is held the more the costs are spread over the life of the loan. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 139) Find the effective mortgage rate for a 30-year, $100,000 mortgage at 6% interest compounded monthly with 2 discount points that is expected to be kept for 5 years. A) 6.24% B) 6.55% C) 6.48% D) 5.98% Answer: C Solve the problem. 140) Consider a 20-year mortgage of $300,000 at 6.8% interest compounded monthly, where the loan is interest only for five years. (a) What is the monthly payment during the first five years? (b) What is the monthly payment during the last 15 years? A) (a) $2200.00 (b) $2763.05 B) (a)$2663.05 (b) $1700.00 C) (a) $1700.00 (b) $2663.05 D) (a) $170.00 (b) $1700.08 Answer: C

141) Consider a 25-year $200,000 5/1 ARM having a 2.8% margin and based on the CMT index. Suppose that the interest rate is initially 6.5% and the value of the CMT index is 3.2% five years later. Assume that all interest rates use monthly compounding. (a) Calculate the monthly payment for the first 5 years. (b) Calculate the monthly payment for the 6th year. A) (a) $1550.41 (b) $1497.63 B) (a) $1350.41 (b) $1297.63 C) (a) $850.41 (b) $1397.63 D) (a) $1083.33 (b) $9056.26 Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 142) How much money would you have to deposit at the end of each year into an annuity paying 5% interest compounded annually in order to have $39,600 at the end of thirty years? [(1.05) 30 ≈ 4.3, (1.5)30 ≈ 191,751, (1.025)60 ≈ 4.4] Answer: $600 143) How much money would you have to place into an account initially at 10% compounded annually to have $10,000 after 5 years? Answer: $6209.21 144) In order to purchase a home, a family borrows $50,000 at an annual interest rate of 6.44%, to be paid back over a 30 year period in equal monthly payments. What is their monthly payment? Answer: $314.06

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Let p denote the statement "Jack committed the crime" and let q denote the statement "Jack is friendly." Which of the following denotes the statement "Jack committed the crime, and he is not friendly"? A) ~(p ∨ q) B) p ∧ q C) ~p ∨ q D) p ∧ ~q E) none of these Answer: D 2) Let p denote the statement "Jack committed the crime" and let q denote the statement "Jack is friendly." Which of the following denotes the statement "Jack did not commit the crime or he is friendly"? A) p ∧ q B) ~(p ∨ q) C) ~p ∨ q D) p ∧ ~q E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 3) Determine which of the following sentences are statements. (a) The youngest student passed the exam. (b) Socrates was a famous pilot. (c) 2y + 7 = 11 (d) If the forecast is accurate, then it will rain. (e) Go away from me. Answer: a, b, and d are statements. 4) Determine which of the following sentences are statements. (a) 2 + 2 = 0. (b) The sky is blue. (c) If the moon is made of cheese, then I enjoy tennis. (d) Alaska is the capital of Texas. (e) What a hard test! Answer: a, c, and d are statements. 5) Determine which of the following sentences are statements. (a) He is clever. (b) 3 + 4 = 8 (c) What a beautiful day! (d) If John works hard, he will pass the class. (e) The ratio of any two integers is an integer. Answer: b, d, and e are statements.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide whether the following is a statement or is not a statement. 6) 9 + 6 = 16 A) Not a statement B) Statement Answer: B 7) 0.8 = .08 A) Not a statement B) Statement Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Identify the simple statements in the given compound statement. 8) The car is loud and fast. Answer: "The car is loud." "The car is fast." 9) I like ice cream or cake for dessert. Answer: "I like ice cream for dessert." "I like cake for dessert." 10) If Anne works hard, then she will make the team. Answer: "Anne works hard." "Anne will make the team." Let p denote the statement " The bank is open today." and let q denote the statement "The post office is open today." Put the given statement into symbolic form. 11) The bank and the post office are open today. Answer: p ∧ q 12) The bank is not open today, or the post office is not open today. Answer: ~p ∨ ~q 13) The bank is open, and the post office is not open today. Answer: p ∧ ~q 14) The bank or the post office is open today but not both. Answer: (p ∨ q) ∧ ~(p ∧ q) Let p denote the statement " It is raining." and let q denote the statement "There are clouds in the sky." Write out the given statement as a proper English sentence. 15) p ∧ q Answer: It is raining, and there are clouds in the sky. 16) ~p ∧ q Answer: It is not raining, but there are clouds in the sky. 17) p ∨ ~ q Answer: Either it is raining or there are no clouds in the sky.

18) ~(p ∨ q) Answer: It is not raining, nor are there clouds in the sky. 19) ~p ∧ ~q Answer: It is not raining, and there are no clouds in the sky. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 20) Given the following truth table: p q * T T F T F T F T F F F F Which of the following statements could replace *? A) ~p ∧ q B) (p ∨ q) ∧ ~q C) (p ∧ q) ∨ ~p D) p ∨ q E) none of these Answer: B 21) Given the following truth table: p q * T T F T F T F T T F F T Which of the following statements could replace *? A) ~(p ∨ q) B) (p ∧ q) ∨ ~p C) p ∨ q D) (p ∨ q) ∧ ~q E) none of these Answer: E SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct a truth table for the statement. 22) (p ∧ q) ∨ (~p ∧ ~q) Answer: p T T F F

q T F T F

(p ∧ q) ∨ (~p ∧ ~q) T F F T

q T F T F

~(p ∧ ~q) T F T T

23) ~(p ∧ ~q) Answer: p T T F F

24) (p ∨ ~q) ∧ (~p ∧ ~q) Answer: p T T F F

q T F T F

(p ∨ ~q) ∧ (~p ∧ ~q) F F F T

q T F T F

(p ∧ ~q) ∨ q T T T F

25) (p ∧ ~q) ∨ q Answer: p T T F F

Construct a truth table for the statement form. 26) (p ∧ q) ⊕ ~(p ∨~q) Answer: p T T F F

q T F T F

(p ∧ q) ⊕ ~(p ∨ ~q) T F T F

27) (p ∧ r) ∨ ~(q ⊕ r) Answer: p T T T T F F F F

q T T F F T T F F

r T F T F T F T F

(p ∧ r) ∨ ~(q ⊕ r) T F T T T F F T

Solve the problem. 28) Compare the truth tables of p ∨(q ∧ r) and (p ∨ q) ∧ (p ∨ r). Answer: They are identical. p q r p ∨ (q ∧ r) (p ∨ q) ∧ (p ∨ r) T T T T T T T T T T F T F T T T T F T T F T T T T F F T F T T T F T T T T T T T F T F F F T F F F F T F F F F T F F F F F F F F (1) (2) (3) (5) (4) (6) (8) (7) Columns 5 and 8 are identical.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Construct a truth table for the compound statement. 29) ~r ∧ ~q A) r q (~r ∧ ~q) T T F T F T F T T F F T B) r q (~r ∧ ~q) T T F T F F F T F F F T C) r q (~r ∧ ~q) T T F T F F F T F F F F D) r q (~r ∧ ~q) T T T T F F F T F F F T Answer: B 30) r ∨ ~(c ∧ q) A) r c q r ∨ ~(c ∧ q) T T T T T T F T T F T T T F F T F T T F F T F T F F T T F F F T B) r c q r ∨ ~(c ∧ q) T T T T T T F T T F T T T F F F F T T F F T F T F F T T F F F F

C) r c q r ∨ ~(c ∧ q) T T T T T T F F T F T T T F F T F T T F F T F T F F T T F F F F D) r c q r ∨ ~(c ∧ q) T T T T T T F T T F T T T F F T F T T F F T F T F F T T F F F F Answer: A 31) (p ∧ ~s) ∧ t A) p s t (p ∧ ~s) ∧ t T T T F T T F F T F T F T F F F F T T F F T F T F F T T F F F T B) p s t (p ∧ ~s) ∧ t T T T F T T F T T F T F T F F F F T T F F T F T F F T T F F F T C) p s t (p ∧ ~s) ∧ t T T T F T T F F T F T F T F F T F T T F F T F T F F T T F F F T 6

D) p s t (p ∧ ~s) ∧ t T T T F T T F F T F T T T F F F F T T F F T F F F F T F F F F F Answer: D 32) ~[(w ∧ q) ∨ s] A) w q s ~[(w ∧ q) ∨ s] T T T F T T F F T F T T T F F F F T T T F T F F F F T T F F F F B) w q s ~[(w ∧ q) ∨ s] T T T T T T F T T F T T T F F F F T T T F T F F F F T T F F F F C) w q s ~[(w ∧ q) ∨ s] T T T F T T F F T F T F T F F T F T T F F T F T F F T F F F F T D) w q s ~[(w ∧ q) ∨ s] T T T T T T F F T F T T T F F F F T T T F T F F F F T T F F F F 7

Answer: C 33) q ∨ (q ∧ ~q) A) q q ∨ (q ∧ ~q) T T F F B) q q ∨ (q ∧ ~q) T F F F C) q q ∨ (q ∧ ~q) T T F T D) q q ∨ (q ∧ ~q) T F F T Answer: A 34) (t ∧ p) ∨ (~t ∧ ~p) A) t p (t ∧ p) ∨ (~t ∧ ~p) T T F T F F F T T F F T B) t p (t ∧ p) ∨ (~t ∧ ~p) T T T T F T F T T F F F C) t p (t ∧ p) ∨ (~t ∧ ~p) T T T T F F F T T F F T D) t p (t ∧ p) ∨ (~t ∧ ~p) T T T T F F F T F F F T Answer: D

35) ~[~(s ∨ q)] A) s q ~[~(s ∨ q)] T T T T F F F T T F F F B) s q ~[~(s ∨ q)] T T T T F T F T F F F F C) s q ~[~(s ∨ q)] T T T T F T F T T F F F D) s q ~[~(s ∨ q)] T T F T F F F T F F F T Answer: C

36) ~(r ∨ t) ∧ ~(t ∧ r) A) r t ~(r ∨ t) ∧ ~(t ∧ r) T T F T F F F T T F F F B) r t ~(r ∨ t) ∧ ~(t ∧ r) T T F T F T F T T F F F C) r t ~(r ∨ t) ∧ ~(t ∧ r) T T F T F F F T F F F F D) r t ~(r ∨ t) ∧ ~(t ∧ r) T T F T F F F T F F F T Answer: D 37) (w ∧ r) ∧ (~r ∨ t) A) w r t (w ∧ r) ∧ (~r ∨ t) T T T F T T F T T F T T T F F T F T T T F T F F F F T T F F F T B) w r t (w ∧ r) ∧ (~r ∨ t) T T T T T T F F T F T F T F F F F T T F F T F F F F T F F F F F

C) w T T T T F F F F

r t (w ∧ r) ∧ (~r ∨ t) T T F T F T F T T F F T T T T T F F F T T F F F

w T T T T F F F F

r t (w ∧ r) ∧ (~r ∨ t) T T F T F F F T T F F T T T T T F F F T T F F T

Answer: B

Construct a truth table using the given information. 38) Define t∣p by the following truth table. t p t∣p T T F T F T F T T F F T Construct a truth table for (t∣t)∣(t∣p). A) t p (t∣t)∣(t∣p) T T T T F F F T F F F T B) t p (t∣t)∣(t∣p) T T F T F T F T T F F T C) t p (t∣t)∣(t∣p) T T T T F T F T F F F F D) t p (t∣t)∣(t∣p) T T F T F F F T T F F T Answer: C 39) Define r Θ s by the following truth table. r s rΘs T T T T F F F T F F F T Construct a truth table for (r Θ ~s) Θ t. A) r s t (r Θ ~s) Θ t T T T T T T F F T F T F T F F T F T T F F T F T F F T T F F F F

B) r s t (r Θ ~s) Θ t T T T T T T F F T F T F T F F T F T T T F T F F F F T F F F F T C) r s t (r Θ ~s) Θ t T T T F T T F T T F T T T F F F F T T T F T F F F F T F F F F T D) r s t (r Θ ~s) Θ t T T T F T T F T T F T T T F F F F T T F F T F T F F T T F F F F Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 40) (a) Draw a tree for the statement form (p ∧ q) ∨ ~r. (b) Assuming p has truth value T, q has truth value F, and r has truth value F, use the tree to find the truth value of the statement form. Answer: (a)

(b) The statement is true.

41) (a) Draw a tree for the statement ~(p ∧ q) ∨ (~p ∧ r). (b) Assuming p has truth value T, q has truth value T, and r has truth value F, use the tree to find the truth value of the statement form. Answer: (a)

(b) The statement is false.

Let p be the TRUE statement "The sun is a star." and q be the TRUE statement "The moon is a planet." Determine the truth value of the given statement. 42) p ∧ q Answer: True 43) p ∧ q Answer: False 44) ~(p ∧ q) Answer: True 45) ~(p ∨ ~q) Answer: False 46) ~p ∨ ~q Answer: False 47) (p ∧ ~q) ∨ ~p Answer: False 48) ~(p ∧ q) Answer: False 49) ~(p ∨ ~q) Answer: False 50) ~p ∨ ~q Answer: False 51) (p ∧ ~q) ∨ ~p Answer: False Write the statement form in symbols using the conditional (→) or the biconditional (↔) connective. If the form is conditional, name the hypothesis and the conclusion. Let p be the statement "It is raining" and q be the statement, "There are clouds in the sky." 52) If it is raining, then there are clouds in the sky. Answer: p → q; hypothesis: p, conclusion: q 53) It is raining if and only if there are clouds in the sky. Answer: p ↔ q 54) It is raining only if there are clouds in the sky. Answer: p → q; hypothesis: p, conclusion: q 55) Clouds in the sky are sufficient for it to be raining. Answer: q → p; hypothesis: q, conclusion: p 56) If there are no clouds in the sky, then it is not raining. Answer: ~q → ~p; hypothesis: ~q, conclusion: ~p 15

Let p denote the TRUE statement "The red/blue/green/yellow spinner is fair" and let q be the TRUE statement "The 1 probability of yellow is ." Write the statement form in symbols, name the hypothesis and the conclusion and determine 4 whether the statement is TRUE or FALSE. 57) If the red/blue/green/yellow spinner is fair, the probability of yellow is

1 . 4

Answer: p → q; hypothesis: p, conclusion: q; true 58) If the red/blue/green/yellow spinner is not fair, the probability of yellow is not

1 . 4

Answer: ~p → ~q; hypothesis: ~p, conclusion: ~q; true 59) The probability of yellow is

1 only if the red/blue/green/yellow spinner is fair. 4

Answer: q → p; hypothesis: q, conclusion: p; true 60) The red/blue/green/yellow spinner is not fair if the probability of yellow is not

1 . 4

Answer: ~q → ~p; hypothesis: ~q, conclusion: ~p; true Insert parentheses in the statement form to show the proper order for the application of the connectives. 61) p ∨ q ∧ r → q ∨ ~p Answer: (p ∨ (q ∧ r)) → (q ∨ (~p)) 62) p → q ↔ r Answer: (p → q) ↔ r 63) ~p ∨ q ↔ r ∧ s Answer: ((~p) ∨ q) ↔ (r ∧ s) 64) ~p ↔ r → q ∧ ~p Answer: (~p) ↔ [r → (q ∧ (~p))] Construct a truth table for the statement form. 65) (~p ∧ q) → p Answer: p q (~p ∧ q) T T T (~p ∧ q) → p F T F F F T 66) p → (q → p) Answer: p T T F F

q T F T F

p → (q → p) T T T T

67) p ∨ q → p ∧ q Answer: p T T F F

q T F T F

p∨q→p∧q T F F T

68) ~p ∨ q → p ∧ ~r Answer: p T T T T F F F F

q T T F F T T F F

r T F T F T F T F

~p ∨ q → p ∧ ~r F F T T F F F F

r T F T F T F T F

(p ∧ r) ↔ (q ∧ r) T T F T F T T T

r T F T F T F T F

(p ∧ r) → (q ⊕ r) F T T T T T T T

69) (p ∧ r) ↔ (q ∧ r) Answer: p T T T T F F F F

q T T F F T T F F

70) (p ∧ r) → (q ⊕ r) Answer: p T T T T F F F F

q T T F F T T F F

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Construct a truth table for the statement. 71) q → ~p A) q p q → ~p T T F T F T F T T F F T B) q p q → ~p T T F T F F F T T F F T C) q p q → ~p T T T T F T F T F F F F D) q p q → ~p T T T T F F F T T F F T Answer: A

72) ~q → (~q ∧ t) A) q t ~q →(~q ∧ t) T T T T F T F T T F F T B) q t ~q →(~q ∧ t) T T T T F T F T T F F F C) q t ~q →(~q ∧ t) T T T T F F F T T F F F D) q t ~q →(~q ∧ t) T T F T F F F T T F F F Answer: B

73) (s → ~q) → (s ∧ ~q) A) s q (s → ~q) → (s ∧ ~q) T T F T F T F T T F F T B) s q (s → ~q) → (s ∧ ~q) T T F T F F F T F F F T C) s q (s → ~q) → (s ∧ ~q) T T T T F T F T F F F T D) s q (s → ~q) → (s ∧ ~q) T T T T F T F T F F F F Answer: D SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Let p be the statement " Maine is one of the 50 United States." and let q be the statement "Germany is one of the 50 United States." Determine the truth value of the given statement form. 74) p → q Answer: False 75) ~q → p Answer: True 76) p ∨ q → p ∧ ~q Answer: True 77) ~p ∧ q → q ∧ ~q Answer: True 78) p → p ∧ q Answer: False Let p be the statement " The moon is made of cheese." and let q be the statement "Albany is the capital of New York." Determine the truth value of the given statement form. 79) p → q Answer: True

80) ~q → p Answer: True 81) p ∨ q → p ∧ ~q Answer: False 82) ~p ∧ q → q ∧ ~q Answer: False 83) p → p ∧ q Answer: True State the converse of the statement. 84) The price of corn increases only if its supply decreases. Answer: The price of corn increases if its supply decreases. 85) Being the home team is a necessary condition for winning the game. Answer: Being the home team is a sufficient condition for winning the game. 86) If the acting is adequate, then the play is successful. Answer: If the play is successful, then the acting is adequate. 87) Tom graduates, if he earns enough credits. Answer: Tom earns enough credits, if he graduates. For the given input values of A and B, use the following program to determine the value of C. IF ((A > 0) AND (B < 0)) OR (B > 4) THEN LET C = ELSE LET C =

A B

B A

88) A = 10, B = -2 Answer: C = -5 89) A = 15, B = 5 Answer: C = 3 90) A = 27, B = 0 Answer: C = 0 91) A = -1, B = 3 Answer: C = -3 92) A = 2, B = -1 Answer: C = -2

Solve the problem. 93) Find values A and B so that the following program gives the output C = 3. IF ((A > 0) AND (A ≤ 5)) AND (B > 1) A THEN LET C = +1 B ELSE LET C = 0 Answer: Possible answer: A = 4, B = 2 (Answers may vary. Any values that satisfy both 2 < A ≤ 5 and B = work.) 94) Write a statement equivalent to ~(p ∨ q) → (~r ∨ p) using only ~ and ∧. Answer: ~((~p ∧ ~q) ∧ r) 95) Using a truth table, show that (~p ∨ q) ∨p is a tautology. Answer: p q T T T F F T F F (1) (2)

(~p ∨ q) ∨p T T F T T T T T (3) (4) From column (4), we see that the implication is a tautology.

96) Using a truth table, show that [(p ∨ q) ∧ ~p] ⇒ q Answer: p q [(p ∨ q) ∧ ~p] ⇒ q T T T F F T T F T F F T F T T T T T F F F F T T (1) (2) (3) (5) (4) (6) From column (6), we see that the implication is a tautology. Determine the truth value of the statement. 97) (~p ∨ q ) ⇔ ( p ⇒ q ) Answer: True 98) p ∨ (q ∧ r) ⇔ (p ∧ q) ∨ (p ∧ r) Answer: not true 99) (p ∧ q) ⇔ ~(~p ∨ ~q) Answer: true 100) ~((p ∨ r) ∧ ~q) ⇔ (~q → ~p ∧ ~r) Answer: true 101) (p → q) ⇔ [(~p ∧ q) → c] Answer: not true

A will 2

Solve the problem. 102) Show that [p → (q ∨ r)] ⇔ [t → (~p ∨ (q ∨ r))] (a) by using equivalences. (b) by constructing a truth table. Answer: (a) [p → (q ∨ r)] ⇔ [ p ∧ (~(q ∨ r)) → c] ⇔ [ t → ~(p ∧ ~(q ∨ r)) ] ⇔ [ t → (~p ∨~(~(q ∨ r))) ] ⇔ [ t → (~p ∨ (q ∨ r)) ] (b) p T T T T F F F F

q T T F F T T F F

(Reductio ad absurdum) (Contrapositve) (De Morgan's law) (Double negation)

r T F T F T F T F

↔ [p → (q ∨ r)] [t → (~p ∨ (q ∨ r))] T T T TT T T T T T TT T T T T T TT T T F F T TF F F T T T TT T T T T T TT T T T T T TT T T T F T TT T F (2) (1) (6) (5) (4) (3) From (6) we see that the biconditional is a tautology.

Negate the statement. 103) Dave's shirt is either red or orange. Answer: Dave's shirt is neither red nor orange. 104) If the probability is more than

2 , I will take the bet. 3

Answer: The probability is more than

2 , but I will not take the bet. 3

105) Susan fixed the car and finished her homework. Answer: Susan did not fix the car, or she did not finish her homework. 106) James will call his mother, if the phone is working. Answer: The phone is working, but James will not call his mother. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the negation of the statement. Remember that the negation of p → q is p ∧ ~q. 107) If c - 4 ≤ 3, then c ≤ 7. A) c - 4 ≤ 3 and c > 7. B) c - 4 ≤ 3 and c ≥ 7. C) c - 4 ≤ 3 so c > 7. D) c - 4 > 3 and c > 7. Answer: A

108) If 7x + 2y > 10, the answer is "Mars". A) 7x + 2y > -10, so the answer is not "Mars". B) 7x + 2y > 10 and the answer is not "Mars". C) 7x + 2y ≤ 10 and the answer is not "Mars". D) If 7x + 2y > 10, the answer is not "Mars". Answer: B 109) If the box is in the mail, then it will be here by Tuesday. A) The box is not in the mail and it will not be here by Tuesday. B) If the box is in the mail, then it will not be here by Tuesday. C) The box is in the mail and it will not be here by Tuesday. D) The box is in the mail, so it will be here by Tuesday. Answer: C 110) If the power drill is on the floor, the baby will get hurt. A) The power drill is not on the floor and the baby does not get hurt. B) If the power drill is on the floor, the baby won't get hurt. C) The power drill is on the floor and the baby does not get hurt. D) The power drill is not on the floor and the baby gets hurt. Answer: C Use De Morgan's laws to negate the statement without using parentheses. 111) ~r ∧ (s ∧ ~t) A) r ∨ s ∨ ~t B) r ∨ ~s ∨ t C) r ∧ ~s ∧ t D) r ∨ ~s ∧ t Answer: B 112) r ∨ s ∨ ~t A) r ∧ ~s ∧ t B) ~r ∨ ~s ∨ t C) ~r ∧ ~s ∧ t D) ~r ∨ ~s ∧ t Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 113) Without using truth tables, show that [(p ∧ q) → r] ⇔ [r ∨ (~p ∨ ~q)]. Answer: [(p ∧ q) → r] ⇔ [~r → ~(p ∧ q)] ⇔ [r ∨ (~(p ∧ q))] ⇔ [r ∨ (~p ∨ ~q)]

(Contrapositive) (Implication) (De Morgan's law)

For the statement, give the contrapositive, converse, and negation. Determine the truth value in each case. 114) "If a triangle has equal sides, it is equiangular." Answer: Contrapositive: "If the triangle is not equiangular, then it does not have equal sides." TRUE Converse: "If the triangle is equiangular, then it has equal sides." TRUE Negation: "The triangle has equal sides and is not equiangular." FALSE

115) "If Richard goes to the museum, then Carol goes to the park." (Assume that this statement is FALSE.) Answer: Contrapositive: "If Carol does not go to the park, then Richard does not go to the museum." FALSE Converse: "If Carol goes to the park, then Richard goes to the museum." TRUE Negation: "Richard goes to the museum, and Carol does not go to the park." TRUE Test the validity of the argument. Tell and show why the argument is valid or invalid. 116) If all four sides of a rectangle are equal, then it is a square. If two adjacent sides of a rectangle are equal, then all four sides are equal. This rectangle has two adjacent sides that are equal. Thus, this rectangle is a square. Answer: Let f = "all four sides of a rectangle are equal" a = "two adjacent sides of a rectangle are equal" s = "the rectangle is a square" 1. f → s Hypothesis 2. a → f Hypothesis → 3. a s Hypothesis syllogism (1, 2) 4. a Hypothesis 5. s Modus Ponens (3, 4) The argument is valid. 117) If Lisa is at home, she is listening to music. She is eating lunch at home. Therefore, she is listening to music. Answer: Let h = "Lisa is at home" m = "Lisa is listening to music" l = "Lisa is eating lunch" 1. h → m Hypothesis 2. l → h Hypothesis 3. h Subtraction (2) 4. m Modus Ponens (1, 3) The argument is valid. 118) If it is warm, Brenda will go to the park or go shopping. It is warm and Brenda goes shopping. Thus, she does not go to the park. Answer: Let w = "It is warm" p = "Brenda will go to the park" s = "Brenda will go shopping" If the argument is valid, the implication is a tautology. The only time that the implication could be false is if the hypothesis is true and the conclusion is false. Consider the case where p, w, and s are true, and fill in the truth values in evaluating the statement. [(w → (p ∨ s)) ∧ (w ∧ s)] → ~p T T T T T F F (1) (3) (2) (5) (4) (7) (6) This leads to a true hypothesis (5) implicating a false conclusion (6): false (7). The implication is not a tautology and the argument is invalid.

119) If Bill is a professional wrestler, then he is large or strong. Bill is not large. He is not strong. Therefore, Bill is not a professional wrestler. Answer: Let p = "Bill is a professional wrestler" l = "Bill is large" s = "Bill is strong" 1. p → (l ∨ s) Hypothesis ∧ 2. ~l ~s Hypothesis 3. ~( l ∨ s) De Morgan's law (2) 4. ~p Modus Tollens (1, 3) The argument is valid. 120) Either I went to the doctor or I went to the dentist. If I went to the dentist, I had a tooth removed. I did not have a tooth removed. Therefore, I went to the doctor. Answer: Let p = "I went to the doctor" q = "I went to the dentist" r = "I had a tooth removed" 1. p ∨ q Hypothesis → 2. q r Hypothesis 3. ~r Hypothesis 4. ~q Modus Tollens (2, 3) 5. p Disjunctive Syllogism (1, 4) The argument is valid. 121) If Hal did all of his homework, he received a grade of A or B. Hal received a B. Thus Hal did all of his homework. Answer: Let h = "Hall did all of his homework" a = "Hal received a grade of A" b = "Hal received a grade of B" If the argument is valid, the implication is a tautology. The only time that the implication could be false is if the hypothesis is true and the conclusion is false. Consider the case where a and b, are true and h is false, and fill in the truth values in evaluating the statement. [(h → (a ∨ b)) ∧ b] → h F T T T F F (1) (4) (2) (3) (6)(5) This leads to a true hypothesis (4) implicating a false conclusion (5): false (6). The implication is not a tautology and the argument is invalid. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the argument is valid. 122) 6 is a rational number. 36 is a rational number. Therefore, 6 is A) Invalid B) Valid Answer: A

36.

123)

16 is less than 16. 8 is less than 16. Therefore,

16 is less than 8.

A) Invalid B) Valid Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 124) Suppose that the following statements are true. (1) If a sedan is a car, then my horse is brown. (2) Either roses are not red or a sedan is a car. (3) Roses are red. Prove: My horse is brown.. Answer: Let p, q, and r denote the following statements: p = "A sedan is a car." q = "My horse is brown." r = "Roses are red." 1. p → q Hypothesis ∨ 2. ~r p Hypothesis 3. r Hypothesis 4. p Disjunctive syllogism (2,3) 5. q Modus Ponens (1,4) Thus, My horse is brown. 125) Suppose that the following statements are true. (1) Axel eats a peach or banana. (2) If Axel eats a peach, then Betty reads today's New York Times. (3) Betty does not read today's New York Times. Prove: Axel eats a banana. Answer: Let p, q, and r denote the following statements: p = "Axel eats a peach." q = "Axel eats a banana." r = "Betty reads today's New York Times." 1. p ∨ q Hypothesis 2. p → r Hypothesis 3. ~r Hypothesis 4. p → ~(~r) Double negation, 2 5. ~p Modus Tollens (4, 3) 6. q Disjunctive syllogism (1,5) Thus, Axel eats a banana. Let U = { 3, 6, 9, 12, 15, ... } and p(x) be the open statement "x is even". Determine the truth value of the given statement. 126) ∀x p(x) Answer: False 127) ∃x p(x) Answer: True

128) [∀x p(x)] → [ ∀x (~p(x))] Answer: True 129) [∃x p(x) ∧[∃x (~p(x))] Answer: True 130) ∀x [~p(x)] Answer: False 131) [∀x p(x)] ∨ [ ∃x (p(x) ∨ ~p(x))] Answer: True Let the universe consist of all positive integers. Let p(x) be the statement "x is even." Let q(x) be the statement "x is a perfect square." Determine the truth value of the indicated statement. 132) ∀x [p(x) ∨ q(x)] Answer: False 133) ∃x [p(x) ∨ q(x)] Answer: True 134) ∀x [p(x) ∧ q(x)] Answer: False 135) ∃x [p(x) ∧ q(x)] Answer: True 136) [∃x p(x)] ∧ [ ∃x q(x)] Answer: True 137) [∀x p(x)] → [ ∀x q(x)] Answer: True Let the universe for both variables x and y be the set {2, 4, 6, 8, 10}. Let p(x,y) = "4 divides the sum x + y". Give the truth value of the indicated statement; explain your answer, and give a counterexample in case the statement is false. 138) ∀x ∀y p(x, y) Answer: False; counterexample: choose x = 2, y = 4, then 4 does not divide 6. 139) ∀x ∃y p(x, y) Answer: True; for x = 2, 4, 6, 8, 10, choose y=10, 8, 6, 4, 2 .respectively. 140) ∃x ∀y p(x, y) Answer: There is no value x in the set such that 4 divides x+y for all y.

141) ∃y ∀x p(x, y) Answer: False. For any y there is a counterexample: choose y = 2, x = 4, 4 does not divide 6. choose y = 4, x = 6, 4 does not divide 10. choose y = 6, x =8, 4 does not divide 14. choose y = 8, x = 10, 4 does not divide 18. choose y = 10, x = 4, 4 does not divide 14. Therefore, there does not exist a y, which for all x satisfies p(x). Let the universe for both variables x and y be the set {1, 2, 3, 4, 5, 6}. Let p(x,y) = "x divides the sum x + y". Give the truth value of the indicated statement; explain your answer, and give a counterexample in case the statement is false. 142) ∀x ∀y p(x, y) Answer: False; counterexample: choose x = 2, y = 3, then 2 does not divide 5. 143) ∀x ∃y p(x, y) Answer: True; for any x, let y = x, then x + y = 2x, and x divides 2x. 144) ∃x ∀y p(x, y) Answer: True; let x = 1, then 1 divides 1 + y for any value of y. 145) ∃y ∀x p(x, y) Answer: False; for any y we choose there is a counterexample: choose y = 1, x = 2, 2 does not divide 3. choose y = 2, x = 3, 3 does not divide 5. choose y = 3, x = 2, 2 does not divide 5. choose y = 4, x = 5, 5 does not divide 9. choose y = 5, x = 2, 2 does not divide 7. choose y = 6, x =4, 4 does not divide 10. Therefore, there does not exist a y, which for all x satisfies p(x). Let the universe be the collection of all college students. Let p(x) be the open statement "x likes classical music." Write the given statement symbolically. 146) All college students like classical music. Answer: ∀x p(x) 147) Not all college students dislike classical music. Answer: ~[∀x (~p(x))] ⇔ [ ∃x (p(x))] 148) No college students like classical music. Answer: ~[∃x p(x)] ⇔ [ ∀x p(x)] 149) Some college students like classical music. Answer: ∃x p(x) 150) All college students dislike classical music. Answer: ∀x [~p(x)]

Let the universe be all the graphic artists. Let p(x) be the statement "x likes mathematics." Assume the statement "all graphic artists like mathematics" is TRUE. Write the given statement symbolically and determine the truth value, if possible. 151) Some graphic designers like mathematics. Answer: ∃x p(x) True 152) Some graphics designers do not like mathematics. Answer: ∃x ~p(x) False 153) There exists no graphic designer who does not like mathematics. Answer: ~(∃x ~p(x)) True 154) Not all graphic designers like mathematics. Answer: ~(∀x p(x)) False Negate the given statement by changing the existential quantifier to a universal quantifier, or vice-versa. 155) Every house is a home. Answer: There is a house that is not a home. 156) No house is a home. Answer: There is a house that is a home. 157) There are people who dislike politics. Answer: Everyone likes politics. 158) All books are blue. Answer: Some books are not blue. 159) Some games do not end. Answer: All games end. 160) Some people are afraid of the dark. Answer: No one is afraid of the dark. 161) Not all TV's have cable. Answer: All TV's have cable. 162) All clouds bring rain. Answer: Some clouds do not bring rain. Solve the problem. 163) Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10} S = {x ∈ U ; x ≥ 4} T = {x ∈ U; x ≤ 9} (a) What implication must be TRUE if and only if S is a subset of T? (b) Is S a subset of T? Answer: (a) ∀x ∈ U [x ≥ 4 → x ≤ 9] (b) No

164) Prove that S is a subset of T, where U = [0, 1, 2, 3, 4, 5] S = {x ∈ U: x is a solution to x2 = 16 } T = {x ∈ U: x is a solution to x2 - 4x = 0} Answer: S = {4} and T = { 0, 4}. Therefore , ∀x ∈ U [x ∈ S → x ∈ T]. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the logic statement represented by the given figure. 165)

A) (p ∧ ~q) ∨ (q ∨ r) B) ~(p ∨ q) ∧ (q ∧ r) C) (p ∧ ~q) ∨ (~q ∨ r) D) (p ∨ ~q) ∧ (q ∧ r) Answer: D 166)

A) (~(p ∨ q) ∧ (p ∨ r)) ∨ (q ∧ ~r) B) ((p ∧ q) ∨ (q ∧ r)) ∧ (q ∨ ~r) C) (~(p ∧ q) ∨ (p ∧ r)) ∧ (q ∨ ~r) D) ((p ∨ q) ∧ (q ∨ r)) ∨ (q ∧ ~r) Answer: C

167)

A) ((p ∨ q) ∧ r) ∨ ~((p ∨ q) ∨ ~r) B) ((p ∧ q) ∨ r) ∧ ~((p ∧ q) ∧ r) C) ((p ∨ q) ∧ r) ∨ (~(p ∨ q) ∨ r) D) ((p ∧ q) ∨ r) ∧ ~((p ∧ q) ∧ ~r) Answer: A Draw the logic circuit that represents the logical statement. Determine the output given the variable values. 168) (p ∧ ~q) ∨ (p ∨ ~r); for p = 1, q = 1, r = 0 A) output = 1

B) output = 1

C) output = 1

D) output = 0

Answer: B 169) ((p ∧ ~q) ∧ ~(q ∨ ~r)) ∧ r; for p =1, q =0, r =1 A) output = 1

B) output = 0

C) output = 0

D) output = 1

Answer: A 170) ((p ∨ q) ∨ ~(p ∧ q)) ∧ r; for p =0, q =0, r = 1 A) output = 1

B) output = 0

C) output = 0

D) output = 1

Answer: D Simplify the logic circuit as much as possible. 171)

A) ((p ∨ q) ∨ r) ∧ ~(p ∧ q) B) ((p ∨ q) ∨ r) ∧ r C) ((p ∨ q) ∨ ~(p ∧ q)) ∧ r D) (~(p ∨ q) ∨ (p ∧ q)) ∧ r Answer: C Design the described logic circuit. 34

172) Design a logic circuit for (p NAND q) NOR (p NAND q), and show the outputs for all values of p and q on a truth table. A) p q

NAND NOR NAND

p q (p NAND q) NOR (p NAND q) 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 (1) (3) (2) B) p q

NAND NOR NAND

p q (p NOR q) NAND (p NOR q) 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 (1) (3) (2) C) p q

NAND NOR NAND

p q (p NAND q) NOR (p NAND q) 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 (1) (3) (2)

D) p q

NOR NAND NOR

p q (p NAND q) NOR (p NAND q) 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 (1) (3) (2) Answer: A

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Consider the difference equation yn = A)

13 3

23 3

8 3

1 y + 6, y0 = 1. Suppose y2 = 6, compute y3 . 3 n-1

D) 8 E) none of these Answer: D 2) Consider the difference equation yn = A)

10 3

19 3

23 3

13 3

1 y + 6, y0 = 1. Suppose y3 = -5, compute y4 . 3 n-1

E) none of these Answer: D 3) Consider the difference equation yn = -2yn-1 + 18. Compute

b . 1-a

2 17

B) 6 17 C) 2 D)

1 9

E) none of these Answer: B 4) Consider the difference equation yn = -2yn-1 + 18, y0 = 5. Compute y7 . A) 640 B) -122 C) -140 D) 134 E) none of these Answer: D

5) Consider the difference equation yn = yn-1 - 3, y0 = 50. Compute y12. A) 38 B) 9 C) 14 D) 8 E) none of these Answer: C 6) Consider the difference equation yn = A)

67 4

87 4

1 y +8, y0 = 64. Compute y6. 2 n-1

C) 87 D) 67 E) none of these Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 7) The solution of a certain difference equation is yn = -3 + 4(2)n . Find y0 , y2 , y3. Answer: y0 = 1, y2 = 13, y3 = 29 8) The solution of a certain difference equation is 1 yn = + 3(2)n . Find y0, y1, y2. 2 Answer: y0 =

7 13 25 ,y = ,y = 2 1 2 2 2

9) Suppose a difference equation is given by yn = 3yn-1 + 2. If y5 = 10, find y7 . Answer: 98 10) If yn = 1.1yn-1 + 0.01, y0 = 1, find y5 . Answer: 1.671561 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Generate y0 , y1 , y2 , y3 , and y4 from the difference equation. 11) y0 = -7, y n = yn-1 + 2 A) -7, 2, 4, 6, 8 B) -5, -3, -1, 1, 3 C) 0, 2, 4, 6, 8 D) -7, -5, -3, -1, 1 Answer: D

12) y0 = 1, yn = 3yn-1 A) 1, 3, 6, 9, 12 B) 3, 9, 27, 81, 243 C) 0, 3, 3, 6, 9 D) 1, 3, 9, 27, 81 Answer: D 13) The population of a certain country is currently 50 million and is declining at the rate of 2% a year due to natural causes. But, the population is increasing by 20,000 people a year due to immigration. Let yn be the population (in millions) after n years. Find a difference equation satisfied by yn . A) yn = 0.98 yn-1 + 0.02, y0 = 50. B) yn = 50(0.98) n . C) yn = -0.02 yn-1 + 0.02, y0 = 50. D) yn = -0.02 yn-1 - 0.02, y0 = 50. E) none of these Answer: A 14) The population of a country is now 60 million with a 2% per year death rate and a 1% per year birth rate. Each year 50,000 people immigrate to the country. Let yn be the population (in millions) of the country at the end of year n. Find a difference equation satisfied by yn . A) yn = 0.99yn-1 - 0.05, y0 = 60. B) yn = 1.01yn-1 - 0.05, y0 = 60. C) yn = 1.01yn-1 + 0.05, y0 = 60. D) yn = 0.99yn-1 + 0.05, y0 = 60. E) none of these Answer: D 15) The population of a country is now 265 million with a 7% per year death rate and a 5% per year birth rate. Each year 100,000 people immigrate to the country. Let yn be the population (in millions) of the country at the end of year n. Find a difference equation satisfied by yn . A) yn = 0.98yn-1 - 0.1, y0 = 265. B) yn = 1.02yn-1 +0.1, y0 = 265. C) yn = 0.98yn-1 + 0.1, y0 = 265. D) yn = 1.02yn-1 - 0.1, y0 = 265. E) none of these Answer: C 16) A car was bought for $7000. Assume that the trade-in value of the car deceases by 20% each year. Let yn be the trade-in value of the car at the end of year n. Find a difference equation satisfied by yn . A) yn = 1.2yn-1 - 1400, y0 = 7000. B) yn = 0.2yn-1, y0 = 7000. C) yn = 0.8yn-1 , y0 = 7000. D) yn = -0.2yn-1 , y0 = 7000. E) none of these Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 17) Rich deposits $1000 into a savings account paying 4% interest compounded annually. At the end of each year he deposits $200 into the account. Let yn be the amount in the account after n years. (a) Find a difference equation satisfied by yn . (b) Solve the difference equation. (c) How much will he have in the account after 6 years? Answer: (a) yn = (1.04)yn-1 + 200 (b) yn = -5000 + (6000)(1.04) n (c) y6 = $2591.91 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) Suppose that certain bacteria can double their size and divide every 30 minutes. Let y0 = 421 represent the initial number of bacteria present. Let yn be the number of bacteria present after n 30-minute intervals. Find a difference equation satisfied by yn . A) y0 = 421; yn = 2 yn-1 B) y0 = 421; yn = 2 y0 1 C) y0 = 421; yn = yn-1 2 D) y0 = 421; yn = 2 yn+1 Answer: A 19) A town has a population of 3000 people and is increasing by 9% every year. What will the population be at the end of 8 years? A) 5160 people B) 5978 people C) 5484 people D) 6516 people Answer: B 20) A town has a population of 30,000 people and is increasing by 10% every year. What will the population be at the end of 4 years? A) 13,923 people B) 43,923 people C) 39,930 people D) 42,000 people Answer: B

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) Consider the difference equation yn = 2yn-1 + 1, y0 = 1. (a) Generate y1 , y2 , y3 , and y4 . (b) Graph y0 through y4 .

Answer: (a) y1 = 3, y2 = 7, y3 = 15, y4 = 31 (b) y 40 35 30 25 20 15 10 5 1

22) Consider the difference equation yn = 2yn-1 + 1, y0 = 1. (a) Solve the difference equation. (b) Use the solution in (a) to find y8 . Answer: (a) yn = -1 + 2(2)n (b) y8 = 511

23) yn = -

2 y + 6; y0 = 18 3 n-1

(a) Generate y0 , y1 , y2 , y3 , y4 . (b) Graph these first few terms. (c) Solve the difference equation.

Answer: (a) 18, -6, 10, -

2 58 , 3 9

(b)

18 72 2 n (c) yn = + 5 5 3 24) Solve the difference equation yn = yn-1 + 10, y0 = 1. Answer: yn = 1 + 10n 25) Solve yn = 0.98yn-1 + 3, y0 = 150 Answer: yn = 150 26) Solve the difference equation yn = 1.05yn-1 - 3. Answer: yn = 60 + (y0 - 60)(1.05) n

27) Consider the difference equation yn = yn-1 + 2, y0 = 5. (a) Generate y1 , y2 , y3 from the difference equation. (b) Solve the difference equation. (c) Use the solution in (b) to obtain y15. Answer: (a) y1 = 7, y2 = 9, y3 = 11 (b) yn = 5 + 2n (c) y15 = 35 28) Consider the difference equation yn = 0.5y n-1 -1, y0 = 6 (a) Generate y1 , y2 , y3 from the difference equation. (b) Solve the difference equation. (c) Use the solution in (b) to obtain y4. Answer: (a) y1 = 2, y2 = 0, y3 = -1 (b) yn = -2 + (8)(0.5)n -3 (c) y4 = 2 29) Consider the difference equation yn = -0.5y n-1 + 6, y0 = 2 (a) Generate y1 , y2 , y3 from the difference equation. (b) Solve the difference equation. (c) Use the solution in (b) to obtain y4. Answer: (a) y1 = 5, y2 =

7 17 ,y = 2 3 4

(b) yn = 4 + (-2)(-0.5)n 31 (c) y4 = 8 30) Consider the difference equation yn = 5yn-1 - 6, y0 = 2. (a) Generate y1 , y2 , y3 from the difference equation. (b) Solve the difference equation. (c) Use the solution in (b) to obtain y4 . Answer: (a) y1 = 4, y2 = 14, y3 = 64 (b) yn = 1.5 + (0.5)5 n (c) y4 = 314

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the difference equation and, by inspection, determine the long-run behavior of the terms (i.e., the behavior of the terms as n gets large). 31) yn = 6yn-1 - 45, y0 = 13 A) yn = 9 + 13(6)n ; yn approaches 9. B) yn = 9 + 4(6)n ; yn gets arbitrarily large and positive. C) yn = 9 - 4(6)n ; yn gets arbitrarily large and negative. D) yn = 9 + 6(4)n ; yn approaches 6. Answer: B 32) yn = 0.2yn-1 - 3.2, y0 = 3 A) yn = -4 + 7(0.2)n ; yn approaches -4. B) yn = -4 + 0.2(7)n ; yn approaches -4. C) yn = -4 - 7(0.2)n ; yn gets arbitrarily large and negative. D) yn = -4 + 3(0.2)n ; yn approaches 3. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 33) A person makes an initial deposit into a savings account paying 5% interest compounded annually. He plans to withdraw $100 at the end of each year. (a) Find the difference equation yn , the amount in the account after n years. (b) How large must the initial deposit be so that the money will not run out? Answer: (a) yn = 1.05yn-1 -100 (b) at least $2000 34) For college, Jen deposits a certain amount of money into the bank at 5.4% interest compounded monthly and withdraws $300 at the end of each month. (a) Set up the difference equation for yn the amount in the bank after n months. (b) How large must the initial deposit be so that the money will never run out? Answer: (a) yn = 1.0045yn-1 - 300 (b) y0 ≥ $66,666.67 35) A person makes an initial deposit into a savings account paying 6% interest compounded annually. He plans to withdraw $120 at the end of each year. (a) Find the difference equation yn , the amount in the account after n years. (b) How large must the initial deposit be so that the money will not run out? Answer: (a) yn = 1.06yn-1 -120 (b) at least $2000

36) Suppose that a consumer loan of $1200 is made with a monthly rate of 2% simple interest and a monthly payment of $200. (a) Write the difference equation for yn , the balance owed after n months. (b) How much is owed on the loan after four payments? (c) How much interest has been paid on the loan after five months? Answer: (a) yn = yn-1 - 176, y0 = 1200 (b) $496 (c) $120 37) How much money will you have after five years if you deposit $100 at 7% compounded semiannually? Answer: $141.06 38) How much money will you have after five years if you deposit $100 at 3.5% compounded semiannually? Answer: $118.94 39) How much money will you have after 10 years if you deposit $2000 at an annual interest rate of 9% compounded monthly? [(1.0075)10 ≈ 1.08, (1.0075)120 ≈ 2.45, (1.09) 10 ≈ 2.37, (1.09) 120 ≈ 30,987.02] Answer: $4900 40) How much money will you have after thirty years if you deposit $10,000 at an annual rate of 6% compounded annually? Answer: $57,434.91 41) How much money will you have after thirty years if you deposit $10,000 at an annual rate of 4.8% compounded annually? Answer: $40,816.76 Find the total amount of simple interest on the given loan. 42) $800 at 5% annually for two years. Interest is paid once a year. Answer: $80 43) $3600 at 2.5% annually for nine months. Interest is paid once a month. Answer: $7.50 Find the final amount on deposit for the given situation. [(1.08) 5 ≈ 1.5, (1.045)10 ≈ 1.6, (1.045)10 ≈ 2.4, (1.045)20 ≈ 2.4, (1.08) 12 ≈ 2.5, (1.09) 20 ≈ 5.6] 44) $2000 deposited at 8% compounded annually for 12 years. Answer: $5000 45) $7500 deposited at 8% simple interest for five years. Answer: $10,500 46) $3200 deposited at 9% compounded semiannually for 10 years. Answer: $7680

Solve the problem. 47) How much money must be put into a savings account now at 4% interest compounded semiannually so as to have $32,500 twelve years from now? [(1.02) 12 ≈ 1.27, (1.02) 24 ≈ 1.61, (1.04) 12 ≈ 1.60, (1.06) 24 ≈ 2.56] Answer: $20,186.34 48) Over time a house worth $60,000 depreciates by 2% annually. What will the value be in 20 years? Answer: $40,056.48 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 49) A deposit of $15,000 is made at 4% interest compounded quarterly. [(1.01) 7 ≈ 1.1, (1.01) 28 ≈ 1.3, (1.04) 28 ≈ 3] How much is in the account after seven years? A) $16,500 B) $19,500 C) $16,050 D) $19,200 E) none of these Answer: B 50) A deposit of $15,000 is made at 4% interest compounded quarterly. [(1.01) 7 ≈ 1.1, (1.01) 28 ≈ 1.3, (1.04) 28 ≈ 3] What is the amount of interest earned after seven years? A) $4200 B) $1050 C) $4500 D) impossible to determine E) none of these Answer: C 1 51) A charge card statement shows an average daily balance of $430. The interest rate is 1 % per month. What is 2 the finance charge? A) $5.38 B) $64.45 C) $53.38 D) $6.45 E) none of these Answer: D 52) Carolyn Johnson borrowed $600 from a finance company at an annual rate of 4% simple interest. She agreed to pay back the loan at the end of nine months. What is the amount of interest she must pay? A) $618 B) $60 C) $18 D) $45 E) none of these Answer: C

Use a difference equation to find how much money a person could borrow given the following conditions. 53) Payments of $590 made annually for 13 years at 6% compounded annually A) $4946.44 B) $5221.21 C) $5223.08 D) $5484.05 Answer: C 54) Payments of $3500 made annually for 25 years at 6% compounded annually A) $44,741.75 B) $44,844.10 C) $45,251.50 D) $44,471.00 Answer: A 55) Payments of $75 made quarterly for 10 years at 4% compounded quarterly A) $2440.16 B) $736.36 C) $2462.60 D) $2017.70 Answer: C Use a difference equation to find how much money will be in an account given the following conditions. 56) $800 deposited at the beginning of each year for 14 years at 6% compounded annually A) $29,345.39 B) $16,012.05 C) $17,820.78 D) $14,305.71 Answer: C 57) $1500 deposited at the beginning of each year for 12 years at 6% compounded annually A) $23,804.91 B) $20,957.46 C) $26,823.21 D) $48,804.91 Answer: C 58) $200 deposited at the beginning of each quarter for 7 years at 4.9% compounded quarterly A) $6154.57 B) $1703.00 C) $6432.41 D) $6713.66 Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 59) Determine the amount of money accumulated after two years when $2 is deposited at 30% interest compounded (a) annually. (b) semiannually. (c) quarterly. (d) monthly. Answer: (a) $3.38 (b) $3.50 (c) $3.57 (d) $3.62 60) Determine the amount of money accumulated after twenty years when $10,000 is deposited at 12% interest compounded (a) annually. (b) semiannually. (c) quarterly. (d) monthly. Answer: (a) $96,462.93 (b) $102,857.18 (c) $106,408.91 (d) $108,925.54 Noah deposits $1,000,000 into a bank account paying 6% interest compounded annually and withdraws $75,000 at the end of each year. 61) (a) Write an appropriate difference equation. (b) How much money will be in the account in 20 years? (c) Would you rather have $1,000,000 now or $75,000 a year for 20 years? Answer: (a) yn = 1,250,000-250,000(1.06) n (b) $448,216.13 (c) $1,000,000 as there is $448,216.13 left in the account versus the $250,000 gained by taking $75,000 a year for twenty years. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 62) The graph of the difference equation yn = -2yn-1 + 3 is A) oscillating and attracted to the line y = 1. B) monotonic and repelled by the line y = 1. C) monotonic and attracted to the line y = 1. D) oscillating and repelled by the line y = 1. E) none of these Answer: D 63) The graph of the difference equation yn = 0.98yn-1 + 2, y0 = 200 is A) oscillating and attracted to the line y = 100. B) increasing and attracted to the line y = 100. C) decreasing and repelled by the line y = 100. D) decreasing and attracted to the line y = 100. E) none of these Answer: D

64) The graph of a difference equation is shown below.

This graph is A) increasing and attracted to the line y = -4. B) monotonic, oscillating, and repelled by the line y = -4. C) increasing, unbounded, and attracted to the line y = -4. D) increasing, unbounded, and repelled by the line y = -4. E) none of these Answer: D 65) The graph of a difference equation is shown below.

This graph is A) monotonic and attached to the line y = -2. B) oscillating and repelled from the line y = -2. C) oscillating, unbounded, and repelled from the line y = -2. D) oscillating and attracted to the line y = -2. E) none of these Answer: D

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Below are graphs of difference equations of the form yn = ayn-1 + b. List the graphs which have the stated property. I.

II.

III.

IV.

66) increasing Answer: II and IV 67) unbounded Answer: I and II 68) oscillating Answer: I and III 69) attracted to the line y =

b 1-a

Answer: III and IV Solve the problem. 70) Is the solution to yn = -2yn-1 + 100, y0 = 0 monotonic or oscillating? Answer: oscillating 71) Is the solution to yn = .5yn-1 + 8, y0 = 48 monotonic or oscillating? Answer: monotonic 72) The graph of yn = Answer: y =

1 y + 1 is asymptotic to what line? 2 n-1

2 3

73) Is the graph of yn = .5yn-1 + 8 attracted to or repelled from the line y = 16? Answer: attracted to 74) Is the graph of yn = -

3 8 y - 2 attracted to or repelled from the line y = - ? 4 n-1 7

Answer: attracted to 75) Consider the difference equation yn =

1 y + 3, y0 = 2. 2 n-1

(a) Classify the difference equation as constant, oscillating, or monotonic. (b) Classify the difference equation as attracting or repelling. Answer: (a) monotonic (b) attracting 76) Consider the difference equation yn = -

1 1 y + , y = -1. 3 n-1 2 0

(a) Classify the difference equation as constant, oscillating, or monotonic. (b) Classify the difference equation as attracting or repelling. Answer: (a) oscillating (b) attracting Make a rough sketch of the given difference equation without generating terms or solving the difference equation. 77) yn = 4yn-1 - 9, y0 = 3

Answer:

78) yn = 0.8yn-1 + 2, y0 = 6

Answer:

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 79) Which of the following could be the graph of yn =

3 y + 1? 2 n-1

Answer: D Sketch a graph having the given characteristics. 80) y0 = 20, monotonic, attracted to y = 10 y

A) y

B) y

C) y

D) y

Answer: C

81) y0 = 10, oscillating, repelled from y = 8 y

A) y

10 8

B) y

10 8

C) y

10 8

D) y

10 8

Answer: A Solve the problem. 82) Under ideal conditions, a bacteria population satisfies the difference equation yn = 2.3yn-1 , y0 = 1.5, where yn is the population, in millions, after n hours. Sketch a graph that shows the growth of the population. y

A) y 60

B) y 60

C) y 60

D) y 60

Answer: A

83) Jeremy makes an initial deposit into a savings fund paying 1.4% compounded quarterly. He plans to withdraw $200 at the end of each quarter. (a) Find the difference equation for yn , the amount of money in the account after n quarters. (b) How large must the initial deposit be so that Jeremy does not run out of money? A) (a) yn = 1.4yn-1 - 200 (b) at least $200.00 B) (a) yn = 1.4yn-1 - 200 (b) no more than $200.00 C) (a) yn = 1.0035yn-1 - 200 (b) no more than $57,142.86 D) (a) yn = 1.0035yn-1 - 200 (b) at least $57,142.86 Answer: D A woman needs $6720 in 12 years to pay for her child's college tuition. She wants to deposit an equal amount at the end of each year into an account paying 10% interest compounded annually. Let yn represent the amount in the account at the end of the year. 13 12 = 2.6, (1.1)12 ≈ 3.1 12 84) Find the difference equation. A) yn = 1.1yn-1 , y0 = 0 B) yn = 1.1yn-1 + D, y0 = 6720 C) yn = 1.1yn-1 + D, y0 = 0 D) yn = 0.1yn-1 + D, y0 = 6720 E) none of these Answer: C 85) Find the solution for the difference equation. A) yn = 6720 + 672n B) yn = -10D + 10D(1.1)n C) yn = 6720(1.1)n D) yn = 10D - 10D(1.1)n E) none of these Answer: B SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Write the appropriate difference equation. (Do not solve the difference equation.) 86) A student's college fund is initially $10,000 invested in a savings account paying an annual interest rate of 6% compounded monthly, but $250 is withdrawn each month. Answer: yn = 1.005yn-1 - 250, y0 = 10,000 87) An annuity requires quarterly payments of $150 and pays interest at an annual rate of 8% compounded quarterly. Answer: yn = 1.02yn-1 + 150, y0 = 0

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 88) Joe wants to retire in 20 years. How much does Joe need to place into an account today to retire with $1,000,000 if the account pays 8% interest compounded annually? A) $214,548.20 B) $46,319.34 C) $463,193.49 D) $21,458.20 E) none of these Answer: A A woman needs $6720 in 12 years to pay for her child's college tuition. She wants to deposit an equal amount at the end of each year into an account paying 10% interest compounded annually. Let yn represent the amount in the account at the end of the year. 13 12 = 2.6, (1.1)12 ≈ 3.1 12 89) The yearly deposit is most near A) $560. B) $181. C) $210. D) $320. E) none of these Answer: D George took out a mortgage for $40,000 at an annual interest rate of 12% compounded monthly for 25 years. (1.01) 25 ≈ 1.3, (1.12) 25 ≈ 17.0, (1.01) 300 ≈ 19.8, (1.01)120 ≈ 3.3, (1.12) 10 ≈ 3.1 90) His monthly payment most closely approximates A) $440. B) $425. C) $421. D) $133. E) none of these Answer: C 91) After making payments for 10 years, the balance on the loan most closely approximates A) $24,000. B) $35,170. C) $0. D) $35,790. E) none of these Answer: B 92) The total amount of interest paid is A) $86,300. B) $132,000. C) $4800. D) $120,000. E) none of these Answer: A

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 93) Suppose that you can afford to pay $450 per month on a mortgage and the yearly interest rate is 9% compounded monthly. The mortgage is to be paid off in 20 years. [(1.09) 20 ≈ 5.60, (1.0075) 20 ≈ 1.16, (1.0075)240 ≈ 6.01] (a) Write the difference equation for yn , the balance on the mortgage after n months. (b) How much can you borrow? Answer: (a) yn = 1.0075yn-1 - 450 (b) $50,116.64 94) How much money would you have to deposit at the end of each year into an annuity paying 5% interest compounded annually in order to have $39,600 at the end of thirty years? [(1.05) 30 ≈ 4.3, (1.5)30 ≈ 191,751, (1.025)60 ≈ 4.4] Answer: $600 95) Suppose on a mortgage you can afford to pay $400 dollars per quarter and the yearly interest rate is 16% compounded quarterly. Exactly how much can you borrow if the mortgage is to be paid off in 30 years? [(1.12) 30 = 30.0, (1.04) 120 = 110.7, (1.04) 30= 3.2] Answer: $9909.67 96) Consider a 30-year mortgage for $9000 at 6% interest compounded monthly. Find the monthly payment. [(1.005)360 = 6.0, (1.06) 30 = 5.7, (1.006)360= 8.6] Answer: $54 97) Suppose $50 is deposited into an annuity at the end of every month with 12% interest compounded monthly. How much money is in the annuity after three years? [(1.12) 36 = 59.1, (1.012)3 = 10.4, (1.01) 36= 1.4] Answer: $2000 98) How much money would you have to place into an account initially at 10% compounded annually to have $10,000 after 5 years? Answer: $6209.21 99) How much money must be deposited at the end of each quarter into an annuity at 8% interest compounded quarterly in order to have $12,000 after 15 years? [(1.02) 60 = 3.28, (1.15) 8 = 3.06, (1.08) 15= 3.17] Answer: $105.26

100) A loan for $35,000 is taken at an annual rate of interest of 8% compounded quarterly. The loan is to be paid off in 30 years. [(1.02) 30 ≈ 1.8, (1.02) 120 ≈ 10.8, (1.08) 30 ≈ 10, (1.08) 120 ≈10,253] (a) Write an appropriate difference equation. (b) Determine the quarterly payment. (c) Determine the total amount of interest paid. Answer: (a) yn = 1.02yn-1 - R, y0 = 35,000 (b) $771.43 (c) $57,571.60 101) Chris and Karen are looking to purchase a home. They have found a lender offering a mortgage at 6% compounded monthly. They can afford to pay $900 per month. (a) Set up the difference equation for yn , the loan balance after n months. (b) How much money can they borrow? Answer: (a) yn = 1.005yn-1 - 900 (b) less than $180,000 Use a graphing calculator to answer the question. 102) Pat buys a computer for $2100 and pays for it with monthly payments of $120. The interest rate is 8.9% compounded monthly. (a) Determine the amount owed after six months. (b) How many months will it take to repay the loan? Answer: (a) $1461.72 (b) 19 months 103) Noah buys a car for $4000 and pays for it with monthly payments of $133. The interest rate is 12% compounded monthly. (a) Determine the amount owed after six months. (b) How many months will it take to repay the loan? Answer: (a) $3427.86 (b) about 36 months MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 104) If $500,000 is to be saved over 30 years, how much should be deposited monthly if the investment earns 4.75% interest compounded monthly? A) $402.56 B) $539.60 C) $465.12 D) $629.07 Answer: D 105) If $100,000 is to be saved over 30 years, how much should be deposited annually if the investment earns 8.75% interest compounded annually? A) $586.14 B) $768.59 C) $1014.00 D) $1348.87 Answer: B

106) In order to purchase a home, a family borrows $75,000 at an annual interest rate of 6.28%, to be paid back over a 20 year period in equal monthly payments. What is their monthly payment? A) $19.63 B) $549.51 C) $392.50 D) $605.77 Answer: B 107) In order to purchase a home, a family borrows $279,000 at 7.6% for 15 yr. What is their monthly payment? Round the answer to the nearest cent. A) $1767.00 B) $3021.52 C) $19,556.28 D) $2602.24 Answer: D A certain bacterial culture grows in such a way that each hour the increase in the number of bacteria in the culture is 60% of the total number present at the beginning of the hour. Let yn be the number of bacteria in the culture after n hours. 108) Find the difference equation for yn . A) yn = 0.6yn-1 B) yn = 1.4yn-1 C) yn = 0.4yn-1 D) yn = 1.6yn-1 E) none of these Answer: D 109) If there were 10,000 bacteria originally, then after two hours the number of bacteria most closely approximates A) 26,000. B) 22,000. C) 19,600. D) 25,600. E) none of these Answer: D 110) If there were 10,000 bacteria originally, then after twenty-four hours the number of bacteria most closely approximates A) 79,228,162 B) 89,228,162 C) 792,281,625 D) 892,291 E) none of these Answer: C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 111) A group of toddlers is standing atop a mountain covered with snow. They make a snowball and roll it downhill. Every foot they roll it, the snowball increases in diameter by 50%. Write the difference equation for the diameter yn after the snowball is rolled n feet. Answer: yn = 1.5yn-1 26

112) A ball is dropped from the top of a building. With every successive bounce, it only reaches

2 of the height of the 3

previous bounce. Write the difference equation for the n th bounce . Answer: yn =

2 y 3 n-1

113) A certain bacterial culture grows in such a way that each hour the increase in the number of bacteria in the culture is 70% of the total number present at the beginning of the hour. (a) Write the difference equation for yn , the number of bacteria in the culture after n hours. (b) If there are 5000 bacteria present initially, approximately how many bacteria are there at the end of three hours? Answer: (a) yn = 1.7yn-1

(b) 24,565

114) A car was bought for $8000. Assume that the trade in value of the car decreases by 30% each year. Let yn be the the trade-in value of the car at the end of year n. (a) Write the difference equation for yn . (b) What is the trade-in value of the car at the end of three years? Answer: (a) yn = 0.7yn-1 , y0 = 8000 (b) $2744 115) The number of bacteria increases by

1 in population every hour. Write a difference equation expressing this 10

relationship. Answer: yn = yn-1 +

1 y = 1.1yn-1 10 n-1

116) Each year a population loses 5% due to emigration. If the population today is 500,000, what will it be in three years? [(0.05) 3 = 0.000125, (0.95) 3 = 0.86, (1.05) 3 = 1.6] Answer: 430,000 117) For each gray hair that a man plucks from his head, two grow back the next day. Write a difference equation for the number of gray hairs, given that he plucks three a day. Answer: yn = yn-1 + 3 118) A bacterial culture increases in population every hour with a rate proportional to the difference between 10,000 and the given population. Write a difference equation for yn , the population after n hours. Answer: yn = yn-1 + k(10,000 - yn-1 ) Use a graphing calculator to answer the question. 119) The 2007 population of a small country is 1.2 million. Suppose the birth rate is 4.5% per year, the death rate is 3.2% per year, and there is a net movement of people into the country at a rate of 400 people per year. (a) Estimate the population in the year 2010. (b) When will the population exceed 1.5 million? (c) When will the population exceed 2 million? Answer: (a) approximately 1.25 million (b) 2024 (c) 2046

Write the appropriate difference equation. (Do not solve the difference equation.) 120) The population of a country is now 20 million with a 2% per year birth rate and a 3% per year death rate. Answer: yn = 0.99yn-1 , y0 = 20,000,000 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 121) If a population of 1000 bacteria doubles in size every 6 minutes, approximately how many minutes will it take for the population to grow to 500,000? A) 9 B) 54 C) 72 D) 90 E) none of these Answer: B 122) A culture of 106 bacteria doubles in size every 12 minutes. How many bacteria are present after 2 hours? A) 2 12(10)6 B) 122 (10)6 10 C) ((10)6 ) D) 2 10(10)6 E) none of these Answer: D 123) A car was bought for $7000. Assume that the trade-in value of the car deceases by 20% each year. Let yn be the trade-in value of the car at the end of year n. After four years, what is the approximate trade-in value for the car? A) $1120 B) $2245 C) $2867 D) $1400 E) none of these Answer: C 124) The value of an investment doubles every month for 10 months. After the tenth month, the investment was worth $1,600,000. What was the investment worth after the third month? A) $1250 B) $12,500 C) $50,000 D) $25,000 E) none of these Answer: B 125) By how much does the value of an investment increase in 3 weeks if it grows at a rate of $25 per day? A) $375 B) $75 C) $525 D) $750 E) none of these Answer: C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 126) Suppose a population gains 2% each year due to the excess of births over deaths. If the population in 2000 was 1 million, what is the population in 2010? Answer: 1.22 million 127) When a cup of coffee is poured, its decrease in temperature each minute is proportional to the difference between the coffee temperature and the room temperature at the beginning of the minute. Suppose that the room temperature is 60°, the temperature of the coffee is 150°, and the constant of proportionality is 0.25. (a) Write the difference equation for yn , the temperature of the coffee after n minutes. (b) Solve the difference equation. (c) What is the temperature of the coffee after two minutes? (d) Make a rough sketch of the graph of the solution of the difference equation in (b).

Answer: (a) yn = 0.75yn-1 + 15, y0 = 150 (b) yn = 60 + 90(0.75) n (c) 110.625° (d)

128) Sunlight is absorbed by water and so, as one descends into the ocean, the intensity of light diminishes. Suppose that at each depth, going down one more meter causes a 15% decrease in the intensity of sunlight. (a) Write the difference equation for yn , the intensity of the light at a depth of n meters. (b) If the intensity of the light at the surface is 500 units, what is the intensity of the light at a depth of two meters? (c) Graph the difference equation.

Answer: (a) yn = 0.85yn-1

(b) 361.25 units

(c)