Math 8 1 st 9 Weeks Benchmark Review - 2008
Name___________________
1) A computer password consists of 5 vowels. The password is case sensitive, which means upper-case and lower-case vowels are different characters. What is the probability of randomly being assigned the password AeiOu? Because the question asks about probability we are looking for a ratio answer. There are 5 vowels (a,e,I,o,u) since upper and lower case are counted as separate characters we have total of 10 outcome choices. There is only 1 way each of the above letters can occur. 1 1 1 1 1 1 ⋅ ⋅ ⋅ ⋅ = 10 10 10 10 10 100, 000 2) North Kell High School requires all staff members to have a 6-character computer password that contains 2 letters followed by 4 numbers. Find the number of possible passwords. Because the question asks about outcomes we are not looking for a ratio answer. Use the counting principle. Ask yourself what are the number of outcomes in each position of the password? 26 x 26 x 10 x 10 x 10 x 10 = 6,760,000 3) If A and B are independent events such that P(A)=
4 3 and P(B)= , what is 5 4
P(A and B)? 4/ 3 3 ⋅ = 5 41 5
1
And means multiply
4) Wyne has two boxes. Box 1 contains 4 bows, 3 clips, 2 headbands, and 3 bands. She also has a second box containing 2 bows and 3 combs. If Wyne selects an item from Box 1 and then an item from Box 2, what is P(bow, comb)? Box 1 contains 12 items with 4 being bows so the probability for 4 box 1 is . Box two contains a total of 5 items with 3 being combs 12 3 so the probability for box 2 is . Because the probability question 5 is separated by a comma it is an implied and then meaning to multiply the probabilities of the two boxes together. 4 1 31 1 ( )⋅ = 12 1 3 5 5
5) The lottery has 20 balls in each machine containing numbers 0-19. There are 2 different machines. What is the probability that each machine would produce a 9? 20 outcomes in each machine with only one outcome being a 9 so 1 each machine’s probability is multiplying the two machines together gives 20 1 1 1 ⋅ = 20 20 400 6) Sari rolls two 6-sided numbered cubes. What is the probability that the two numbers added together will equal 3? Making a table of the sums is the best way to find the outcomes dice 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 The table shows 36 outcomes with 2 sums of three. This gives a probability 2 1 = of 36 18 7) The Black and White party is being catered. The caterers offer 3 appetizers, 2 salads, and 3 main courses for each student to choose for dinner. If the caterers would like 36 different combinations of dinners, how many desserts should they offer? Use the counting principle 3 x 2 x 3 x ? = 36; therefore x = 2 desserts 8) There are 52 cards in a deck. What is the probability of drawing an King or a spade? There are 4 kings and 13 spades totaling 17 cards. One of the kings is a spade and must be subtracted from the total. Therefore the answer 16 4 = would be 52 13 9) There are 52 cards in a deck. What is the probability of drawing an three,
replacing it, and then drawing a Jack? 4 4 Probability of three is and the probability of a jack is multiplying 52 52 10) If A and B are independent events such that P(A)=0.27 and P(B)=0.32, what is the P(A or B)? Or means add so 0.27 + 0.32 = 0.59 11) Akala is building a wooden planter for her flowers in the shape of a right triangle. She has calculated the longest board to be 12 feet and one of the other boards to be 10 feet . What would be the length of the other side?
10
Draw a picture. 12
Understand the Pythagorean theorem and that 12 would be the hypotenuse a 2 + b2 = c2 a 2 + 102 = 122 a 2 + 100 = 144 a 2 = 44 a 2 = 44 a ≈ 6.6
12) To the nearest meter, how long is the diagonal of a 6 m. by 9 m.
rectangular court yard ? Draw a picture and use the Pythagorean Theorem
a 2 + b2 = c2 62 + 92 = c 2
c
36 + 81 = c
6
117 = c 2 117 = c 2 10.8 ≈ c
9
13) Which point on the number line is closest to 28 ? A B C D -1
0
1
2
3
4
5
28 falls between the perfect squares of 25 and 36. the square root of 25= 5 and the square root of 36 = 6. Therefore 28 is between 5 and 6 so point D.
14) Solve: Estimate the ± 179 169 13 179 is between 169 and 196, but closer to 169. The square root of 169 is 13 so estimated 13.3. Since the question asks for the positive and negative I must answer 13.3 and -13.3
15) Is 42 rational or irrational? Is 169 rational or irrational? 42 is not a perfect square and therefore an irrational number. 169 is a perfect square so it is a rational number.
17) Simplify: 6 12 ⋅ When multiplying you multiply coefficients by coefficients and radicands by radicand. In this problem the coefficients are both one. Once I multiply 6 and 12 I must factor to simplify.
6 ⋅12 NO 2 ⋅ 3⋅ 2 ⋅ 2 ⋅3 I see a pair of 2’s and a pair of 3’s that I must move outside the radical This leaves a 2 under the radical. My final answer would be 6 2
18) Subtract: 3 2 − 7 2 When subtracting or adding I can only add or subtract the coefficient of like radicands. In this case both radicands are 2 so I can subtract 3-7 giving me -4. My final answer would be −4 2 19) Simplify:
5 10
It is easiest to start by simplifying the fraction under the radical first. This 5 1 1 = = gives me The problem is not in simplest form when there is 10 2 2 a radical in the denominator. To “rationalize” the denominator I multiply both the numerator and denominator by the radical in the denominator. This creates a perfect square in order to eliminate the radical in the denominator. 1 2 2 2 ⋅ = = 2 2 2 4 20) What is 7( n −3) when n=1? Replace the n with 1 and you will get 7-2. The negative exponent must 1 1 be moved to the denominator and then evaluated. This give 2 = 7 49 21) When is an exponent expression equal to 1? Whenever the exponent is 0.
22) Simplify: 7-3(74) 7
(-3+4)
We are multiplying with like bases so we add the exponents. This give which is 71 or 7
23) A lab technician measured the bacteria and found it to be 7.3x 10−3 inches. What is this number in standard form? The decimal is moved 3 places to the left
0.0073 inches
24) What is the product of 20,000 and 30 written in scientific notation? The short method is to multiply the front digit and then use the zeros to determine the exponent. 2 x 3 = 6 with 5 zeros. The answer would be 6 x 105 25) Simplify the quotient and write the answer in scientific notation: ( 9 x105 ) á (3 x 103) Again the short method is to divide the front digit and then subtract the exponents since the bases are alike. 9 divided by 3 = 3 and the exponents of 5-3 gives 2. This would yield an answer of 3 x 102 a 9 x105 ; In this format we can see b 3 x103 how to divide the 9 and 3 and then subtract the exponent of the like bases. Sometimes if helps to see it written in the