Математическая часть

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Contents Course Syllabus………………………..………………………………………………………………………………………………………………....4 Introduction ........................................................................................................................................................ 5 Lesson 0. Tips and tricks in GMAT ................................................................................................................... 6 Lesson № 1. Arithmetic ................................................................................................................................... 14 Numbers ....................................................................................................................................................... 14 Divisibility.................................................................................................................................................... 20 Decimals and Fractions ................................................................................................................................ 26 Digits ............................................................................................................................................................ 35 Powers and Roots ......................................................................................................................................... 40 Home assignment ......................................................................................................................................... 49 Test№1. Number properties, fractions, digits, powers and roots, ratios, percent, divisibility ..................... 61 Lesson № 2. Arithmetic ................................................................................................................................... 64 Progressions ................................................................................................................................................. 64 Statistics ....................................................................................................................................................... 71 Home assignment ......................................................................................................................................... 82 Test № 2. Progression and statistics ............................................................................................................. 90 Lesson № 3. Arithmetic ................................................................................................................................... 94 Factorization................................................................................................................................................. 94 LCM and GCD ............................................................................................................................................. 97 Division with Remainder ........................................................................................................................... 103 Home assignment ...................................................................................................................................... 110 Test № 3. Prime factorization, LCM, GCD, division with remainder ....................................................... 118 Lesson № 4. Algebra ...................................................................................................................................... 121 Functions. Substitution. Symbolism........................................................................................................... 121 Simplifying algebraic expressions ............................................................................................................. 126 Linear equations. Systems of linear equations. .......................................................................................... 130 Absolute Value. Equations and Inequalities with Absolute Value............................................................. 135 Quadratic Equations ................................................................................................................................... 139 Exponential equations ................................................................................................................................ 145 Inequalities. Exponential inequalities. ....................................................................................................... 148 Home assignment. ...................................................................................................................................... 153

Test № 4. Algebra. ..................................................................................................................................... 161 Lesson № 5. Word Problems ......................................................................................................................... 165 Word translations ....................................................................................................................................... 166 Rate problems............................................................................................................................................. 171 Work problems ........................................................................................................................................... 179 Mixture Problems ....................................................................................................................................... 183 Percentage and Profit Problems ................................................................................................................. 188 Overlapping Sets Problems ........................................................................................................................ 195 Test № 5. Word problems .......................................................................................................................... 222 Lesson № 6. Geometry. .................................................................................................................................. 228 Triangles..................................................................................................................................................... 232 Quadrilaterals ............................................................................................................................................. 242 Polygons ..................................................................................................................................................... 247 Circle .......................................................................................................................................................... 250 Volume geometry ....................................................................................................................................... 256 Coordinate geometry .................................................................................................................................. 261 Home assignment ....................................................................................................................................... 265 Test № 6. Geometry ................................................................................................................................... 278 Lesson № 7. Combinatorics & Probability .................................................................................................... 285 Combinatorics ............................................................................................................................................ 285 Probability .................................................................................................................................................. 298 Home assignment. ...................................................................................................................................... 307 Test № 7. Combinatorics, Probability ........................................................................................................ 321 Practice test .................................................................................................................................................... 327 Key Glossary .................................................................................................................................................. 343 Appendix. Arithmetic to Memorize ............................................................................................................... 350

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Arithmetic

Algebra

3

Exp equations

Inequalities

Quadratic equations Absolute value

Algebraic expressions Linear systems

Functions

Remainders

LCM & GCD

Factorization

Statistics

Progressions

Powers & roots

Digits

Decimals & Fractions

Divisibility

Numbers

Introduction

lesson 1

lesson 2

lesson 3

lesson 4

lesson 5

lesson 6

Course syllabus lesson 7

lesson 8

lesson 9

lesson 10

lesson 11

Word pr.

W. p.

Geometry

4

Test Day

Final test analysis

Final Test

IR

Probability

Combinatorics

Coordinates

3D Geometry

Circles

Quadrilaterals

Triangles

Sets

Percent

Mixture

Work

Rate

Word translat.

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Lesson 6

Lesson 7

Lesson 8

Lesson 9

Lesson 10

Lesson 11

Introduction (GoGMAT, Intro) The quantitative section of the GMAT contains 37 multiple choice-questions in a 75-minute period. The math topics include arithmetic, basic algebra and geometry (no proofs). The test writers carefully choose questions to eliminate biases toward candidates with specific majors: all test takers will be on a level playing field. The section is designed to test your ability to solve problems, rather than your mathematical knowledge. Questions lean heavily toward word problems and applying mathematical formulas in typical real-world applications. The handbook covers all themes with formulas, definitions, main types of questions and algorithms for solving. The questions include two formats: a) standard multiple choice b) data sufficiency Instructions for Data Sufficiency Questions: Directions: Each of the following Data Sufficiency problems contains a question followed by two statements, numbered (1) and (2). You need not solve the problem; rather you must decide whether the information given is sufficient to solve the problem. The correct answer to a question is A if statement (1) ALONE is sufficient to answer the question but statement (2) alone is not sufficient; B if statement (2) ALONE is sufficient to answer the question but statement (1) alone is not sufficient; C if the two statements TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient; D if EACH statement ALONE is sufficient to answer the question; E if the two statements TAKEN TOGETHER are still NOT sufficient to answer the question. Drawings: The drawings are drawn to scale according to the information given in the question, but may conflict with the information given in statements (1) and (2). You can assume that a line that appears straight is straight and that angle measures cannot be zero. You can assume that the relative positions of points, angles, and objects are as shown. All drawings lie in a plane unless stated otherwise. Abbreviations:

OG - Official Guide GC- GMAT Club MH -McGraw-Hill PR - The Princeton Review MG - Master The GMAT

SC - ScoreTop MathOG - Official Guide GC- GMAT Club MR -Manhattan Review GP - GMAT Problems Math Pack

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Lesson 0. Tips and tricks in GMAT The main goal in GMAT questions is not as to solve a problem as to find the correct answer. This could be done in different ways and of course, one of them is to solve a problem in usual way. But sometimes it could be very complicated and take a lot of time. Time is one of the main resources on the exam, so in this chapter we want to introduce some technics that could provide the right answer omitting the whole solution. The first trick corresponds to “algebraic problems”. A question is “algebraic problem” if it contains some variables (even if we are dealing with geometrical problems because this technic is also applied there) The main idea is to substitute a number for a variable.

Plug-in (Pl) 1. (GC) If P# = P/(P - 1), what is the value of P##? (A) P/(P - 1) (B) 1/P (C) P (D) 2 - P (E) P - 1 Let’s consider P=3. In this particular case we should find 3## 3# = 3/2 3## = (3/2)# = (3/2) / (3/2 – 1) = 3/2 : ½ = 3 Let’s see what happen with our answers: (A) P/(P - 1) = 3/2 (B) 1/P = 1/3 (C) P = 3 (D) 2 – P = -1 (E) P – 1 = 2 So, the right answer is (C). Note, that we must check each answer from (A) till (E) before we chose the correct one, because there is a chance to have some identical arithmetical solutions. Why did we chose P=3? We could have considered P=2, but in this case answer (A) is equivalent to the answer (C), so they both could be correct. It’s not a big problem with this kind of situation. In such case if P=2 we must eliminate definitely incorrect (B), (D) and (E) and the next time when we apply plug-in P=3, check only (A) and (C). On the one hand, we recommend using “simple numbers” such as 0, 1, -1, 2… On the other hand, it’s better to use 3 or 4 at once because some answers are identical with such

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numbers as 0 and 1, so just make a “jump” from the origin of the number line and use some numbers a little bit greater than 0 and 1. 2. (A) (B) (C) (D) (E)

If n is a positive integer number, calculate

1 1 2

1 1 1 ... 2 3 3 4 n(n 1)

1

n 1 n (n 1) 2 n n n 1 n 2 n 1

1 1 1 3 = . 1 2 2 3 3 4 4 Answer checking would give us: (A) 1, (B) 1/2, (C) 4/3, (D) ¾, (E) 3/10 (D) is correct answer. As it was in previews problem, the plug-in n = 2 gives 2 identical answers (B) and (C). Well, let’s consider n=3:

It’s not true to say that the plug-in in the form variable = 3 is a panacea but it really works in many algebraic problems. The main advantage of this method is little time you spend when solve the problem and very simple calculations. Don’t do any algebraic calculation if you can replace them by arithmetical ones. It’s always faster and easier! When does this method work? It works always, but sometimes it’s very hard to apply the plug-in. What do two previews problem have in common? There aren’t almost any restrictions. In 1st problem P could be any real number except P=1 and in the 2nd one n>0 and n is an integer are the only constraints of n. Let’s consider the next example: 3. (MR) If P2 - QR = 10, Q2 + PR = 10, R2 + PQ = 10, and R ≠ Q, what is the value of P2 + Q2 + R2 ? (A) 15 (B) 20 (C) 25 (D) 30 (E) 10

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We can try to consider P = 3. Then - QR = 10, Q2 + R = 10, R2 + Q = 10. But there is no such numbers R and Q satisfying conditions below (check this). We can even try P = 2 and P = 1, but it doesn’t  give  us  any  result.  In  this  problem  there  are  some  really  essential  constraints,  so  we  can’t  substitute  any  number  we  want. This is a type of problems where plug-in  works  but  it’s  very  hard to find the valid example. Before you start to solve using mathematic method you can try plug-in with some simple numbers, and  if  it  doesn’t  work,  resolve  the  problem  as  usual. Here Q = 0  works.  It’s  easy  to  check  that  P  =  Q  =  âˆš10 satisfies our conditions. Thus, P2 + Q2 + R2 = 10 + 10 = 20. (B) is the answer. 4.

(DP) In the square to the right, 12w = 3x = 4y. What fractional part of the square is shaded?

(A) (B) (C) (D) (E)

2/3 14/25 5/9 11/25 3/7

This  problem  is  geometrical.  But  it’s  also  â€œalgebraicâ€?  problem  because it contains some variables. Let’s  use  plug-in. w = 1. Then x = 4, y = 3 and we have a square with side = x + w = 5 and area = 5Ă—5 = 25:

Areas of the corner triangles are 1Ă—4/2 = 2, 1Ă—2/2 = 1, 3Ă—4/2 = 6, 1Ă—4/2=2  clockwise.  That’s  why  bounded  area = 25 – 2 – 1 – 6 – 2 = 14 and the ratio we need = 14/25. (B) is the answer. Check the general conclusion: “S(bounded)  =  S(square)  â€“ S(triangle 1) - S(triangle 2) - S(triangle 3) - S(triangle 4) = (đ?‘¤ + đ?‘Ľ) – wx/2 –đ?‘¤ – xy/2 – wx/2 = đ?‘Ľ + wx – xy/2 = 14đ?‘¤ , When S(square) = (đ?‘¤ + đ?‘Ľ) = 25đ?‘¤ , so the ratio = 14đ?‘¤ /25đ?‘¤ =  14/25â€? It has the same idea but is more complicated than the arithmetical solution.

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Geometric Plug-in (gPl) Let’s  start  with  the  next  problem: 5. Point E lies inside the rectangle ABCD. If EA= a , EB= b and EC = c then ED is equal to

(A) (B)

a 2 4b2 c 2

a 2 b2 c 2

(C) a 2 2b 2 c 2 (D) a b c (E)

2a 2 b2 2c 2

Let’s  consider  P=C.  In  this  type  of  questions  we  shift  the  geometrical  position  of  the  point  (the  variable  is  this  position)  in  such  a  way  that  it’s  easy  to  express  our  length  ED.

Using Pythagorean rule we obtain  đ??¸đ??ˇ = đ?‘Ž − đ?‘? , so đ??¸đ??ˇ = √đ?‘Ž − đ?‘?  , If P=C then c=0, so (B) is the one answer that satisfies our conditions.

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Last digit (LD) 6.

Electricity usage in a certain household on May 1: Number of Hours in Use

Number of Watts of Electricity Used per Hour

TV

4

145

Computer

3

155

VCR

2

45

Stereo

2

109

Appliance

According to the table above, what was the total number of watts of electricity used for the four appliances in the household on May 1? (A) (B) (C) (D) (E)

454 860 1,100 1,230 1,353

We don’t need to calculate the exact value if the problem is an arithmetical question (i.e. there aren’t any variables) and the answers end with different digits. We have to find only the last digit of all operations. The answer would be 4×145 + 3×155 + 2×45 + 2×109 4×145 ends with 0, 3×155 ends by 5, 2×45 ends with 0 and finally 2×109 ends with 8. 0 + 5 + 0 + 8 = 13, Thus, the whole sum ends with 3. (E) is the answer.

Answers picking (AP) 7. (A) (B) (C) (D) (E)

Find the least positive number, which leaves, if divided by 3, 5, and 12, a remainder of 2. 62 42 12 2 122

Let’s check our answers: (A) 62 gives remainder of 2 when is divided by 3, 5 or 12. (B) 42 is divisible by 3, (C) 12 is divisible by 12, (D) 2 satisfies all of the conditions and (E) also is “good”. But (D) has the smallest number among all 3 valid options. Thus, (D) is the answer. Of course, when we pick the least value we should start with the least, so (D) must be the 1st option we check.

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Odd/even tricks There are 4 operations in elementary arithmetic: +, - ,  âˆ™  and  : Three  of  them  are  â€œgoodâ€?  (+,  - ,  âˆ™)  and  one  (:)  is  bad  because  if  we  apply  this  operation  on  integer  numbers we can obtain non-integer  one.  So,  division  is  one  of  the  4  operation  that  don’t  save  some  â€œgoodâ€?  properties  of  the  integer  numbers.  For  example,  division  doesn’t  keep  some  odd/even  properties. I.

There is one important hint when working with odd/even problems and good operations: If we want just to find whether an expression is odd or even, every even number in this expression could be substituted by 0, and every odd number could be replaced by 1. The reason of this mathematical fact is the behaviour of the remainders (which could be 0 or 1 only) if numbers are divided by 2 when numbers are multiplied or added. Let’s  see  the  example: n is an even number and m is an odd one. Is 3nđ?‘š +(n-m)(n+2m) odd ? We can consider m->1, n->0 then 3nđ?‘š +(n-m)(n+2m)  =  3∙0∙1  +  (0-1)(0+2) = -2 -2 is even, so 3nđ?‘š +(n-m)(n+2m) is always even (that what the theorem says) Another odd/even trick can help solve the problem II.        đ??źđ?‘“  đ?‘›  đ?‘–đ?‘  đ?‘Žđ?‘›  đ?‘–đ?‘›đ?‘Ąđ?‘’đ?‘”đ?‘’đ?‘&#x;  đ?‘Žđ?‘›đ?‘‘  đ?‘˜  đ?‘–đ?‘  đ?‘Ž  đ?‘?đ?‘œđ?‘ đ?‘–đ?‘Ąđ?‘–đ?‘Łđ?‘’  đ?‘–đ?‘›đ?‘Ąđ?‘’đ?‘”đ?‘’đ?‘&#x;  đ?‘Ąâ„Žđ?‘’đ?‘›  đ?‘› and đ?‘› are  both  even  or  both  odd.  So,  it  doesn’t  matter  what  do  we  use  đ?‘›  đ?‘œđ?‘&#x;  đ?‘› to find evenness of the expression, we can just eliminate the power k. Example: đ?‘š + đ?‘› = 2đ?‘š + đ?‘›, is m even? Let’s  eliminate  the  powers.  We  obtain  đ?‘š + đ?‘›  =  2đ?‘š + đ?‘› or đ?‘š = 0 – even, so m is always even. Note: pay attention if k could be equal to 0 in the expression đ?‘› . đ?‘› =1  (if  n≠0)  and  is  always  odd.

Similar answer (SA) Every question in GMAT has exactly one correct answer. So if you find 2 identical answers you must immediately eliminate them. Of course they are not symbol-by-symbol equivalent in any problem you choose,  but  sometimes  they  are  equivalent  by  the  sense  of  the  question.  Let’s  see  some  examples: 8. (A) (B) (C) (D) (E)

(OG) If positive integers x and y are not both odd, which of the following must be even? xy x+y x-y x+y+1 2(x+y)-1

Expressions x+y and x-y are both even or both odd. There is no difference in odd/even sense what sign has a number: + or - . So, there is no difference in evenness of x+y or x-y. We can eliminate

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(B) and (C). If (B) is a correct answer then (C) is a correct as well. But GMAT question has exactly one right answer. If z=y+1 then we can eliminate (D) x+z using the same idea. (E) is also odd, so (A) is the right answer. 9.

(OG) If đ?‘&#x; and đ?‘ are integers and đ?‘&#x;đ?‘ Â + Â đ?‘&#x; is odd, which of the following must be even?

(A) (B) (C) (D) (E)

đ?‘&#x; đ?‘ đ?‘&#x;+đ?‘ đ?‘&#x;+đ?‘ đ?‘&#x; +đ?‘

As we already know, we can erase all the powers. After this we obtain: (A) đ?‘&#x; (B) đ?‘ (C) r+s (D) r+s (E) r+s (C), (D) and (E) are identical, so we can eliminate them. r=1 and s=0 satisfy the condition (plug-in) thus (A) 1 is the wrong and (B) 0 is the right answer. 10.

If x2 – y2 < 0, which of the following must be true?

I. x+y<0 II. x – y < 0 III. x2 – y3 < 0 (A) None (B) I only (C) II only (D) I, and III only (E) II and III only In this question there is no difference whether y is positive or negative number (we are dealing with y or (-y) ) because the only condition is x2 – y2 < 0 and y2 = (-y)2 . So, we can conclude x2 – (- y)2 < 0 as well. It means I and II are identical: x-y = x + (-y). If  they  are  true  then  â€œI  and  IIâ€?  must  be  in  the  right  answer.  So,  the  answer is (A).

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Special property (SP) Sometimes it’s hard to find the right solution because the problem has one special idea and one special answer differs from the others. Let’s take a look at the example: 11. If x and у аrе factors of 165165, where x > y > 0, which of the following cannot be the value of x – y? (A) 2 (B) 4 (C) 8 (D) 15 (E) 28 Only one answer is odd ( D. 15) and the others are even. May be the main idea is odd/even property in this problem. 165165 is odd number that has odd factors only. x-y is even since x and y are odd. So, x-y ≠ 15. (D) is the answer. 12. (MR) Given that a, b, c, and d are different nonzero digits and that 10d + 11c < 100 – a, which of the following could not be a solution to the addition problem below?

abdc + dbca (A) 3689 (B) 6887 (C) 8581 (D) 9459 (E) 16091 Let’s take a look at hundreds digits of the answers. They are all even except (C). In (C) hundreds digit is 5. May be this is the main idea.

abdc + dbca * x** Let’s see the picture below. When is x odd digit? b+b = 2b – even, so we have additional unit when dc and ca are added. Thus, dc + ca > 100. It implies 10d + c + 10c + a > 100 10d + 11c > 100 – a. Contradiction. (C) could not be the value of our sum. (C) is the answer. As you see, some really hard problems could be done by using special tricks in 1-2 steps. In this book we will mark all the problems (by Pl, LD, SA, AP, SP) in the answer tables, so you can resolve such question using testing techniques even if you solved them in usual way before. Good luck!

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Lesson № 1. Arithmetic Numbers (GoGMAT, Session 1) Numbers are classified according to their type. The first type is the one you have known since elementary school: Natural numbers (whole positive numbers) – are the numbers used for counting: 1, 2, 3, 4, 5, … Adding 0 and the negatives of the naturals, we obtain Integers - numbers from the set {…, -2, -1, 0, 1, 2, 3, 4, 5, … } Note: The integer 0 is neither positive nor negative. Integers

Positive (greater than 0) {1, 2, 3, …}

1.

0

Negative (less than 0) {…, -5, -4, -3, -2 -1}

Learn more about (OG) The value of -3 – (-10) is how much greater than the value of -10 – (-3)? (A) -14 (B) -7 (C) 0 (D) 7 (E) 14

2.

×(

)

=

(A) -524 (B) 0 (C) 125/13 (D) 49 (E) 491 Integers

Even (divisible by 2) {…, -4, -2, 0, 2, 4, …} N=2k Note: The integer 0 is even: 0=2×0.

Odd (not divisible by two) {…, -5, -3, -1, 1, 3, 5, …} N=2k+1

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Properties of even/odd integers even × even = even even × odd = even odd × odd = odd even +/− even = even odd +/− odd = even odd +/− even = odd even = even or not integer odd even = even or odd or not integer even odd = odd or not integer odd 3. (OG) If n is a member of the set {33, 36, 38, 39, 41, 42}, what is the value of n? (1) n is even (2) n is a multiple of 3 4. (OG) If positive integers x and y are not both odd, which of the following must be even? (F) xy (G) x+y (H) x-y (I) x+y+1 (J) 2(x+y)-1 5. (OG) If n is a positive integer, is n odd? (1) 3n is odd (2) n + 3 is even The numbers –2, –1, 0, 1, 2, 3, 4, 5 are consecutive integers. Consecutive integers can be represented by n, n + 1, n + 2, n + 3, . . . , where n is an integer. The numbers 0, 2, 4, 6, 8 are consecutive even integers. Consecutive even integers can be represented by 2n, 2n + 2, 2n + 4, . . . , where n is an integer. The numbers 1, 3, 5, 7, 9 are consecutive odd integers. Consecutive odd integers can be represented by 2n + 1, 2n + 3, 2n + 5, . . . , where n is an integer. Note: if the numbers are given in the problem in a raw like a, b, c it does not mean that a<b<c. Properties: for every two consecutive integers n and n + 1 product n ∙ (n + 1) is divisible by 2; for every three consecutive integers n, n + 1, n + 2 product n ∙ (n + l) ∙ (n + 2) is divisible by 3 and by 2; for every k consecutive integers their product is divisible by all numbers from 1 to k. 6. (PR) The set consists of five consecutive integers. If the first member of this set is a, find the sum of all its members in terms of a. (A) 10 (B) a+5 (C) a+10 (D) 5a+5 (E) 5a+10

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7. (OG) If n is a positive integer, then n(n +1)(n + 2) is (A) even only when n is even (B) even only when n is odd (C) odd whenever n is odd (D) divisible by 3 only when n is odd (E) divisible by 4 whenever n is even 8.

(OG) If a, b, and с are consecutive positive integers and a < b < c, which of the following must be true? I. c - a = 2 II.abc is an even integer III.(a + b + c)/3 is an integer (A) I only (В) II only (С) I and II only (D) II and III only (E) I, II, and III

9. (OG) If x, y, and z are three integers, are they consecutive integers? (1) z – x = 2 (2) x < y < z A prime number is a positive integer that has exactly two different positive divisors: 1 and itself. For example, 2, 3, 5 are prime numbers, but 15 is not, since 15 has four different positive divisors, 1, 3, 5, and 15. First prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47... Note: If the number less than 100 is not divisible by 2, 3, 5, 7, it is prime. Note: The number 1 is not a prime number since it has only one positive divisor. Note: 2 is the only even prime. Indeed, if a prime greater than 2 were even, it would have at least three different divisors: 1, itself and 2. 10. The sum of prime numbers that are greater than 60 but less than 70 is (A) 67 (B) 128 (C) 191 (D) 197 (E) 260 11. (MH) Is the positive integer y a prime number? (1) 80 < y < 95 (2) y = 3x +1, where x is a positive integer

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GoGMAT Problem

17

Answers and explanations 1. Since -3 – (-10) = -3 + 10 = 7 and -10 – (-3) = -10 + 3 = -7, the first expression is greater than the second one by 7 – (-7) = 14. The answer is E. 2. At first, find the value of the expression in brackets: 372+128 = 500. Then, multiply it by 5: 5 × 500 = 2500. Subtracting 1863 gives us 2500-1863 =637. Then, divide this result by 13: 637/13 = 49. The answer is D. 3. From statement (1) we know that n belongs to the subset {36, 38, 42}. Thus, we cannot uniquely determine the value of n. Therefore (1) is not sufficient to answer the stem. Then, the answer must be B, C, or E. Statement (2) alone is again insufficient since it states that n belongs to the subset {33, 36, 39, 42}. The answer must be C or E. Statements (1) and (2) together are insufficient since from (1) and (2) we can derive that n belongs to {36, 42}. The answer is E. 4. If integers are not both odd, at least one of them is even. xy must be even since x and y are not both odd. A is the answer. B, C, D, E are wrong since it is unknown whether x and y are both even or not. 5. (1) alone is sufficient since the product of integers is odd only when all factors are odd. Odd = 3× n=3× odd. n is odd. (2) is also sufficient, since if n + 3 is even. The sum of two integers is even if they are both even or both odd. 3 is odd, and n must be odd. D is the answer. 6. If the first member of a set is a, the second will be a+1, the third = a+2 and so on. So, the sum can be found as: a+(a+1)+(a+2)+(a+3)+(a+4) = 5a+ (1+2+3+4) =5a+10. The answer is E. 7. n(n + l)(n + 2) is always divisible by 2 and 3, so A, B, C, and D are wrong. E is the answer. Indeed, if n is even, n(n + l)(n + 2) contains two even numbers: n and (n+2). So, their product is divisible by 4. 8. If a, b, c consecutive integers, a = a, b = a + 1, c = a + 2. Then c − a = a + 2 − 2 = 2. Also, abc must be an even integer since there is at least one even integer is among a, b, and с. Moreover, sum a + b + с = a + a + 1 + a + 2 = 3a + 3 = 3(a + 1) is also divisible by 3. You can also check it by picking the numbers. Thus, E is the answer. 9. From statement (1) we know that z – x = 2. Nothing is said about y, thus, we cannot make any inference concerning y. (1) is not sufficient to answer the stem. Then, the answer must be B, C, or E. Statement (2) alone is again insufficient since it states that x < y < z, but we still don’t know whether z – y = 1 and y – x = 1. Thus, statement (2) alone is insufficient to answer the stem question. The answer must be C or E. Statements (1) and (2) together are sufficient since from x < y < z, z - x = 2 we get z – y ≥ 1, y – x ≥ 1, and z – x ≥ 2. Thus, z – y = 1 and y - x = 1. The answer is C. 10. Numbers that are greater than 60 and less than 70 are: 61, 62, 63, 64, 65, 67, 68, 69. Note that 62, 64, 66, and 68 are divisible by 2; 63, 66, and 69 are divisible by 3; and 65 is divisible by 5. The only numbers that are left are 61 and 67. They are not divisible by 2, 3, 5, 7, so 61 and 67 are prime numbers. As 61 + 67 = 128, the correct answer is B. 11. (1) alone is insufficient because the range mentioned contains both prime numbers (e.g., 83, 89) and nonprime numbers (e.g., 84, 85). (2) alone is insufficient because it provides both prime (e.g., 7=3×2+1) and nonprime (e.g., 16 =3×5+1) values for y. If the statements are combined, the numbers that meet both criteria are 82, 85, 88, 91, and 94. None of them is prime: 82=2×41, 85=5×17, 88=2×44, 91=7×13, 94=2×47. So, the answer on a question is "No". Both statements together are sufficient. The answer is C.

18

GoGMAT Problem Explanation

19

Divisibility GoGMAT, Session 1 The fact, that 9 when divided by 3 yields an integer can be expressed in several ways: 9 is divisible by 3 9 is a multiple of 3 3 is a divisor of 9 3 is a factor of 9. There are three groups of criterions of divisibility: Divisibility is determined by last digits of a number (divisibility by 2, 4, 5, 8, 10, …);; Divisibility is determined by algebraic sum of all digits (divisibility by 3, 9 and 11); Divisibility is determined by mix of some criterions from first two groups (divisibility by 6, 12, 15, 18, 22…). Let’s consider an integer N=a,bcd. Divisor (N is a multiple of…)

Criterion

Examples

2

The last digit d is divisible by 2

84 is a multiple of 2 since 4 is a multiple of 2.

4

333 is NOT a multiple of 2 since 3 is NOT a multiple of 2. 3,124 is a multiple of 4 since 24 is a Two last digits cd form the integer multiple of 4. divisible by 4 1,333,334 is NOT a multiple of 4 since 34 is NOT a multiple of 4

8

Three last digits bcd form the integer divisible by 8

88,863,024 is a multiple of 8 since 24 is a multiple of 8. 17,723,001 is NOT a multiple of 8 since 1 is NOT a multiple of 8.

1.

Which of the following numbers, if added to 1379, would give the sum that is a multiple of 4? (A) 1503 (B) 19784 (C) 123760 (D) 1756323 (E) 189281001

20

Divisor (N is a multiple of…) 3

9

Criterion

The sum of all digits a+b+c+d is divisible by 3

The sum of all digits a+b+c+d is divisible by 9

Examples 177 is a multiple of 3 since 1+7+7=15 is a multiple of 3. 162,346 is NOT a multiple of 3 since 1+6+2+3+4+6=22 is NOT a multiple of 3. 1,737 is a multiple of 9 since 1+7+3+7=18 is a multiple of 9. 1,333,334 is NOT a multiple of 9 since 1+3+3+3+3+3+4=20 is NOT a multiple of 9.

2. Which one of the following is divisible by both 2 and 3? (A) 1005 (B) 1296 (C) 1351 (D) 1406 (E) 1414 Divisor (N is a multiple of…) 5

10

Criterion

Examples

The last digit d is 5 or 0

95 is a multiple of 5 since the last digit is 5.

The last digit d is 0

813 is NOT a multiple of 5 since the last digit is neither 5 nor 0. 7,120 is a multiple of 10 since the last digit is 0. 1,333,334 is NOT a multiple of 10 since the last digit is NOT 0.

11

The sum of digits on odd places minus the sum of digits on even places ((a+c)-(b+d)) is divisible by 11

824,472 is a multiple of 11 since (8+4+7)-(2+4+2)=11 is a multiple of 11. 17,723,001 is NOT a multiple of 11 since (1+7+3+0)-(7+2+0+1)=1 is NOT a multiple of 11.

If a number is not divisible by 5, the remainder when the number is divided by 5 is the same as the remainder when the last digit is divided by 5.

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If a number is not divisible by 10, the remainder when the number is divided by 10 is the same as the remainder when the last digit is divided by 10. Examples: The number 145,632 is not divisible by 5, since its last digit, 2, is neither 5 nor 0. When 145,632 is divided by 5, the remainder is 2, since 2 divided by 5 is 0 with a remainder of 2. The number 7,332,899 is not divisible by 10, since its last digit, 9, is not 0. When 7,332,899 is divided by 10, the remainder is 9, since 9 divided by 10 is 0 with a remainder of 9. Note: Divisibility by other integers can be checked using combinations of criterions listed above. Examples: The number 724,560 is divisible by 12, since the number formed by its last two digits, 60, is divisible by 4, and the sum of its digits is 30, which is a multiple of 3. The number 7,145,580 is divisible by 15, since its units digit is 0 and the sum of its digits, 30, is divisible by 3. 3.

Is number x divisible by 15? (1) x is a multiple of 6. (2) 35 is a factor of x.

4. Is 9,5a8 divisible by 6, where a represents any number of {0,1,2,3,4,5,6,7,8,9}? (1) a is even (2) a is prime

22

GoGMAT Problem

23

Answers and explanations 1. As criterion of divisibility by 4 bases on analysis of the last two digits, we only need to sum up 2 last digits of 1379 with the 2 last digits of the answer choices. 79+03 =82, which is not divisible by 4. 79+84 = 163, two last digits of this number form 63, which is not divisible by 4. 79+60 = 139, two last digits of this number form 39, which is not divisible by 4. 79+23 =102, two last digits of this number form 02, which is not divisible by 4. 79+01 =80, which is evenly divisible by 4. The answer is E. 2. Obviously, A and C cannot be correct answers, since they are odd numbers. Among B, D and E we choose B, because 1+2+9+6=18, which is divisible by 3. Hence, 1296 itself is divisible by 3. The answer is B. 3. To be divisible by 15, x needs to be divisible by its prime factors 3 and 5. From the first statement, if x is a multiple of 6, it as to be a multiple of both 2 and 3. So, we know that x is divisible by 3. But there is no information about 5. Insufficient. From the second statement, x is divisible by 35, so, it is divisible by both 5 and 7. But we have any information about divisibility by 3. When statements are taken together, we get enough info: x is divisible by both 3 and 5, so, it is divisible by 15 as well. The answer is C. 4. Since 6=2Ă—3, we will use both divisibility tests for 2 and 3. The last digit, 8, is even, so 95a8 is divisible by 2 for any a. The sum of the digits is 9+5+a+8=22+a. The number 9,5a8 will be divisible by 6 if 22+a would be divisible by 3. Statement 1 does not provide enough information, as if a=2, 22+a=24 is divisible by 3. From the other side, if a =4, 22+a=26 is not divisible by 3. Statement 2 does not provide enough information as well. If a=5, 22+a=27 is divisible by 3. If a =3, 22+a=25 is not divisible by 3. Two statements taken together tell us that a is even prime number, so, a=2. Then 22+a=24 which is divisible by 3. The answer is C.

24

GoGMAT Problem Explanation

25

Decimals and Fractions GoGMAT, Session 2 GMAT math goes beyond an understanding of the properties of integers or whole numbers. Let’s consider the numbers that fall in between the integers. They can be expressed by fractions or decimals.

Fractions In a fraction

n is the numerator and d is the denominator.

Note: The denominator of a fraction can never be 0, because division by zero is not defined. Simplifying a fraction The process of simplifying (or cancelling) is governed by one simple rule: multiplying or dividing both the numerator and the denominator by the same number does not change the value of the fraction. Example: 20 20 ÷ 5 4 = = 15 15 ÷ 5 3 12 12 ÷ 12 1 = = 60 60 ÷ 12 5 1. Which of the following is equal to 3/4? (A) 82/100 (B) 15/18 (C) 40/15 (D) 12/16 (E) 18/27 Fraction in lowest term A fraction is irreducible or in lowest term if a and b have no any positive common divisors except 1. For example,

is irreducible (in contrast to ), since no integer greater than 1 is a factor

of both 2 and 9 ( but = , so 3 is a common factor of both 6 and 9). Every fraction can be reduced to lowest terms by dividing the numerator and the denominator by their greatest common devisor (GCD) (see Lesson 3) Adding and Subtracting fractions In order to add or subtract fractions you have to: 1. Find a common denominator – a number, divisible by each denominator of given fractions. 2. Change each fraction so, that it is expressed using this common denominator. 3. Add or subtract the numerators only.

26

Example: A common denominator is 24. Thus, =

+

=

Express each fraction using the common denominator 24.

+ +

and

Finally, add the numerators to find the answer.

=

2. (OG) 1 – 3/7 – 1/3 = (A) 3/5 (B) 6/7 (C) 2/21 (D) 5/21 (E) 16/21 3. (OG) A straight pipe 1 yard in length was marked off in fourths and also in thirds. If the pipe was then cut into separate pieces at each of these markings, which of the following gives all the different lengths of the pieces, in fractions of a yard? (A) 1/6 and 1/4 only (B) 1/4 and 1/3 only (C) 1/6, 1/4 and 1/3 only (D) 1/12, 1/6 and 1/4 (E) 1/12, 1/6 and 1/3 Multiplication of fractions To multiply fractions, first multiply the numerators together, multiply the denominators together. Example: ×

=

× ×

=

× × × × × ×

=

Dividing fractions In order to divide fractions: 1. Change the divisor into its reciprocal. 2. Multiply the fractions. The reciprocal to is ; the reciprocal to 5= is Note: The product of a number and its reciprocal always equals 1. Example: 3 7 3 11 3 × 11 33 ÷ = × = = 4 11 4 7 4×7 28 4. (OG) In which of the following pairs are the two numbers reciprocals to each other? I. 3 and 1/3 II. 1/17 and -1/17 III. √3 and -√3 (A) I only (B) II only (C) I and II

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(D) I and III (E) II and III

Ratio and Proportion The ratio of the number a to the number b (b ≠ 0) is . A ratio may be expressed or represented in several ways. For example, the ratio of 2 to 3 can be written as 2 to 3, 2:3, or

. The order of the terms of a ratio is important.

A proportion is a statement that two ratios are equal. For example, =

is a proportion.

To solve a proportion involving an unknown is to cross multiply, obtaining a new equality. 5. Find the value of the positive integer n, if the ratio of 2 to 3 has the same value as the ratio of n to 12. (A) 2 (B) 4 (C) 6 (D) 8 (E) 12 6. (OG) The ratio of two quantities is 3 to 4. If each of the quantities is increased by 5, what is the ratio of these two new quantities? (A) 3/4 (B) 8/9 (C) 18/19 (D) 23/24 (E) It cannot be determined from the information given. 7. (OG) What is the ratio of 3/4 to the product 4×(3/4)? (A) 1/4 (B) 1/3 (C) 4/9 (D) 9/4 (E) 4 Comparing fractions Among the fractions with equal denominators the greatest will be the one with the greatest numerator:

<

Among the fractions with equal numerators the greatest will be the one with the least denominator, so, > If neither the denominators, nor the numerators are equal instead of finding a common denominator we can just cross multiply.

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Example. To compare 6/7 and 7/8, cross-multiply the numerators and denominators, and write the results on the respective sides:

6 × 8 = 48 7 × 7 = 49 48 < 49, 6/7 < 7/8.

8. (OG) Which of the following is greater than 2/3? (A) 33/50 (B) 8/11 (C) 3/5 (D) 13/27 (E) 5/8 9. (GC) Which one of the five fractions is the largest? (A) 25038876541/25038876543 (B) 25038876543/25038876545 (C) 25038876545/25038876547 (D) 25038876547/25038876549 (E) 25038876549/25038876551

Decimals Any number can be expressed as a decimal. Note: On the GMAT it is the decimal point that determines the place value of the digit, not comma. Example: 5 = 5.0 3 = 0.6 5 1 = 0.3333 … = 0. (3) 3 Example: Here are some examples of transforming decimals and fractions to each other. 321 0.321 = 1000 156 1.56 = 100 3 = 3 ÷ 8 = 0.375 8 Addition and subtraction of decimals Example: 1) 0.287-0.5 = 0.287-0.500 = -0.213. 2) 1445 + 203.78 =

29

Line them up:

1445.00 + 203.78 And add each pair of digits: 1648.78 10.

(PR) 0.163+ 24.15 -0.313 = (A) 2.265 (B) 23.865 (C) 23.99 (D) 24 (E) 25

Multiplying and dividing decimals There is a simple trick to remember when multiplying decimals: the product has the same number of  decimal  points  as  the  two  multipliers  have  together.  e.g.  0.5  Ă—  0.25  =  0.125.  The  first  multiplier  has one decimal, the second one has two, so the product has three. Dividing decimals also has a trick: if you multiply the dividend and the divider by the same number, the answer will not change. So when we have decimals, we can multiply by the powers of ten,  10,  100,  1000,  so  that  we  are  dealing  with  divisions  of  whole  numbers:  0.75  á  0.15  =  75  á  15  =  5. 11. (A) (B) (C) (D) (E) 12. (A) (B) (C) (D) (E)

. .

+

. .

is equal to which of the following?

3.245 0.3245 32.45 69 69.54 (OG) 3.003/2.002 = 1.05 1.50015 1.501 1.5015 1.5

All decimals are divided into three groups: Terminating (they have only a finite number of non-zero digits: 0.2, 1.00456) Periodic (their digits are periodically repeated: 0.3333..., 12.545454..) Infinite (these contain infinite number of digits after the decimal point: 0.6666.., đ?œ‹, √2) The fraction expressed as a decimal will be the terminating decimal if it can be presented as

,

where a is an integer, m=0,1,2,3... , and n=0,1,2,3... .

30

13.

(MH) Which of the following is terminating decimal? (A) 1/120 (B) 1/630 (C) 1/143 (D) 1/250 (E) 1/225

14. (OG) Any decimal that has only a finite number of non-zero digits is a terminating decimal. If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a terminating decimal? (1) 90 < r < 100 (2) s = 4 15. (OG) If each of the following fractions were written as a repeating decimal, which would have the longest sequence of different digits? (A) 2/11 (B) 1/3 (C) 41/99 (D) 2/3 (E) 23/37

Percent Percent means per hundred or number out of 100. A percent can be represented as a fraction with a denominator of 100, or as a decimal. Example: 37% =

= 0.37

To find a certain percent of a number, multiply the number by the percent expressed as a decimal or fraction. Example: 20% of 90 = 0.2 × 90 = 18 75 3 75% of 200 = × 200 = × 200 = 150 100 4 16. (OG) What is 45% of 7/12 of 240? (A) 63 (B) 90 (C) 108 (D) 140 (E) 311 17. (OG) What is 15 percent of x? (1) 18 is 6% of x (2) 2/3 of x is 200

31

GoGMAT Problem

32

Answers and explanations 1. Let’s simplify fractions from answer choices, so that they are in their simplest forms and see which one will turn out to be ¾. Choice A:

=

÷ ÷

=

. 41 and 50 have no common factors (besides

1) so this fraction cannot be further simplified. It does not equal 3/4.Choice B gives us

÷

= ≠ . Choice C:

÷

=

÷ ÷

= ≠ . Choice D:

=

÷ ÷

=

= . This is the correct answer and

on the test you do not need to go any further, but just for the sake of completeness let’s check choice E.

=

÷ ÷

= ≠ . So, our answer is D.

2. Since 3/7 + 1/3 = (9 + 7)/21 = 16/21, 1= 21/21, 1 – 3/7 – 1/3 = 21/21 - 16/21 = (21-6)/21 = 5/21. The answer is D. 3.

4. 5.

1/4 is necessarily one of the answers. Thus, answer choices A, C, and D are under consideration. Choice C is wrong since no pieces of length 1/3 exist (length of every piece does not exceed 1/4, the minimal marking). Piece 1/12 = 1/3 - 1/4 necessarily belongs to the answer choice set. The answer is D. It is evident that 3×1/3 = 1, therefore, 3 and 1/3 are reciprocals to each other. 1/17 × -1/17 ≠1, so, this is not a pair of reciprocals. So is pair {√3, -√3}: √3×-√3 = -3 ≠1. The answer is A. The problem can be written down as = . Cross multiply, obtaining 2 × 12 = n × 3, or 24 =

3n; then divide both sides by 3, to get n = 8. The answer is D. 6. Let the quantities be P and Q. From the stem it can be inferred that P/Q = 3/4. This information is insufficient to determine the value of (P + 5)/(Q + 5). Thus, the answer is E. / 7. ×( / ) = , which can be found when the numerator and the denominator are both divided by (3/4). A is the answer. 8. Use cross-multiplying. 33/50 < 2/3 since 99 < 100; 8/11 > 2/3 as 24 > 22; 3/5 < 2/3 as 9 < 10; 13/27 < 2/3 is equivalent to 39 < 54; 5/8 < 2/3 since 15 < 16. B is the answer. 9. We can see that in any fraction the difference between denominator and numerator is 2, so each fraction can be represented as x/(x+2). Evidently, 1/3 < 2/4=1/2. In general x/(x + 2) < (x + 2)/(x + 4) if x > 0. Thus, A < B < C < D < E. E is the answer. 10. 0.163+ 24.15= 0.163+ 24.150 = 24.313. Then, 24.313 - 0.313 = 24. The answer is D. 11. Each space to the right of the decimal point means we need to multiply by an increased power of 10:

. .

=

. .

×

, ,

=

= 50. Similarly,

. .

×

=

=

= 19.We can now add 50 +

19 = 69. And the correct answer is D. 12. 3.003/2.002 = (3×1.001)/(2×1.001) = 3/2 = 1.5. E is the answer. 13. Let's look at the denominator of each answer choice in order to determine if it can be represented as a product of powers of 2 and 5. 120 = 2 × 5 × 3, it contains 3, so A is not our answer.630 =3 × 7 × 2 × 5, it contains 3 and 7, so B is not our answer. 143= 13× 11, it contains 13 and 11, so C is not our answer. 250 =5 × 2, so, it's a terminating decimal. Let's look at the last answer choice to be sure: 225=3 × 5 , it contains 3, so E is not our answer. The answer is D. 14. Any fraction r/s, where r and s are mutually prime integers and s = 2k×5m, can be expressed as a terminating decimal. So, if s=4=22×50 , our decimal will be terminating. Thus, the answer is B.

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15. If we divide numerator by denominator in each fraction, will find that2/11 = 0.(18), 1/3 = 0.(3), 41/99 = 0.(41), 2/3 = 0.(6), 23/37 = 0.(621). Thus, the answer is E. 16. 45%×7/12×240 = 45%×140 = 45/100×140 = 45×7/5 = 9×7 = 63. Thus, the answer is A. 17. From statement (1) we know that 18 = 6%×x, x = 18×100/6 = 300, 15%×x = 45. Therefore, the answer must be A or D. From statement (2) we can infer that 2/3×x = 200 and x = 300, and, similarly, 15%×x = 45. The answer is D. GoGMAT Problem Explanation

34

Digits GoGMAT, Session 1 A digit is an integer from 0 to 9 inclusive: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. 1. (OG) If a and b represent single digits in the correctly worked computation below, what is the value of a +b? 4 a 7 b 2 3 + 1 6 2 1 2 2 2 (A) 8 (B) 9 (C) 10 (D) 11 (E) cannot be uniquely determined Every digit in a number has a particular place value depending on its location within the number. You should memorize the names of all the places value.  Let’s  consider  all  the  digits  in  the  number  91356.2804: 9 – is a ten-thousands digit 1 – is thousands digit 3 – is a hundreds digit 5 – is a tens digit 6 – is a ones or a units digit . - is a decimal point 2 – is a tenths digit 8 – is a hundredths digit 0 – is a thousandths digit 4 – is a ten-thousandths digit

2. Is a 3-digit integer divisible by 3? (1) The hundreds' digit of the number is equal to the tens' digit of the number. (2) The units' digit of the number is 6 greater than the tens' digit of the number đ?&#x;?

đ?&#x;?

Note: đ??šđ??›đ??œđ???. đ??žđ??&#x; = đ?&#x;?đ?&#x;Žđ?&#x;Žđ?&#x;Žđ??š + đ?&#x;?đ?&#x;Žđ?&#x;Žđ??› + đ?&#x;?đ?&#x;Žđ??œ + đ??? + đ?&#x;?đ?&#x;Ž đ??ž + đ?&#x;?đ?&#x;Žđ?&#x;Ž đ??&#x; Example: 527  =  5 Ă— 100 + 2 Ă— 10 + 7 Ă— 1. Thus, in the number 527 the digit 7 is in the ones (units) place, the digit 2 is in the tens place, and the digit 5 is in the hundreds place. The name of each  location  corresponds  to  the  â€œvalueâ€?  of  the  place.   Note:   Notation  â€œabâ€?  can  have  two  different  meanings:  a  product  of  a  and  b  or  a  two-digit number. If it is given that a and b are integers, than you deal with a product. If a and b are said to be digits treat ab as a two-digit number.

35

3. A and B are both two-digit numbers. If A and B contain the same digits, but in reverse order, and A>B, what integer must be a factor of (A-B)? (A) 5 (B) 7 (C) 9 (D) 11 (E) cannot be determined 4. (OG) If the two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N? (A) 181 (B) 165 (C) 121 (D) 99 (E) 55 5.

(OG) The product of the two-digit numbers below is the three-digit number ada, where a, b, and d are three different nonzero digits. If a×b < 10, what is the two-digit number ab? ab ×ba ada (A) (B) (C) (D) (E)

6.

11 12 13 21 31

(GC) If 1050 – 74 is written as an integer in decimal notation, what is the sum of the digits in that integer? (A) 412 (B) 423 (C) 433 (D) 440 (E) 461 Knowing the place value is important because the GMAT occasionally requires you to round a number to a specific place value. Example: What is 3.194 rounded to the nearest tenths? First, find the digit located in the specified place value. The digit 1 is in the tenths place. Second, look at the right-digit-neighbour (the digit immediately to the right) of the digit in question. In this case we have 9. If the right-digit-neighbour if 5 or greater, round the digit in question UP. Otherwise, the digit in question remains the same. In this case since 9 is greater than 5, the digit in question (1) must be rounded up to 2. Thus, 3.192 rounded to the nearest tenth equals 3.2. Note: all the digits to the right of the right-digit-neighbour are irrelevant when rounding. In the example the digit 4 is not important.

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7.

(OG) If d = 2.0453 and d* is the decimal obtained by rounding d to the nearest hundredth, what is the value of d* - d? (A) -0.0053 (B) -0.0003 (C) 0.0007 (D) 0.0047 (E) 0.0153 GoGMAT Problem

37

Answers and explanations 1.

Perform this computation. Since 2 + 6 + 1+a = 12, a = 3. Since 4 + b + 1 + 1 = 12, b = 6. Thus, a + b = 9. Thus, the answer is B. 2. A number is divisible by 3 if the sum of its digits is divisible by 3. So 3-digit integer abc will be divisible by 3 if a+b+c is a multiple of 3. From the first statement we know, that a=b, therefore a + b + c = 2b + c, which may or may not be a multiple of 3. Hence, the first statement is not sufficient. From the second statement we know, that c=b+6. Therefore, a + b + c = a + 2b + 6, which again may or may not be a multiple of 3. Hence, statement 2 is not sufficient. Finally, having a=b and c=b+6 we obtain a + b + c = b + b + b + 6 = 3b + 6 = 3(b + 2) which is divisible by 3. The correct answer is C.

3. To solve this problem, assign A=xy and B=yx. Using your knowledge of place value, you can express A as 10x+y, where x is the digit in the tens place and y is the digit in the unit place. B, therefore, can be expressed as 10y+x. Hence, the difference of A and B can be expressed as follows: A − B = 10x + y − 10y − x = 9x − 9y = 9(x − y). Clearly, 9 must be a factor of A-B. The answer is C. 4. If M = XY = 10X + Y (where X and Y are digits), then N = YX = 10Y + X and M + N = 11(X + Y). Thus, M + N is divisible by 11. Among answer choices given, only A does not satisfy this condition. Don’t forget about those divisibility shortcuts for 2, 3, 5, 9, 10, and 11! 5. Since a×b < 10 and a×b=a, solving last equation gives b=1 or a=0. Only b=1 remains as a root, because a is not equal to 0. Using this information, ab = 21 or 31, the shortest way to check is to multiply 21 by 12 or 31 by 13: 21×12=252. D is the answer. 6. 1050 – 74 = 9999....99926, where the last integer contains 48 9’s. Thus, the sum of its digits equals to 9×48 + 2 + 6 = 440. The answer is D. 7. d* = 2.05, d* - d = 0.0047. Thus, the answer is D.

38

GoGMAT Problem Explanation

39

Powers and Roots GoGMAT, Session 3 An exponent, or power, is a short way of writing the value of a number multiplied several times by itself. For example, 4∙4∙4∙4∙4=4 . So, an exponent refers to the number of times a number (referred to as the base) is used as a factor. In the number 4 , the base is 4 and the exponent is 5. There several rules to remember about exponents: Multiplying numbers with the same base: when you multiply numbers that have the same base, you simply add their exponents. đ?‘Ž ∙đ?‘Ž

=đ?‘Ž

For example, 6 ∙ 6 = 6 =6 . Dividing numbers with the same base: when you divide numbers that have the same base, you simply subtract the bottom exponents from the top exponents. đ?‘Ž =đ?‘Ž đ?‘Ž

For example, 6 : 6 =

=6

=6 .

Raising a power to power: when you raise a power to power, you can simply multiply the exponents. (đ?‘Ž ) = đ?‘Ž

∙

For example, (6 ) = 6 ∙ = 6 . Combining bases with the same exponents: when you multiply or divide numbers that have the same exponents, you simply combine the bases. đ?‘Ž ∙ đ?‘? = (đ?‘Ž ∙ đ?‘?) đ?‘Ž đ?‘Ž = đ?‘? đ?‘?

For example, 5 ∙ 6 = (5 ∙ 6) = 30 . Raising to a negative power: any number to the negative power đ?‘› is equal to the reciprocal of the same number to the positive power đ?‘› đ?‘Ž

=

1 đ?‘Ž

For example, 2 = = = 0.25. Raising to 1: any number to the first power is equal to itself đ?‘Ž =đ?‘Ž

Raising to 0: any number to the 0 power is equal to 1 đ?‘Ž =1

Other important rules for đ?‘› > 0:

40

1 =1 0 =0 0  đ?‘–đ?‘  đ?‘˘đ?‘›đ?‘‘đ?‘’đ?‘“đ?‘–đ?‘›đ?‘’đ?‘‘

If you raise a positive fraction that is less than 1 to power, the fraction gets smaller. For example,

= ∙ = .

If you raise a negative number that is less than -1 to an odd power, the number gets smaller. For example, (−3) = (−3)(−3)(−3) = −27 < −3. 1. (GC) What is the value of 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 ? (A) 2 (B) 2 (C) 2 (D) 2 (E) 2 2. (GC) If 2 = 128, then 2 (A) 10 (B) 5 ∙ 10 (C) 2 ∙ 10 (D) 5 ∙ 10 (E) 2 ∙ 10 3. (GC) If đ?‘Ľ equals 10 (A) 10 (B) 10 (C) 10 (D) 10 (E) 100

∙5

=

, and đ?‘Ľ  equals 10 , what is the value of đ?‘˜?

On the flip side of exponents and powers are roots and radicals. A square root of a number đ?‘› is a number that, when squared, is equal to đ?‘›. only non-negative real numbers have square roots; 0 has only one square root; any positive real number has two square roots (one negative and one positive). Positive root of a real number is sometimes called arithmetic square root and √    is a symbol for such root. A number inside the √    is called a radical. For example, √9 denotes 3. The two square roots of 9 are 3 and −3. By definition we have two main properties of square roots: √đ?‘Ž

=đ?‘Ž

đ?‘Ž = |đ?‘Ž|

and You should be pretty familiar with following properties of roots:

41

√đ?‘Ľ ∙ đ?‘Ś = √đ?‘Ľ đ?‘Ś

=

đ?‘Ľâˆ™đ?‘Ś đ?‘Ľ đ?‘Ś

To simplify a radical, try factoring. For example, √32 = √2 ∙ 16 = √2 ∙ √16 = √2 ∙ 4 = 4√2. 4. (OG) (16 Ă— 20  +  8 Ă— 32) = (A) 4√20 (B) 24 (C) 25 (D) 4√20 + 8√2 (E) 32 5. (OG) How many bits of computer memory will be required to store the integer đ?‘Ľ, where đ?‘Ľ = −√810,000, if each digit requires 4 bits of memory and the sign of đ?‘Ľ requires 1 bit? (A) 24 (B) 25 (C) 17 (D) 13 (E) 12 6. (OG)  √16  +  16 = (A) 4√2 (B) 8√2 (C) 16√2 (D) 8 (E) 16 The following problems will give you deeper insight into conditions under which roots are integers. 7. (OG) If đ?‘‘ is a positive integer, is √đ?‘‘ an integer? (1) đ?‘‘ is the square of an integer. (2) √đ?‘‘ is the square of an integer. 8. (OG) Is the positive square root of đ?‘Ľ an integer? (1) đ?‘Ľ = đ?‘› and đ?‘› is an integer (2) đ?‘Ľ = 16 9. (OG) If đ?‘Ľ is an integer, is √x an integer? (1) √4x is an integer (2) √3x is not an integer

42

You should remember approximations of roots: √2 ≈ 1.4 √3 ≈ 1.7, √5 ≈ 2.2 10.

(OG ) Of the following numbers which one is third greatest? (A) 2√2 − 1 (B) √2 + 1 (C) 1 − √2 (D) √2 − 1 (E) √2

11. (OG) The value of √−89 is between (A) -9 and -10 (B) -8 and -9 (C) -4 and -5 (D) -3 and -4 (E) Undefined 12. (A) (B) (C) (D) (E)

(OG) The expression 4 3 2 5 6

.

∙ . √

is approximately equal to

A radical can be rewritten as a fractional exponent and vice versa: đ?‘Ž = √đ?‘Ž

.

In general case đ?‘Ž

= (đ?‘Ž ) = √đ?‘Ž

.

Check yourself: 1.

2.

1 125

810.75

15 6 1 2

2 3 2 3

3. 8 : 8

2 1 6

1 32

1 5

4.

5.

1.5

27 27

6.

2 3

27

8 1 9

1

5

1

12 2 7

3 2

4 3

3 4

1

2 3

9

81 3 9

2

1 3

5

1 3

32 7 3 1 2

8

1 9

3 4

1 6

4 3

43

Write in radicals:

2

1.

4

2 3

3

5 6

33 9 4 27

2.

3.

4.

27

72

2 5

2 3

2 1 2

1 5

2 1 6

5 6

36 : 2

4 3

Formulae of short multiplication: đ?‘Ž − đ?‘? = (đ?‘Ž − đ?‘?)(đ?‘Ž + đ?‘?) (đ?‘Ž + đ?‘?) = đ?‘Ž + 2đ?‘Žđ?‘? + đ?‘? (đ?‘Ž − đ?‘?) = đ?‘Ž − 2đ?‘Žđ?‘? + đ?‘? 13. (A) (B) (C) (D) (E) 14.

(OG) (1 − √5)(1 + √5) = -4 2 6 2√5 -2√5 (GC) = √

√

(A) √3 + √2 (B) √3 + √2 (C) 1 (D) 1 − √3 (E) 1 − √2 15. (OG) (√7 + √7) = (A) 28 (B) 14 (C) 49 (D) 98 (E) 7 More complex problems may appear on the GMAT. Consider following example: 16.

6 + 2√5 = (A) √5 (B) √5 − 1 (C) √5 + 1 (D) 3 − √5 (E) 3

44

GoGMAT Problem

45

Answers and explanations 1. (B) Let’s  add  the  numbers  step  by  step:  (2 + 2 ) + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 2∙2 +2 +2 +2 +2 +2 +2 +2 = (2 + 2 ) + 2 + 2 + 2 + 2 + 2 + 2 = 2∙2 +2 +2 +2 +2 +2 +2 = (2 + 2 ) + 2 + 2 + 2 + 2 + 2 = (2 + 2 ) + 2 + 2 + 2 + 2 = â‹Ż = 2 The answer is B. 2. (E) Since 2 = 128, đ?‘› = 7. Then 2 ∙5 = 2 ∙ 5 = 2 ∙ 2 ∙ 5 = 2 ∙ (2 ∙ 5) = 2 ∙ 10 . The answer is E. 3. (C) Since đ?‘Ľ = 10 and đ?‘Ľ = 10 , (10 ) = 10 or 10 = 10 . Therefore, 100đ?‘Ľ = đ?‘˜, or 100 ∙ 10 = đ?‘˜. Thus, đ?‘˜ = 10 . 4. (B) Since 16 ∙ 20 + 8 ∙ 32 = 576 = 24 , √16 ∙ 20  +  8 ∙ 32 = 24. The answer is B. But it is easier to split off expression inside √    : 16 ∙ 20 + 8 ∙ 32 = 16 ∙ (20 + 16) = 16 ∙ 36 = 4 ∙ 6 = (4 ∙ 6) . 5. (D) 810,000 = 81 ∙ 10,000 = 9 ∙ 100 = 900 . Therefore, đ?‘Ľ = −√810,000 = −√900 = −900 and  it  takes  4  bits  for  each  of  the  digits  and  one  bit  for  sign  â€œ-â€?:  3 ∙ 4 + 1 = 13 bits to store the information. Thus, the answer is D. 6. (A) 16 + 16 = 16 ∙ 2 = 2 ∙ 2. Therefore, √16 + 16 = √2 ∙ 2 = 2 ∙ √2 = 4√2. A is the answer. 7. (D) Statement (2) alone is sufficient: the square of an integer is always an integer. (1) is sufficient too, since there exist some positive p such that đ?‘‘ = đ?‘? and √đ?‘‘ = đ?‘?. The answer is D. 8. (D) Clearly, (1) alone is sufficient: √x = n and đ?‘› is an integer. (1) is sufficient too, since √16 = 4. The answer is D. 9. (A) (1) is sufficient to answer the stem question: √4x = 2√x is an integer and thus √x must be integer. (2) is not sufficient (examples are đ?‘Ľ = 1 and đ?‘Ľ = 3). A is the answer. 10. (E) It is evident that 1 < √2 < 2. Having this, we can prove that 2√2– 1 > √2 + 1 > √2 > √2– 1 > 1 − √2. Also we can use here approximation of √2 ≈ 1.4. The answer is E. 11. (C) Since−4 = √−64 > √−89  >  âˆšâˆ’125 = −5, C is the answer. 12. (B) When you are asked to find approximate solution it is easier to round factors than round the result. We know that 0.998 is close to 1, √403 is close to 20, and the result is close to 3. The answer is B. 13. (A) If we use first formula of short multiplication then (1 − √5)(1 + √5) = 1 – √5 = −4. A is the answer. 14. (A) Here roots are in the denominator, and if we need rational numbers in denominator formula of short multiplication will be useful: √3 − √2 √3 + √2 = 3 − 2 = 1. Lets multiply numerator and denominator by √3 + √2 : √

√

=

√ √

√

√ (√

√

= )

√

√

=

√3 + √2. A is the answer. 15. (A) √7 + √7 = 2√7 and (2√7) = 4 ∙ 7 = 28. Or we can use the formula: (√7 + √7) = √7 + 2√7√7 + √7 = 7 + 2 ∙ 7 + 7 = 28. A is the answer. 16. (C) The key to such problems is to find that 6 + 2√5 = (1 + 5) + 2 Ă— 1 Ă— √5 = 1 + 2 Ă— 1 Ă— √5 + √5 = (1 + √5) . Having this, we can find that C is the answer.

46

Check yourself 1. 81

.

+ ∙

2.

= 81 + 125 − 32 = (3 ) + (5 ) − (2 ) = 3 + 5 − 2 = 30

− ∙

=

∙

∙

3. 8 ÷ 8 ∙ 9

(

= (

4.

=

.

∙

5.

∙

∙

=

∙

6.

∙

)

∙

.

)

(

∙

=

= 5 = 25

=

=

) ∙

(

= (3 )

)

∙

)

∙

∙(

27 ∙

=

) ∙(

)

∙(

= (5

= ∙

= ∙

=

)

∙

∙

= =

=3

∙2

∙7

= 3 ∙ 1 ∙ 7 = 21

∙ ∙

∙

=3 ∙3

=3

Write in radicals: 1. 2.

= √ √ √

=

=

3.

27 ∙ 2 ∙ 2

4.

72

∙

=

= 3 = (3 )

∙

= 3 ∙2

∙ 36 ÷ 2 = (2 )

∙

∙

∙ (3 )

=3

=3∙2=6 ∙

∙3

∙

∙2

∙

÷2 =2

∙3

=3

47

GoGMAT Problem Explanation

48

Home assignment I. 1.

(A) (B) (C) (D) (E) 2.

Numbers. If a and b are integers such that a + b = 5, which of the following must be true? I. The product of a and b is odd. II. If a is odd, b is even. III. If a is negative, b is positive. I only II only I and II only II and III only I, II, and III (GC) What is the product of 6 consecutive integers? (1) The greatest integer is 4 (2) The sequence has both positive and negative integers

3. (OG) If p is an even integer and q is an odd integer, which of the following must be an odd integer? (A) p/q (B) pq (C) 2p+q (D) 2(p+q) (E) 3p/q 4. (OG) Is x an integer? (1) x/2 is an integer (2) 2x is an integer 5.

(GC) Is a × (a + 5) even number? (1) a is positive integer (2) a is odd

6.

If n is an even number, which of the following must be even?

I. II. III.

(A) (B) (C) (D) (E)

4n 4 4 4n 2 2n 4 4n 2

4n 4

I only II only III only I and III only II and III only

49

7. If x is an integer, is y an integer? (1) The average (arithmetic mean) of x, y, and y – 2 is x (2) The average (arithmetic mean) of x and y is NOT an integer 8. If n is a positive integer, what is the maximum possible number of prime numbers in the following sequences: n + 1, n + 2, n + 3, n + 4, n + 5, and n + 6? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 9. If the sum of five consecutive positive integers is a, then the sum of the next five consecutive integers in terms of a is (A) a+1 (B) a+5 (C) а + 25 (D) 2a (E) 5a 10. (A) (B) (C) (D) (E)

(OG) If đ?‘&#x; and đ?‘ are integers and đ?‘&#x;đ?‘ Â + Â đ?‘&#x; is odd, which of the following must be even? đ?‘&#x; đ?‘ đ?‘&#x;+đ?‘ đ?‘&#x;+đ?‘ đ?‘&#x; +đ?‘

11. (OG) If m is an integer, is m odd? (1) m/2 is not an even integer (2) m - 3 is an even integer 12. (GC) Are integers a and b both even? (1) ab is evenly divisible by 4 (2) ab = 0 13. If x, Ńƒ and z are positive integers such that 0 < x < y < z and x is even, Ńƒ is odd, and z is prime, which of the following is a possible value of x + Ńƒ + z ? (A) 4 (B) 5 (C) 11 (D) 15 (E) 18 14. If n is an integer greater than 1, and n is not a prime number, then which of the following must be true? (A) n is the sum of three prime numbers (B) n is the difference between 2 even numbers (C) n is the difference between one even number and one odd number (D) n is the product of one even number and one odd number (E) n is the product of prime numbers

50

15. (1) (2) II.

1. (A) (B) (C) (D) (E) 2. (1) (2)

If x is a positive integer, is x a prime number? x + 1 is a prime number. x – 5 is a prime number

Divisibility Which of the following is the smallest number that can be divisible by 6? 102,000 101,010 110,100 100,010 110,010 If x and y are positive integers, is x – y divisible by 4? xy is divisible by 16. x is divisible by 4.

3. (GC ) What is the sum of the least and the greatest positive four-digit multiples of 4 that can be written each using the digits 1, 2, 3, and 4 exactly once? (A) 5555 (B) 5658 (C) 5636 (D) 4312 (E) 1324 4. & represents the tens digit in integer A=1543&2. What is &? (1) A is divisible by 9 (2) A is divisible by 4 5. (A) (B) (C) (D) (E)

If l×2×3× … ×n is divisible by 990, what is the least possible value of n? 10 11 12 18 22

6. (GC ) Let N be the greatest integer with distinct digits and multiple of 8 as well. What is the remainder when N is divided by 1000? (A) 120 (B) 12 (C) 320 (D) 240 (E) 210

51

III. Decimals

and fractions

1. (OG) If the numbers 17/24, 1/2, 3/8, 3/4, and 9/16 were ordered from greatest to least, the middle number of the resulting sequence would be (A) 17/24 (B) ½ (C) 3/8 (D) ¾ (E) 9/16 2. (OG) 3/100 + 5/1000 + 7/100000 = (A) 0.357 (B) 0.3507 (C) 0.35007 (D) 0.0357 (E) 0.03507 3. (OG) Which of the following fractions is equal to 0.0625? (A) 5/8 (B) 3/8 (C) 1/16 (D) 1/18 (E) 3/80 4. (OG) The number 2 – 0.5 is how many times the number 1 – 0.5? (A) 3 (B) 6 (C) 1.5 (D) 0.5 (E) 2.5 5. (GC) Find the value of (p-1)3 if p = 15/18 + 5/16 - 20/24. (A) (B)

−

(C) (D)

−

(E)

6. (OG) What number when multiplied by 4/7 yields 6/7 as the result? (A) 2/7 (B) 2/3 (C) 3/2 (D) 24/7 (E) 7/2

52

7. (OG) What is the value of (-1.5×1.2 – 4.5×0.4)/30? (A) 0 (B) -0.12 (C) 0.12 (D) 1.2 (E) -0.012 8. (OG) What is the value of (0.3)5/(0.3)3? (A) 0.03 (B) 0.09 (C) 0.27 (D) 0.003 (E) 0.009 9. (OG) 3×0.072/0.54 = (A) 0.04 (B) 0.3 (C) 0.4 (D) 0.8 (E) 4 10. (OG) 31/125 = (A) 0.248 (B) 0.252 (C) 0.284 (D) 0.312 (E) 0.320 11. (OG) What percent of 30 is 12? (A) 2.5% (B) 3.6% (C) 25% (D) 40% (E) 250% 12. (OG)If x is equal to one of the numbers 1/4, 3/8, or 2/5, what is the value of x? (1) 1/4 < x < 1/2 (2) 1/3 < x < 3/5 13. (OG) Which of the following fractions has greatest value? (A) 6/(22×52) (B) 1/(23×52) (C) 28/(22×53) (D) 62/(23×53) (E) 122/(24×53)

53

14. (OG) If 1 < d < 2, is the tenths’ digit of the decimal representation of d equal to 9? (1) d + 0.01 < 2 (2) d + 0.05 > 2 15. (OG) What is the decimal equivalent of (1/5)5? (A) 0.00032 (B) 0.0016 (C) 0.00625 (D) 0.008 (E) 0.03125 16. (OG) If 18 is 15% of 30% of a certain number, what is the number? (A) 9 (B) 36 (C) 40 (D) 81 (E) 400 17. (OG) If x > 0, x/25 + x/50 is what percent of x? (A) 6% (B) 25% (C) 37.5% (D) 60% (E) 75% 18. (OG) If r and s are positive integers, r is what percent of s? (1) r = 3/4s (2) r + s = 75/100 19. (OG) The manager of a theatre noted that for every 10 admission tickets sold, the theater sell 3 bags of popcorn at $2.25 each, 4 sodas at $1.50 each, and 2 candy bars at $1.00 each. To the nearest cent, what is the average (arithmetic mean) amount of these snack sales per ticket sold? (A) $1.48 (B) $1.58 (C) $1.60 (D) $1.64 (E) $1.70

20. (OG) In the fraction x/y, where x and y are positive integers, what is the value of y? (1) The least common denominator of x/y and 1/3 is 6 (2) x = 1

54

21. (OG) If the successive marks on the line below are equally spaced and if x and y are the numbers designating the end points of intervals as shown, what is the value of y?

(1) x = 1/2 (2) y – x = 2/3 22. If a and b are integers, is a a terminate decimal? b

(1) a = 5r, where r is an integer. (2) b = 2s, where s is an integer. 23. (OG) On the number line segment [0, 1] has been divided by tick marks into fifths and into sevenths What is the least possible distance between any two of the tick marks? (A) 1/70 (B) 1/35 (C) 2/35 (D) 1/12 (E) 1/7 IV. Digits 1. (PR) If x is a positive number and x=0.abc, is the tenths’ digit of the decimal representation of x equal to 7? (1) 0.5 +2x < 1.74 (2) 2x - 0.035 > 0.001 2. (OG) If a and b each represent single digits in the decimal 3.2ab6, what does digit a represent? (1) When the decimal is rounded to the nearest tenth, 3.2 is the result (2) When the decimal is rounded to the nearest hundredth, 3.24 is the result 3. (OG) What is the value of the two-digit integer x? (1) The sum of the two digits is 3 (2) x is divisible by 3 4. (GC) The sum of the digits of a standard two-digit numeral is 9. If the digits are reversed, the numeral represents a number that is 45 less than original number. What is the original numeral? (A) 72 (B) 27 (C) 54 (D) 63 (E) 81 5.

(GC) M is a positive integer less than 50. The units digit of M is 5. What is the value of M? (1) Thousands digit of M2 is 1 (2) Hundreds digit of M2 is 2

55

6. (OG) If Q is an integer between 10 and 100, what is the value of Q? (1) One of Q’s digits is 3 more than the other, and the sum of its digits is 9 (2) Q < 50 7. (OG) A cashier mentally reversed the digits of one customer’s correct amount of change and thus gave the customer an incorrect amount of change. If the cash register contained 45 cents more than it should have as a result of this error, which of the following could have been the correct amount of change in cents? (A) 14 (B) 45 (C) 54 (D) 65 (E) 83 8. (OG) If x = 0.rstu, where r, s, t, and u each represent a nonzero digit of x, what is the value of x? (1) r = 3s = 2t = 6u (2) The product of r and u is equal to the product of s and t 9. (GC) A person travels in a car with uniform speed. He observes a mile-stone, which has 2 digits. After one hour he observes another mile-stone with same digits reversed. After another hour he observes another mile-stone with same 2 digits containing a 0. Find the speed of the car in miles per hour, if mile-stones tell you how far one has travelled from the starting point. (A) 40 (B) 45 (C) 50 (D) 55 (E) 60 10. (GC) S = 0.abc, where a, b, and c are any decimal digits. Is S > 2/3? (1) a + b > 14 (2) a + c > 15 11. (GC) x, y, z and w are distinct decimal digits. Is 0.xy+0.zw > 1? (1) The smallest among the 4 digits is 4 (2) The product of the decimals 0.xy and 0.zw is greater than 1/2 12. (GC) A, B, C are digits where A×B is not zero. If AB, BA are both two-digit numbers and AAC is a three-digit number, what is the value of B? (1) AB+BA = AAC (2) A=1 V. 1.

Powers and roots

(GC ) What is the value of √x + 24 − √x? (1) √x is an integer (2) √x + 24 is an integer

2. (OG ) Of the following, the closest approximation to (A)

.

× .

.

is

5

56

(B) 15 (C) 20 (D) 25 (E) 225 3. (OG) √453 is between (A) 21 and 22 (B) 22 and 23 (C) 23 and 24 (D) 24 and 25 (E) 25 and 26 4. (OG) What is the value of the positive integer n? (1) n4 < 25 (2) n ≠ n2 5. (OG) (A)

√

=

√6 (B) 2√6 (C) 1 + √6 (D) 1 + 2√6 (E) 2 + √6

6. (OG) (√2 + 1)(√2 − 1)(√3 + 1)(√3– 1) = (A) 2 (B) 3 (C) 2√6 (D) 5 (E) 6 7. (OG) (√3 + 2)(√3 − 2) = (A) √3 - 4 (B) √6 - 4 (C) -1 (D) 1 (E) 2 8.

3√80 + (A) (B) (C) (D) (E)

√

=

3√5 3 3√3 3 + 2√5 9 + 4√5

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9. (GC) If x and y are integers, is k the square of an integer? (1) k = x2y2 (2) √k = 4 10. Find the value of (A) 3 (B) 3 (C) 3 (D) 3 (E) 3

√3

11. Which of the following is not equal to the square of integer? (A) (B) (C) (D) (E)

√1 √4 18/2 41-25 36

12. Which of the following ratios is most nearly equal to the ratio 1 + √5 to 2? (A) 8 to 5 (B) 6 to 5 (C) 5 to 4 (D) 2 to 1 (E) 1 to 1 13.

9 + √80 + 9 − √80 (A) 1 (B) 9 − 4√5 (C) 18 − 4√5 (D) 18 (E) 20

=

14. If đ?‘Ľ is a positive number, is đ?‘Ľ less than 1? (1) đ?‘Ľ > √đ?‘Ľ (2) −√đ?‘Ľ > −đ?‘Ľ 15. If đ?‘Ś is a positive integer is đ?‘Ś an integer? (1)

4đ?‘Ś is not an integer

(2)

5đ?‘Ś is an integer

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# of question

Topic of question (type) Your answer ARITHMETIC Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Correct Answer

D D C (Pl) A B (Pl) C A C C (Pl) B (Pl) B E E (SP) E C Divisibility

1 2 3 4 5 6

B E C (LD+SP) A B A Decimals and fractions

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

E E C A B C B B C A D E D B A (LD) E A A

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19 20 21 22 23

A E D B B Digits

1 2 3 4 5 6 7 8 9 10 11 12

A E E A (SP) A C E (SP) A B (SP) D D A Powers and Roots

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

E B A C C A C C D A B A E D D

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Test№1. Number properties, fractions, digits, powers and roots, ratios, percent, divisibility 1. (A) (B) (C) (D) (E) 2.

If N=1/3+(1/3)2+(1/3)3, then N is between 1

0 and 9 1 9 1 3 8 9 4 3

1

and 3 8

and 9 4

and 3 and 2

When x is which of the following numbers,

1 3

x

has the largest value?

(A) – 5 (B) – 4 (C) 0 (D) 4 (E) 5 3. What is the sum of digits of number 1028 – 28? (A) 227 (B) 228 (C) 236 (D) 237 (E) 243 4. If m, n, p, and q are consecutive positive even numbers, m < n < p < q, in terms of m, what is the sum of four numbers? (A) 4m + 3 (B) 4m + 6 (C) 4m + 8 (D) 4m + 12 (E) 4m + 16 5. There are 135 employees in Company X. If the number of employees who were born in June is the greatest, what is the least possible number of the employees in the company who were born in June? (A) 10 (B) 11 (C) 12 (D) 13 (E) 14

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For positive integer x, if x2 has 4 digits, which of the following must be true? I. x must be a 2-digit integer II. 2x must be a 3-digit integer III. 3x must be a 3-digit integer (A) I only (B) II only (C) III only (D) I and II only (E) I and III only 6.

7. In a certain positive two-digit integer, the ratio of the units digit to the tens digit is 2 to 3, what is the integer? (1) The tens digit is 3 more than the units digit. (2) The product of the two digits is 54. 8.

x

For numbers x, y, z, and w, if x is 50% greater than y, and z is 20% greater than w, y is what z

percent greater or less than w ? (A) 20 % greater (B) 20 % less (C) 25 % greater (D) 25 % less (E) 50 % greater 9. (1) (2)

If xn = l, where x and n are both integers, what is the value of the x? n is a multiple of 5. n is an odd number.

10. The ratio of number of teachers to number of students in school X is equal to that of school Y. What is the ratio of number of students in school X to number of students in school Y? (1) The number of the students in school X is 2,000 more than the number of the students in school Y. (2) The ratio of number of teachers to number of students in school Y is 1:20. 11. If a and b are positive integers, what is the value of ab? (1) ab is divisible by 6. (2) a and b are prime numbers. 12. (1) (2)

If a and b are positive integers, is a even? ab + b is an even number. ab + a is an even number.

13. (1) (2)

Is 49 in the set S? All numbers in set S are multiple of 7. All numbers in set S are square numbers.

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14. (1) (2)

If z and x are integers with absolute values greater than 1, is zx less than 1? x < 0. zz < 1.

15. (1) (2)

If the product of digits of a 2-digit positive integer n is 20, what is the value of n? n is greater than 50. n is an even number.

16. (1) (2)

Integer n is greater than 20 and less than 80, is n odd? The two digits of n both are prime numbers. The sum of two digits is prime number.

17. If m and n are positive integers and m + n is prime, is m odd? (1) n is an odd number. (2) n is greater than 2. 18. (1) (2)

If x < y < z, is xyz > 0? xy > 0. xz > 0.

19.

If m and n are integers, is

(1) (2)

m n

an infinite decimal?

m is a prime factor of 100. n is a prime factor of 50. Answers: 1. C 2. B 3. E 4. D 5. D 6. A 7. D 8. C 9. B 10. E

11. C 12. E 13. E 14. A 15. D 16. E 17. C 18. E 19. B

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Lesson  â„–  2.  Arithmetic Progressions GoGMAT, Session 7 An arithmetic progression (sequence) is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. This constant is called progression difference (d). 1. Determine whether the sequence below is an arithmetic progression and find its difference. a) -3, -2, -1,  0,  1,  2,  â€Ś b) -7, -4, -1,  2,  5,  8,  â€Ś c) 10, 6, 2, -2, -6,  â€Ś d) 3,  3,  3,  3,  3,  3,  â€Ś On GMAT arithmetic progression can be described in different ways: 1) Sequence đ?‘Ž , đ?‘Ž , đ?‘Ž , ‌ such that đ?‘Ž = đ?‘Ž + đ?‘‘, 2) Sequence đ?‘Ž , đ?‘Ž , đ?‘Ž , ‌ such that đ?‘Ž = đ?‘Ž + đ?‘‘, 3) Equally spaced numbers, 4) Sequence đ?‘Ž , đ?‘Ž , đ?‘Ž , ‌ such that đ?‘Ž = , Key facts about arithmetic progression: 1) An arithmetic progression is uniquely defined by its first term and difference. 2) The definition of the arithmetic sequence can be expressed as the following recursive formula: đ?‘Ž =đ?‘Ž + đ?‘‘,      đ?‘› = 2, 3, 4, ‌  . 3) Since the first term and the difference of the arithmetic progression are given, any term with number n can be found using the formula of the nth term: đ?‘Ž = đ?‘Ž + đ?‘‘ Ă— (đ?‘› − 1). Indeed,  from  the  figure  below  it’s  clear,  that  đ?‘‘

đ?‘‘ đ?‘Ž

đ?‘Ž

đ?‘‘ đ?‘Ž

đ?‘Ž

‌

to get to the second element you have to make one step from the first term, or đ?‘Ž = đ?‘Ž + đ?‘‘, To get to the third element - two steps đ?‘Ž = đ?‘Ž + 2 Ă— đ?‘‘, Etc. So  the  number  of  steps  is  always  one  less  than  the  number  of  the  element  you  want  to  obtain.  That’s  why in the formula of the nth term you have to add đ?‘‘ Ă— (đ?‘› − 1), not đ?‘‘ Ă— đ?‘›. Be sure to understand how the same principle works in following situations:

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2. a) b) c)

Answer the following questions: If a rope is cut in 5 places, how many pieces will there be? How many spaces are there in a text consisting of 100 words? How many pieces of cake will we get, if we make 4 cuttings from its center to the edge?

4) Number of elements between the first and the nth terms inclusive is called the length of progression. The length of progression can be easily found using the formula: đ?‘› Â = Â

(�  –  � )  +  1. �

Let’s  look  at  it  a  little  closer:

đ?‘› Â = Â

(�  –  � )  +  1 �

Number of intervals

Adding 1 to the number of intervals, we

between đ?‘Ž and đ?‘Ž

get the number of elements.

5) The sum of n terms of the arithmetic progression is equal to � =

đ?‘Ž +đ?‘Ž Ă— đ?‘›, 2

Or, applying the formula of the nth term, 2đ?‘Ž + đ?‘‘(đ?‘› − 1) đ?‘›. 2 Looking closer at the formula of the sum of n terms: đ?‘† =

� =

đ?‘Ž +đ?‘Ž Ă—đ?‘› 2

Arithmetic average

Multiplying the average by the

of n terms

number of terms, we get their sum

3. (OG) If p, q, r, s, t is an arithmetic progression, which of the following must be also arithmetic progression? I. 2p, 2q, 2r, 2s, 2t II. p – 3, q – 3, r – 3, s – 3, t - 3 III. p2, q2, r2, s2, t2 (A) I only (B) II only (C) III only (D) I and II (E) II and III

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4. (OG) How many multiples of 4 are there between 12 and 96, inclusive? (A) 21 (B) 22 (C) 23 (D) 24 (E) 25 5. (GC) How many integers between 324700 and 458600 have tens digit 1 and units digit 3? (A) 10300 (B) 10030 (C) 1353 (D) 1352 (E) 1339 6.

(OG ) If x is equal to the sum of the even integers from 40 to 60, inclusive, and y is the number of even integers from 40 to 60, inclusive, what is the value of x + y? (A) 550 (B) 551 (C) 560 (D) 561 (E) 572

7. In the first week of the year, Nancy saved $1, in each of the next 51 weeks, she saved $1 more than she had saved in the previous week. What was the total amount that Nancy saved during the 52 weeks? (A) $ 1,326 (B) $ 1,352 (C) $ 1,378 (D) $ 2,652 (E) $ 27,560 8. (OG) What is the sum of the integers in the table below? 1 2 3 4 5 6 7 -2 -4 -6 -8 -10 -12 -14 3 6 9 12 15 18 21 -4 -8 -12 -16 -20 -24 -28 5 10 15 20 25 30 35 -6 -12 -18 -24 -30 -36 -42 7 14 21 28 35 42 49 (A) (B) (C) (D) (E) 9.

28 112 336 448 784

(OG) If an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence? (A) 585 (B) 580

66

(C) (D) (E)

575 570 565

10. (OG) If the sum of n consecutive integers is 0, which of the following must be true? I. n is an even number II. n is an odd number III. The average (arithmetic mean) of the n integers is 0. (A) I only (B) II only (C) III only (D) I and III (E) II and III Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. Example: the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. On the GMAT you do not need any formula to solve problems concerning geometric progression. Consider the following examples: 11. A certain series is defined by the following recursive rule: S = k Ă— S , where k is a constant. If the first term of the series is 4 and the 5th term is 244, what is the 7th term? (A) 312 (B) 488 (C) 976 (D) 2196 (E) 6588 12. A certain culture of bacteria quadruples every hour. If a container with these bacteria was half full at 10:00 a.m., at what time was it one-eighth full? (A) 9:00 a.m. (B) 7:00 a.m. (C) 6:00 a.m. (D) 4:00 a.m. (E) 2:00 a.m.

67

GoGMAT Problem

68

Answers and explanations 1. Actually, all of these sequences are arithmetic progressions with respective differences a) d = 1 (consecutive integers form an arithmetic progression); b) d = 3; c) d = -4 (note the sign of the difference); d) d = 0. 2. a) 6 pieces b) 99 spaces c) 4 pieces. 3. Difference between any two consequent numbers from I is constant. Thus, I is an example of arithmetic progression. The same is true for II. III is not necessarily true (it is easy to see given the example: p = 1, t = 5). Thus, D is the answer. 4. It is easy to see that multiples of 4 constitute arithmetic progression with difference d = 4. Thus, n = (96 - 12)/4 + 1 = 22. B is the answer. 5. For example, consequent members of this progression are 324713 and 324813, and we can find out that numbers with given tens and units digits constitute arithmetic progression with difference d = 100.The first member of this progression is 324713, the last member is 458513. Thus, the progression has (458,513 – 324,713)/100 + 1 = 4585 – 3247 + 1 = 1339 members. 6. Even integers from 40 to 60, inclusive, constitute arithmetic progression with difference d = 2. Then y = (60 – 40)/2 + 1 = 11. X = (40 + 60) Ă—  11/2  =  550.  Thus,  x  +  y  =  561. 7. Here we deal with the arithmetic progression, for which đ?‘Ž = 1, đ?‘‘ = 1. We are asked about the sum of 52 terms, but we do not know đ?‘Ž . That is why we should use the 2nd formula for the sum of n terms of the arithmetic progression. đ?‘†

=

2 Ă— 1 + 1 Ă— (52 − 1) Ă— 52 = 53 Ă— 26. 2

You can easily count the number, but first determine the last digit of this product – it is 8. The only such answer is C. 8. Integers in each row constitute arithmetic progression. Thus, we can easily find the sum in first row: S1 =  (7  +  1)Ă—7/2  =  28.  If  we  look  at  two  consecutive  rows  then  second  one  is  product  of  first  row  and integer 2, the third row is a product of first row and integer 3, etcetera. The sums in rows can be found: S2 = -2Ă—S1, S3 =  3Ă—S1, S4 = -4Ă—S1, S5 =  5Ă—S1, S6 = -6Ă—S1, S7 =  7Ă—S1. The sum of all integers in the table is equal to S1Ă—(1  â€“ 2 + 3 – 4 + 5 – 6 + 7) = S1Ă—4  =  112.  The  best  answer  is  B. 9. Let the first integer be x, then the last integer in the sequence is x + 9. Thus, the sum of the first 5 integers  is  equal  to  (x  +  x  +  4)Ă—5/2  =  5x  +  10.  The  sum  of  the  last  5  integers  is  equal  to  (x  +  5  +  x  +  9)Ă—5/2  =  5x  +  35.  These  two  sums  differ  by  25,  and  the  second  sum  is  greater  than  the  first  one.  Thus, the answer is 565 + 25 = 585. The best choice is A. 10. If the sum of consecutive numbers is 0, then for each positive x in the sequence must be negative – x. For example, if 2 in sequence then -2 is also there. There are k positive numbers, k negative numbers, and there should be zero: n = k + k + 1 = 2k + 1  â€“ odd number. The arithmetic average of some set is equal to the sum of its elements divided by the number of elements in this set. Therefore we can conclude that the average is equal to 0 if and only if the sum is equal to 0. Given this, III is also true. E is the answer. 11. Using recursive rule, S = S Ă— k , 244 = 4 Ă— k ,we can find value of k: k = 81, k = 3.  To find the 7th term: S = S Ă— k , S = 4 Ă— 3 = 2916 . The answer is D. 12. One-eighth is four times less, than one half. But from the problem we know, that the number of bacteria quadruples every hour. So, if at 10 a.m. the container is half full, one hour ago it was oneeighth full. The answer is A.

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GoGMAT Problem Explanation

70

Statistics GoGMAT, Session 4 I.

Mean or arithmetic average 1) Mean (arithmetic average) of a set of n numbers is the sum of those numbers divided by n. đ?‘šđ?‘’đ?‘Žđ?‘› = Â

đ?‘ đ?‘˘đ?‘š  đ?‘œđ?‘“  đ?‘›  đ?‘œđ?‘?đ?‘—đ?‘’đ?‘?đ?‘Ąđ?‘ . đ?‘›

Example 1: the average of 6, 4, 7, 10, and 4 is

=

= 6.2

2) The mean of the arithmetic progression can be found using the following formula: � +� ×� � +� � 2 ���� = = = . � � 2 So, to calculate the mean of the arithmetic progression there is no need to add all the terms, but only the first and the last. 3) If you are given the average A of a set of n numbers, multiply A by n to get their sum. 4) If all numbers in a set are the same, then that number is the average. 5) If the numbers in the set are not all the same, then the average must be greater than the smallest number and less than the largest number. 6) The sum of the differences between the average and the terms above the average equals the sum of the differences between the average and the terms below the average. Example: Consider the set {83, 84, 86, 98, 99}. Its mean equals 90. The differences between 90 and the terms that are below 90 are 90 – 83 = 7 The total difference is 7 + 6 + 4 = 17. 90 – 84 = 6 90 – 86 = 4 The differences between 90 and the terms that are above 90 are 98 – 90 = 8 The total difference is 8 + 9 = 17. 99 – 90 = 9 These differences are always equal. 7) If there are two groups of objects: Group A

Group B

m objects,

n objects,

mean of x

mean of y The mean for two groups combined: ��+�� �+�

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Example 2: Assume that there are two groups of students: group A consisting of 15 students and group B with 10 students in it. If both groups passed the test and the average scores are 70 and 80 respectively, what is the average score for the two groups combined? To answer the question, use the formula above: 15 Ă— 70 + 10 Ă— 80 1050 + 800 = = 74. 15 + 10 25

8) Note, that if there are two sets A (with average of x) and B (with average of y), then to find the mean for the combined set, you do not actually need to know exact numbers of terms in those sets, but only the ratio of number of terms in A to the number of terms in B. Example: Assume again that there are two groups of students A and B with respective average test scores 70 and 80. If the ratio of number of students in A to the number of student in B is 3:2, then what is the average for the sets A and B combined? To answer the question, use the same formula above, but express the numbers of students as 3s for A and 2s for B: 3đ?‘ Ă— 70 + 2đ?‘ Ă— 80 đ?‘ (210 + 160) 370 = = = 74. 3đ?‘ + 2đ?‘ 5đ?‘ 5

Thus, we obtained the same result as in the example above, but without actual values for the numbers of students. 9) If there are two groups of objects, A and B with respective averages m and n, then the average of the two sets combined will be closer to the average of the set with a greater number of objects. Hence, if the numbers of objects are equal, then the average of the two sets combined will be equidistant from the individual averages of the sets A and B. Example 3: In the example 2 we found the average for the two groups with 15 and 10 students respectively, where the individual average scores were given: 70 for A and 80 for B. Here group A has more students than group B, and the average 74 is closer to the average of this group A. Let’s  switch  the  numbers  of  students: Group A has 10 students with the average score 70 Group B has 15 students with the average score 80. The average for the two groups combined is: 10 Ă— 70 + 15 Ă— 80 700 + 1200 = = 76. 10 + 15 25

76 is closer to 80, the average of the group B with greater number of students. Now assume there are 10 students in both groups: 10 Ă— 70 + 10 Ă— 80 700 + 800 = = 75. 10 + 10 20

75 is equidistant form 70 and 80.

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II.

Median 1) To calculate the median of n numbers, ďŹ rst order the numbers from least to greatest. If  n  is  odd,  the  median  is  deďŹ ned  as  the  middle  number If n is even, the median  is  deďŹ ned  as  the  average  of  the  two  middle  numbers. Example 4: In order to find the median of the set {6,4,7,10,4}, put them in order from least to the greatest: {4,4,6,7,10}, and take the central number: 6. Thus the median is 6. For the numbers 4, 6, 6, 8, 9, 12, the median is (6+8)/2=7. 2) Arithmetic progression has its mean equal to its median.

III.

Range 1) The range of the set {x1, x2, x3, ..., xn} is equal to the difference between its greatest and lowest elements.

2) The range is always non-negative: đ?‘&#x;đ?‘Žđ?‘›đ?‘”đ?‘’ ≼ 0. And it is equal to 0 in two cases: all the elements  are  equal  to  each  other  {a,  a,  a‌}, the set consists of only one element {a}. 3) Adding the same number to all the terms of the set does not change the range. 4) Multiplying all terms of the set by the same number (m) changes the range. The new range will be m times greater than the initial range.

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IV.

Mode

Mode is a most frequent element in the set. Example: in the set {1, 2, 3, 4, 3, 5, 7, 2, 1, 2} mode is equal to 2. In the set {1, 2, 3, 2, 3} there are two modes, 2 and 3.

V.

Standard deviation 1) If x* is the mean of some finite set S = {x1, ..., xn}, then (đ??ą đ?&#x;? − đ??ą ∗ )đ?&#x;? + (đ??ą đ?&#x;? − đ??ą ∗ )đ?&#x;? +. . . +(đ??ą đ??§ − đ??ą ∗ )đ?&#x;? đ??§ is called standard deviation of S. đ?›”=

In fact, you do not need this formula on the GMAT. What you do need is the general (and rough) understanding of the standard deviation as the average of deviations of all the terms from the mean.

đ?œŽ

mean

đ?œŽ

(x − x ∗ )

High standard deviation

Low standard deviation

đ?œŽ

mean

đ?œŽ

mean

đ?œŽ

đ?œŽ

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2) The standard deviation is always non-negative:đ?‘ đ?‘Ąđ?‘Žđ?‘›đ?‘‘đ?‘Žđ?‘&#x;đ?‘‘  đ?‘‘đ?‘’đ?‘Łđ?‘–đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› ≼ 0. And it is equal to 0 in two cases:

3) 4) 5) 6)

all the elements  are  equal  to  each  other  {a,  a,  a‌}, the set consists of only one element {a}. The standard deviation equals zero if and only if range equals zero. If the same constant a is added to each number of the set {x1, x2, x3,‌,  xn} with standard deviation s, the deviation of the new set {x1 + a, x2 + a, ‌,  xn + a} will still be s. If each number of the set {x1, x2, x3,‌,  xn} with standard deviation s is multiplied by the same number b, the deviation of the new set {bx1, bx2,  â€Ś,  bxn} will equal bs. The value of standard deviation is independent from other characteristics, such as mean, median or mode. To obtain the standard deviation we need to know all the values of set members.

Sample problems: 1. Last week, two classes took a test. If the average score for all students in two classes was 82, which class has more students? (1) The average score for the students in class A is 80. (2) The average score for the students in class Đ’ is 86. 2. In a certain group of people, the average weight of the males is 170 pounds and of the females, 120 pounds. What is the average weight of the people in the group? (1) The group contains twice as many females as males. (2) The group contains 15 more females than males. 3. (OG) If m is the average of the first 10 positive multiples of 5 and if M is the median of the first 10 positive multiples of 5, what is the value of M – m? (A) -5 (B) 0 (C) 5 (D) 25 (E) 27.5 4. (OG) For a certain set of n numbers, where n > 1, is the average equal to the median? (1) If the n numbers in the set are listed in increasing order, then the difference between any pair of successive numbers in the set is 2 (2)  The  range  of  n  numbers  in  the  set  is  2Ă—(n  â€“ 1) 5.

(OG) If the average of four numbers K, 2K + 3, 3K – 5, and 5K + 1 is 63, what is the value of K? (A) 11 (B) 63/4 (C) 22 (D) 23 (E) 253/10

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6. (OG) If S = {0, 4, 5, 2, 11, 8}, how much greater than the median of the numbers in S is the mean of the numbers in S? (A) 0.5 (B) 1.0 (C) 1.5 (D) 2.0 (E) 2.5 7. (OG) If Jill’s average score for three games of bowling was 168, what was her lowest score? (1) Jill’s highest score was 204 (2) The sum of two Jill’s highest scores was 364 8. (OG) The average of 6, 8, and 10 equals the average of 7, 9, and (A) 5 (B) 7 (C) 8 (D) 9 (E) 11 9. (OG) If x books cost $5 each and y books cost $8 each, then the average cost, $ per book, is (A) (5x + 8y)/(x + y) (B) (5x + 8y)/(xy) (C) (5x + 8y)/13 (D) 40xy/(x + y) (E) 40xy/13 10. (OG) The temperatures in degrees Celsius recorded at 6 in the morning in various parts of a certain country were 100, 50, -20, -10, -50, and 150. What is the median of these temperatures? (A) -20C (B) -10C (C) 20C (D) 30C (E) 50C 11. (OG) If the median of the numbers in the list {3, 6, 8, 19} equals to the median of {x, 3, 6, 8, 19}, what is the value of x? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 12. (OG) If n is an integer between 1 and 10, inclusive, then the median of the set {2, 4, 6, 8, n, 3, 5, 7, 9} must be (A) either 4 or 5 (B) either 5 or 6 (C) either 6 or 7 (D) N (E) 5.5

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13. (OG) Which of the following numbers is greater than three-fourths of the numbers but less than one-fourth of the numbers from the set {38, 69, 22, 73, 31, 47, 13, 82}? (A) 56 (B) 68 (C) 69 (D) 71 (E) 73 14. (OG) S is a set containing 9 different numbers, T is a set containing 8 different numbers, all of which are members of S. Which of the following CANNOT be true? (A) The mean of S is equal to the mean of T (B) The median of S is equal to the median of T (C) The range of S is equal to the range of T (D) The mean of S is greater than the mean of T (E) The range of S is less than the range of T 15. (OG) If S is a set of four integers w, x, y, and z, is the range of the numbers in S greater than 2? (1) w – z > 2 (2) z is the least number in S 16. In a certain class, boys' heights range is 10 inches, and girls' height range is 8 inches. What is heights' range for all of the students? (1) The shortest boy is 2 inches taller than the tallest girl. (2) The tallest girl in the class is 72 inches in height. 17. What is the value of x ? (1) x is a mode of {3, 0, 1, -1, 0, 5, 1} (2) x is neither positive nor negative 18. (OG) I = {72, 73, 74, 75, 76} II = {74, 74, 74, 74, 74} III = {62, 74, 74, 74, 89} The data sets I, II, and III are ordered from greatest standard deviation to least standard deviation in which of the following? (A) I, II, and III (B) I, III, and II (C) II, III, and I (D) III, I, and II (E) III, II, and I 19. (GC) Is mean of set A greater than mean of set B? (1) The median of set A is greater than the median of set B (2) The standard deviation of set A is greater than the standard deviation of set B 20.

9.4, 9.9, 9.9, 9.9, 10.0, 10.2, 10.2, 10.5 The mean and the standard deviation of the 8 numbers shown are 10 and 0.3 respectively. What percent of the 8 numbers are within 1 standard deviation of the mean? (A) 90% (B) 85% (C) 80%

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(D) 75% (E) 70% 21. Set T consists of a certain number of even integers divisible by 3. Is standard deviation of T positive? (1) All elements of set T are positive (2) The range of set T is positive

GoGMAT Problem

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Answers and Explanations 1. Use the point 9) from the mean paragraph, which tells us, that there are more objects in a set with a mean closer to the mean of the combined set. Statement (1) gives us the average for class A (80). But we do not know, whether the average of class B is closer to 82 or not. So it is insufficient. Statement (2) provides us with similar information about class B, so it is insufficient as well. From both statements together we understand, that the mean of A (80) is closer to 82, than the mean of B (86). Thus, we conclude, that class A has more students, than class B. The answer is C. 2. Here the property 8) from the mean paragraph is tested. From statement (1) we know the ratio of women to men. Hence, we can determine the mean of the two groups of people combined. From statement (2) we know the difference, but we have no clue about the ratio. So, it is insufficient. The answer is A. 3. Multiples of 5 constitute numbers is an arithmetic progression, for which mean and median are equal. Thus their difference is equal to 0. The answer is B. 4. Statement (1) essentially says that this set of numbers is arithmetic progression with difference d = 2. For arithmetic progression we have mean equal to median. Therefore, (1) alone is sufficient and the answer must be A or D. Second statement gives no clue about question asked in the stem and thus is insufficient. The answer is A. 5. Since (K + 2K + 3 + 3K – 5 + 5K + 1)/4 = 63, (11K – 1)/4 = 63, 11K = 253, K = 23. D is the answer. 6. Clearly, mean of S is equal to (4 + 5 + 2 + 11 + 8)/6 = 30/6 = 5. To find the median we should order the numbers in the set: {0, 2, 4, 5, 8, 11}. Median of S is equal to (4 + 5)/2 = 4.5. The difference between mean and median is 5 – 0.5 = 0.5. The answer is A. 7. Statement (1) is insufficient to answer the stem question since the other two scores are unknown. (2) is alone sufficient to answer since the lowest score is 3×168 – 364 = 504 – 364 = 140. B is the answer. 8. The average of 6, 8, and 10 is 8, and if x is missing number, (7 + 9 + x)/3 = 8 and x = 8. C is the answer. 9. Total cost of the books is 5×x + 8×y, and total number of books is x + y. Thus, A is the answer. You can use the formula from the property 7) for the mean. 10. Since the set contains even number of elements, the median is (-1 + 5)/2 = 20C. The answer is C. 11. It is evident that the median of the first set equals to 7 ( = 7). Median of the second set is x, 6, or 8. Thus, x = 7. The answer is B. 12. Note that if n = 5, the median is equal to 5, and if n = 6, the median is equal to 6. Otherwise n < 5 or n > 6 and the median is anyway in the set {5, 6}. Thus, B is the answer. 13. The ranking of the numbers is as follows: {13, 22, 31, 38, 47, 69, 73, 82}. Thus, answer choice should be greater than 6 numbers - three-fourths of set and less than 2 numbers - onefourth of numbers - strictly between 69 and 73. Of the choices given, only 71 satisfies these conditions. D is the answer. 14. A can be true if S = {-4, -3, -2, -1, 0, 1, 2, 3, 4} and T = {-4, -3, -2, -1, 1, 2, 3, 4}. The same is true for B and C. D can be true if S = {-4, -3, -2, -1, 0, 1, 2, 3, 4} and T = {-4, -3, -2, -1, 0, 1, 2, 3}. E CANNOT be true anyway since range can only shrink when some elements are taken out from the set. Thus, E is the answer.

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15. (1) is sufficient to answer the question from the stem, since range is equal to the maximum difference between any two elements of the set. If there are some members from the set S that have difference greater than 2, then the range is greater than 2. (2) alone is not sufficient to answer the stem question since no information about any other members is given. A is the answer. 16. Let’s make an illustration to this problem. From statement (1) we get the following picture: Girls

2 in.

8 inches

Boys 10 inches

Range = 20

Clearly, the range equals 20. Statement (2) does not provide any information about how girls’ and boys’ height relate to each other, therefore it is insufficient. The answer is A. 17. Statement (1) suggests that we should find the mode of the set {3, 0, 1, -1, 0, 5, 1}. Be attentive here: there are two modes, i.e. 0 and 1. So, the 1st statement is not sufficient. Statement (2) alone is sufficient, because the only number, that is neither positive, nor negative is 0. The answer is B. 18. Standard deviation of the second set is 0. Even from this fact we can infer that the answer is B or D. Let’s compare standard deviations I and III. It can be assumed that standard deviation of I is less than standard deviation of III (elements in I are closer to their mean m = 74, than elements in III). In fact one can calculate s2(I) = (-2)2 + (-1)2 + 02 + 12 + 22 = 10, while s2(III) = (-12.6)2 + (...)2 + (...)2 + (...)2 + (...)2 >> s2(I). The answer is D. 19. Statements (1) and (2) together are insufficient to answer the stem question: if A = {4, 7, 10} and B = {3, 5, 7} both (1) and (2) are satisfied and the mean of A is greater than the mean of B; if A = {1, 7, 7} and B = {6, 6, 6} both (1) and (2) are again satisfied and the mean of A is less than the mean of B. Thus, E is the answer. 20. The words “within one standard deviation” mean, that you should consider numbers both, less than the mean and greater than the mean. One standard deviation below the mean: 10 - 0.3 = 9.7 One standard deviation above the mean: 10 + 0.3 = 10.3 So, the numbers that are within one standard deviation should be between 9.7 and 10.3. These numbers are: 9.9 × 3 times, 10, 10.2 × 2 times. All in all, there are 6 such numbers. 6 is 75% of 8. The answer is D. 21. Do not be confused by the question: standard deviation is not always positive, but can equal zero. Statement (1) does not allow us to understand, whether the numbers are different, or not, which is crucial. So, the 1st statement is not sufficient. Statement (2) gives, that the range is not zero, hence, not all of the numbers are the same, which means, that the standard deviation cannot be zero. So it is positive. The answer is B.

80

GoGMAT Problem Explanation

81

Home assignment

Progressions 1. (1) (2)

In an arithmetic sequence, if a5 = 16 and d = an – an-1, what is the value of d? �  =  �  +  4. d > 0.

2. What is the sum of all positive multiples of 6 that are less than or equal to 714? (A) 3,600 (B) 7,200 (C) 14,000 (D) 28,800 (E) 42,840 3. What is the largest prime factor of the sum of the series of consecutive integers from 1 to 30, inclusive? (A) (B)

(C) (D) (E)

31 17 13 11 5

4. (GoGMAT) In a pile of bricks each layer contains one more brick than the layer on top of it. If there are 38 layers and the top layer has only one brick, how many bricks are there in the pile? (A) 147 (B) 471 (C) 526 (D) 652 (E) 741 5. In the arithmetic sequence đ?‘Ą , đ?‘Ą , đ?‘Ą , ‌ , đ?‘Ą , ‌ , đ?‘Ą = 23 and đ?‘Ą What is the value of n when đ?‘Ą  = −4? (A) -1 (B) 7 (C) 10 (D) 14 (E) 20

 Â

=đ?‘Ą

Â

− 3 for each đ?‘› > 1.

6.    Of  the  five  numbers  in  a  sequence,  the  first  term  is  10,000,  and  each  of  the  following  terms  is  20%  of  the  previous  term.  What  is  the  value  range  of  the  five  numbers? (A) 9,375 (B) 9,750 (C) 9,975 (D) 9,984 (E) 10,736

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7.

(GPrep)

�  , � , , �  , ‌ , �

Â

In the sequence shown, đ?‘Ž = đ?‘Ž  + đ?‘˜, where 2 ≤ đ?‘› ≤ 15, and k is a non-zero constant. How many of the terms of the sequence are greater than 10? (1) đ?‘Ž  = 24 (2) đ?‘Ž  = 10 8.

For any positive integer n, the sum of the first n positive integers equals n(n 1) . If m and n 2

are positive integers and m > n, what is the sum of all the integers between m and n, inclusive? (A)

m(m 1) 2

n(n 1) 2

(B)

m(m 1) 2

n(n 1) 2

(C)

m(m 1) 2

n(n 1) 2

(D)

m(m 1) 2

n(n 1) 2

(E)

m(n 1) 2

n(m 1) 2

9. How many integers from 1 to 200, inclusive, are divisible by 3 but not divisible by 7? (A) 38 (B) 57 (C) 58 (D) 60 (E) 66 10. The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression? (A) 300 (B) 120 (C) 150 (D) 170 (E) 270 11. How  many  integers  between  100  and  150  inclusive  can  be  evenly  divided  by  neither  3  nor  5? (A) 22 (B) 24 (C) 26 (D) 27 (E) 28

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Statistics 1. (OG) What is the value of the sum of a list of n odd integers? (1) n = 8 (2) The square of the number of integers on the list is 64 2. (OG) A certain bakery has 6 employees. It pays annual salaries of $14,000 to each of 2 employees, $16,000 to 1 employee, and $17,000 to each of the remaining 3 employees. The average annual salary of these employees is closest to which of the following? (A) $15,200 (B)

$15,500

(C)

$15,800

(D)

$16,000

(E)

$16,400

3. (OG) If x, y, and z are numbers, is z = 18? (1) The average of x, y, and z equals to 6 (2) x = - y 4. (OG) Is the average of x and y greater than 20? (1) The average of 2x and 2y is 48 (2) x = 3y 5. (OG) Will the first 10 volumes of a 20-volume encyclopaedia fit upright in the bookrack of width x? (1) x = 50 cm (2) Twelve of the volumes have an average thickness of 5 cm 6. (GC) What is the average of a, b, and c? (1) (a + b) + (c + d) = 17 (2) d = 5 7. (OG) If a basketball team scores an average of x points per game for n games and then scores y in its next game, what is the team’s average score for the n + 1 games? (A)

(nx + y)/(n + 1)

(B)

x + y/(n + 1)

(C)

x + y/n

(D)

n(x + y)/(n + 1)

(E)

(x + ny)/(n + 1)

8. (GC ) Are the integers a, b, and c consecutive? (1) b – a = c - b (2) The average of a, b, and c equals to b

84

9. If the mode of 5 numbers is unique, is the mode of the 5 numbers equal to the mean of the 5 numbers? (1) The mode is not the largest number in the set. (2) Three of the numbers in the set are the same. 10. (OG) What is the average of j and k? (1) The average of j + 2 and k + 4 is 11 (2) The average of j, k, and 14 is 10 11. (GC ) What is the median of the set with 9 positive integers? (1) The mean of 4 smallest ones is 150 (2) The mean of 4 largest ones is 500 12. (OG) On 3 sales John has received commissions of $240, $80, and $110, and he has one additional sale pending. If John is to receive an average commission of $150 on the 4 sales, then the fourth commission must be (A) (B) (C) (D) (E)

$164 $170 $175 $182 $185

13. What is the range of 10 numbers 2, 3, 4, 6, 7, 9, 10, 12, 20 and x? (1) The average of the 10 numbers is 8. (2) x is equal to the median of the 10 numbers. 14. (GC) A set S has a range 24.7 and following operations are performed on S: 6 is added to all elements of the set and further all the elements of the set S are divided by 10. What will be the new range of the set? (A) (B) (C) (D) (E)

24.7 2.47 3.07 3.47 30.7

15. (OG) The average of six numbers is 8.5. When one number is discarded, the average of the remaining numbers becomes 7.2. What is the discarded number? (A) (B) (C) (D) (E)

7.8 9.8 10.0 12.4 15.0

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16. (GC) A secondary school teacher conducted Math test to 2 groups of students. Is the range for the first group same as the second group? (1) The average score was 60 for each group (2) The number of students in each group taking the test was 30, and the lowest score was 48 17. (OG) If p, q, x, y, and z are different positive integers, which of the five integers is the median? (1) p + x < q (2) y < z 18. (GC ) Two groups in the class have passed the test. If the average grade for the first group was 68.2, and the average grade for the second group was 73.5, what was the average grade in two groups? (1) The first group had 20 more students than the second one (2) The first group has 3 times more students than the second one 19. (GC) Two numbers are removed from the list {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}. What is the standard deviation of the remaining 8 numbers? (1) The median of the eight numbers is 10 (2) The mean of the eight numbers is the same as the original mean 20. (OG) If the average of 4 numbers is 50, how many of the numbers are greater than 50? (1) None of the four numbers is equal to 50 (2) Two of the numbers are equal to 25 21. What is the standard deviation of given set of numbers with average of 5? (1) None of the numbers is greater than the average. (2) If the value of each of the given numbers is increased by 7, the standard deviation is equal to zero. 22. The mean for four positive distinct integers is 24. If the median of four numbers is 18, what is the largest possible range of the four numbers? (A) 60 (B) 58 (C) 54 (D) 52 (E) 48 23. Is the range of a non-empty set S greater than its mean? (1) All members of S are negative (2) The median of set S is negative 24. (GC) What is the value of k? (1) The median of the set {n, k, 2, 8, 12} is 7 (2) k > n

86

25. (OG) Which of the following CANNOT be the median of the three positive integers x, y, and z? (A) x (B) z (C) x+z (D) (x+z)/2 (E) (x+z)/3 26. (OG) The average of the set of numbers {3, k, 2, 8, m, 3} is 4. If k and m are integers and đ?‘˜  ≠ đ?‘š, what is the median of the set? (A) (B) (C) (D) (E)

2.0 2.5 3.0 3.5 4.0

27. (GC) Is n > 5? (1) The median of the set {-2, 1, 5, 8, n} is greater than 1 (2) The median of the set {-2, 1, 5, 8, n} is less than 4 28. 500 records have a distribution that is symmetric about the mean which is 10. If 68 percent of the distribution lie within one standard deviation of the mean, and the standard deviation is 2, then how many of the records are less than 12? (A) 160 (B) 320 (C) 340 (D) 420 (E) 460 29. (GC) If the average height of three people is 68 inches, is the shortest person not less than 60 inches tall? (1) The height of the tallest person is 72 (2) One of the persons is 70 inches tall. 30. (OG) Last year department store X had a sales total for December that was 4 times the average of the monthly sales totals for January through November. The sales total for December was what fraction of the sales total for the year? (A)

1/4

(B)

4/15

(C)

1/3

(D)

4/11

(E)

4/5

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31. Which of the following data sets has the third largest standard deviation? (A) {1, 2, 3, 4, 5} (B) {2, 3, 3, 3, 4} (C) {2, 2, 2, 4, 5} (D) {0, 2, 3, 4, 6} (E) {-1, 1, 3, 5, 7} 32. (GC) In a certain company, the average wages of employees in Town A is X, the average wages of employees in Town B is Y. If both types of employees are added together, is the new average salary smaller than (X + Y)/2? (1) There are more employees in Town A than in town B (2) Y – X = 4200

88

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

C E A E C D B B (Pl) B C D Statistics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

E C C A E C A (Pl) E E D E B D B E E E B E E D B D C C (Pl) C B D A B (Pl) A C

89

Test  â„–  2.  Progression  and  statistics 1. If sequence x1, x2, x3,‌  ,  xn is such that xn+1 is 5 more than xn for n > 0, does x99 have  a  unit’s  digit  of 9? (1) The 100 term plus 1 is a multiple of 5. (2) The first term is 4. 2. On a given day, 50 students in a certain class took a math test. Some of the students took the test on test paper A, and the remaining of the 50 students took the test on test paper B. What is the average score on the test for the 50 students? (1)The average score for the students who take the test A was 75. (2)The average score for the students who take the test Đ’ was 77. 3. What is the average of n – x , n – y, n – z, and n – w? (1) The average of x, Ńƒ, z and w is 4n. (2) n = 120. 4. If the range of the six numbers 4, 3, 14, 7, 10, and x is 12, what is the difference between the greatest possible value of x and least possible value of x? (A) 2 (B) 7 (C) 12 (D) 13 (E) 15 5. If the average of positive integers x, y, and z is 10, what is the greatest possible value of z? (A) 8 (B) 10 (C) 20 (D) 28 (E) 30 6. (GoGMAT) The sequence of numbers đ?‘?, đ?‘?đ?‘ , đ?‘?đ?‘ , đ?‘?đ?‘ , đ?‘?đ?‘ , đ?‘?đ?‘ is a geometric progression. The sum of the first three terms in the series is one-ninth the sum of the first six terms. What of the ratio of the fifth term to the second term? (A) 1:1 (B) 2:1 (C) 3:1 (D) 8:1 (E) 9:1

90

a 7. The infinite sequence a1, a2, …, an, … is such that a1 = 1, a2 = 2, and an an 1 , what is a5 – a4? (A) 4 (B) 12 (C) 28 (D) 60 (E) 124 n 2

8. Which of the following is terminating decimal? 5 I. 120 33 II. 625 37 III. 512

(A) (B) (C) (D) (E)

I only II only III only I and II only II and III only

9. 233 + 234 – 235 = (A) – 3×233 (B) – 233 (C) 233 (D) 2×233 (E) 3×233 10. If 60 students took a test and the median score was 80, which of the following must be true? I. At least 29 scores are greater than 80 II. At least 30 scores are equal to or greater than 80 III. At most 30 scores are equal to or less than 80 (A) I only (B) II only (C) III only (D) I and II only (E) I and III only

11. Is 3 in set M? (1) 13 is in the set M. (2) If x is in the set, then x – 2 is also in the set M.

12. If p and q are prime numbers, where p is no more than q, is sum of p and q a prime number? (1) p is not equal to q. (2) p is greater than 2.

91

13. If the terms of a arithmetic sequence are 2, 4, 6, 8, 10, ..., xn, what is the value of n? (1) The sum of n terms is 2,550. (2) The average of the n terms is 51.

14. If x and Ńƒ are integers and xy2 is a positive odd integer, which of the following must be true? I. xy is positive II. xy is odd III. x + Ńƒ is even (A) I only (B) II only (C) III only (D) I and II only (E) II and III only 15. Is mx < m + x? (1) 0 < x < 1. (2) m is a positive integer. 16. What is the range of numbers in Set S? (1) The standard deviation of the numbers is 3.5. (2) The average of all numbers is 7. 17. The average of positive integers x, y, and z is 12. If x < y < z, and the median of three numbers is 10 greater than the smallest number, what is the greatest possible value of z? (A) 11 (B) 15 (C) 22 (D) 24 (E) 33 m

18. If 8 < m < 120, how many integer m are possible for 3 is a square of integer? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 19. If the median of five different positive integers is 20, which of the following is the least possible sum of these five positive integers? (A) 62 (B) 66 (C) 69 (D) 72 (E) 100

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20. Is the standard deviation of the scores of students in class X greater than the standard deviation of the scores of students in class Y? (1) The median score of students in class Y is greater than the median score of students in class X. (2) The average (arithmetic mean) score of students in class Y is greater than the average score of students in class X.

Answers: 1. B 2. E 3. C 4. D 5. D 6. D 7. B 8. E 9. B 10. B 11. C 12. B 13. D 14. E 15. C 16. E 17. D 18. D 19. B 20. E

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Lesson  â„–  3.  Arithmetic Factorization GoGMAT, Session 1 Positive integer that has exactly two different positive factors is said to be prime. Examples: 7 is an example of prime number since it has only two positive integer factors: 1 and 7. 1 is not prime since it has only one positive integer factor: 1. Every positive integer can be uniquely factored into product of powers of primes. Example:  Let’s  find  all  prime  factors  of  132.  Obviously,  the  least  prime  it  can  be  divided by is 2. We write 2 in the right column and the result of division (66) in the left one. As we see, that 66 is also divisible by 2, we repeat the previous step. 33 can by evenly divided by 3 with 11 as a result. 11 is prime itself. Thus, 132 = 2 Ă— 3 Ă— 11. 72 2 Consider one more example: 72 = 2 3

36

2

18

2

9

3

3

3

132

2

66

2

33

3

11

11

1

1 In general, in order to find all prime factors of a certain number N, first find out whether it is divisible by the least primes you know, e.g. 2, 3 or 5. If it is not divisible by any of them, try 7, 11 and so on. Each time when you find a prime factor, divide N by it and continue with the newly obtained number until you get 1. As a result, the integer N will be expressed as a product of n primes p , p , ‌ , p , each in the respective power s , s , ‌ , s . đ??Ź đ??Ź đ??Ź đ??? = đ??Šđ?&#x;?đ?&#x;? ∙ đ??Šđ?&#x;?đ?&#x;? ∙ ‌ ∙ đ??Šđ??§đ??§ 1.

(OG) If the product of the integers x, y, z, and w is 770, and if 1<w<x<y<z, what is the value of w+ z? (A) 10 (Đ’)       13 (ĐĄ)       16 (D) 18 (E) 21 More complex problems may ask for the total number of factors of a given integer. Example: Find the number of factors of 8. The only prime factor of 8 is 2, but the total number of its positive different factors is 4: they are 1, 2, 4 and 8. In most cases such procedure is too complicated and takes a lot of time. There is an easy way: đ??Ź đ??Ź đ??Ź for đ??? = đ??Šđ?&#x;?đ?&#x;? ∙ đ??Šđ?&#x;?đ?&#x;? ∙ ‌ ∙ đ??Šđ??§đ??§ the total number of positive factors (Q) can be found using the formula đ??? = (đ??Źđ?&#x;? + đ?&#x;?) ∙ (đ??Źđ?&#x;? + đ?&#x;?) ∙ ‌ ∙ (đ??Źđ??§ + đ?&#x;?)

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Example: 8 = 2 , hence 8 has 3+1=4 positive factors. 72 = 2 3 , and 72 has (3 + 1) ∙ (2 + 1) = 4 ∙ 3 = 12  positive factors. 132 = 2 Ă— 3 Ă— 11, and 132 has (2 + 1) ∙ (1 + 1) ∙ (1 + 1) = 3 ∙ 2 ∙ 2 = 12 positive factors. 2.

(OG) How many different positive integers are factors of 441? (A) 6 (B) 9 (C) 4 (D) 12 (E) 8

3.

(OG) If n is an integer, then n is divisible by how many positive integers? (1) n is the product of two different prime numbers (2) n and 23 are each divisible by the same number of positive integers Properties If a positive integer N is expressed as a product of primes đ??Ź đ??Ź đ??Ź đ??? = đ??Šđ?&#x;?đ?&#x;? ∙ đ??Šđ?&#x;?đ?&#x;? ∙ ‌ ∙ đ??Šđ??§đ??§ then N is expressed as follows: đ?&#x;?đ??Ź

đ?&#x;?đ??Ź

đ?&#x;?đ??Ź

đ???đ?&#x;? = đ??Šđ?&#x;? đ?&#x;? ∙ đ??Šđ?&#x;? đ?&#x;? ∙ ‌ ∙ đ??Šđ??§ đ??§ , In other words, the prime factors of đ?‘ are the same as of N, and the powers of those primes are doubled. Analogically, the powers of primes of any perfect cube are divisible by 3, and so on.

4.

(OG) If n is a positive integer and n2 is divisible by 72, then the largest positive integer that must divide n is (A) 6 (Đ’) Â Â Â Â 12 (ĐĄ) Â Â Â Â 24 (D) 36 (E) 48

5.

(OG) If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer? (A) (x+z)z (B) (y+z)/x (C) (x+y)/z (D) xy/z (E) yz/x (OG) If r and s are positive integers, is r/s an integer? (1) Every factor of s is also a factor of r (2) Every prime factor of s is also a prime factor of r

6.

7.

(OG) If the integer n is greater than 1, is n equal to 2? (1) n has exactly 2 positive factors (2) The difference of any two positive factors of n is odd.

95

8. (OG) If у is an integer, is y3 divisible by 9? (1) у is divisible by 4 (2) у is divisible by 6 Answers and explanations 1.

The prime factorization of 770 is 2×5×7×11. Since 1<w<x<x<y<z, the values for the variables must be w = 2, x = 5, у = 7, and z = 11, so w + z = 13. Thus, В is the answer.

2.

441 = 3×147 = 3×3×49 = 3×3×7×7 = 32×72. Substituting this into the formula, we have (2 + 1) × (2 + 1) = 9 factors. Thus, B is the answer

3.

(1) is sufficient to answer the stem question since n = p × p , it has (1 + 1) × (1 + 1) - 4 factors. (2) is also sufficient since the number of integer factors of 23 is (3 + 1) = 4. Thus, D is the answer.

4. If n2 is divisible by 72=2 × 3 , then n must be divisible by 2 and 3 . So, n must be divisible by 2 = 4 and by 3 = 3 (given the formula from the property). Thus, n must be divisible by 12. В is the answer. 5.

Evidently, x and у are divisible by z and everything that has z as its denominator is necessarily an integer. Thus, the only answer choices possible are В and E. z/x is NOT necessarily an integer since from the stem it is only known that x/z is always an integer. Thus, В is the answer.

6.

(1) is sufficient to answer the stem question since s has at least two factors - s itself and 1. Both s and 1 are factors of r, according to the stem. Thus, r is divisible by s and r/s is an integer. Thus, the answer must be A or D. (2) alone is insufficient to answer the stem question since if r = 18 and s = 16 r/s is not an integer, while if r = 18 and s = 2 r/s is an integer. Thus, A is the answer.

7. Statement (1) is insufficient to answer the stem question since it actually states that n is prime. So, n can be 2, or 3, or 5, etc. Statement (2) says that the difference of any two positive factors of n is odd. Thus, n does not have odd factors except for 1. Therefore, no prime greater than 2 is a factor of n. Having this, we can state that n = 2. Thus, В is the answer. 8. (1) is not sufficient. (2) is sufficient, since у is divisible by 6 and consequently, у is divisible by 3. Thus, y3 is divisible by 27 and consequently, y3 is divisible by 9. В is the answer.

96

LCM and GCD GoGMAT, Session 1 Every pair of integers has one or more common divisors (i.e. numbers, which are the factors of both). The least of them is always 1 and the greatest one is called the GCD (Greatest Common Divisor). Numbers a and b are called mutually prime if they do not have common factors greater than 1: GCD(a,b)=1. Example: GCD(3, 6) =3 GCD(5,7) =1 (mutually prime) GCD(10, 15)=5 GCD(8,9) =1 (mutually prime). There is infinite number of common multiples for any given pair of integers (i. e. numbers, which are divisible by both the integers). The greatest of them cannot be found and the least of them is called the LCM (Lowest Common Multiple). Example: LCM(3, 6) =6 LCM(5,7) =35 LCM(10, 15)=30 LCM(8,9) =72.

1) 2) 3) 4)

In general, to find the GCD and LCM of two numbers Find prime factorizations of two numbers; Find the overlap of these factorizations; Multiply the primes from the overlap to get the GCD. In other words, GCD(a, b) is a product of the lowest powers of each prime in the factorizations of a and b. The LCM(a,b) will be the product of the highest power of each prime in the factorizations of a and b together. Example: Find the value of LCM(48, 180) and GCD(48,180). First, find the prime factorizations of the two numbers: 48 = 2 × 2 × 2 × 2 × 3 = 2 × 3 × 5 , 180 = 2 × 2 × 3 × 3 × 5 = 2 × 3 × 5 . What they share in common is two "2"s and a "3":

GCD(48, 180) = 2 × 3 × 5 = 12 (the lowest powers of primes). LCM(48, 180) = 2 × 3 × 5 =720 (the highest powers of primes).

97

1. Find the value of the lowest common multiple of 8, 9, 21. (A) (B) (C) (D) (E)

168 252 504 756 1512

2. The sum of the greatest common divisor of 120 and 50 and the greatest common factor of 30 and 75 equals to (A) 10 (B) 15 (C) 17 (D) 25 (E) 35 Properties: 1) LCM(a,b) and GCD(a,b) are defined for only positive integers a and b. GCD(a, b) ≤ a LCM(a, b) ≥ a 2) ; . GCD(a, b) ≤ b LCM(a, b) ≥ b Therefore, if the LCM of two numbers is equal to their GCD, then the numbers are equal to each other. 3) GCD(a, b) ∙ LCM(a, b) = a ∙ b. 4)

5)

⟹ d is a factor of GCD(a, b)

⟹ X is a multiple of LCM(a, b)

6) GCD(a, b, c) = GCD(GCD(a, b), c) 7) GCD(n, n + 1) = GCD(n, 1) = 1 . Any two consecutive integers are mutually prime. In general: GCD(a, a + b) = GCD(a, b) 3. (OG) Which of the following CANNOT be the greatest common divisor of two positive integers x and y? (A) 1 (B) x (C) y (D) x-y (E) x+y 4.

(GC) Is a two-digit number PQ odd? (1) The lowest common multiple of P and Q is even (2) P is odd

5. (GC) What is the value of ab? (1) a = 4m, b = 4n, and the greatest common divisor of m and n is 1 (2) The lowest common multiple of a and b is 24

98

6.

(OG) If a positive integer n is divisible by both 5 and 7, the n must also be divisible by which of the following? I. 12 II. 35 III. 70 (A) None (В) I only (С) II only (D) I and II (E) II and III

7. (OG) If n is a positive integer less than 200 and 14n/60 is an integer, then n has how many different positive prime factors? (A) Two (В) Three (С) Five (D) Six (E) Eight 8.

What is the greatest common divisor of the integers z and x? (1) Greatest common factor of x and y equal to 5. (2) z = 3x+y

9. (OG) If к and n are integers, is n divisible by 7? (1) n - 3 = 2k (2) 2k - 4 is divisible by 7 10. (OG) Is integer x divisible by 36? (1) x is divisible by 12 (2) x is divisible by 9 11. (OG) Which of the following is the least positive integer that is divisible by 2, 3, 4, 5, 6, 7, 8, and 9? (A) 15,120 (B) 3,024 (C) 2,520 (D) 1,890 (E) 1,680

99

GoGMAT Problem

100

Answers and explanations 1. First, factor out each number and express it as a product of prime number powers. 8= 2 ∙3 ∙7 9= 2 ∙3 ∙7 21 = 2 ∙ 3 ∙ 7 Out of the 4 prime factor categories 2, 3, and 7, the highest powers from each are 23, 32, and 71. Thus, LCM(8,9,21) = 2 ∙ 3 ∙ 7 = 8 ∙ 9 ∙ 7 = 504. The answer is C. 2. 120=2 ∙ 3 ∙ 5 , 50=2 ∙ 3 ∙ 5 , finding the minimum powers of primes, we get: GCD(120,50) =2 ∙ 3 ∙ 5 = 10. 30=2 ∙ 3 ∙ 5 , 75=2 ∙ 3 ∙ 5 , minimum powers give us GCD(30,75)= 2 ∙ 3 ∙ 5 =15. So, GCD(120,50)+GCD(30,75) = 10+ 15 = 25.The answer is D. 3. Evidently, x + у > x and x+y>y, so, x+y > GCD(x, у). Thus, x + у CANNOT be the greatest common divisor of x and у. Е is the answer. 4.

Statement (1) actually says that at least one digit (P or Q) is even. (1) itself is insufficient to answer the stem question. Statement (2) alone is insufficient since no information about Q is given. Thus, the answer must be С or E. Statements (1) and (2) together are sufficient since P is odd and Q must be even to satisfy information from (1). Therefore, Q is even and PQ is even, and С is the answer.

5. Statement (1) actually says that the greatest common divisor of a and b is 4. Statement (2) states that the lowest common multiple of a and b is 24. As we already know, LCM(a, b)xGCD(a, b) = axb. Therefore, С is the answer. 6.

If n is divisible by both 5 and 7, then it must be divisible by LCM(5, 7) = 35. Thus, only II is satisfied and the answer is C.

7. If 14n/60 is an integer, then 7n/30 is also an integer. Thus, n is divisible by 30. Therefore, n can be {30, 60, 90, 120, 150, 180}. All these numbers have exactly three prime factors: 2, 3, and 5. В is the answer. 8.

(1) alone gives us any information about z. Insufficient. (2) tells us z=3x+y, Then GCD(z,x)=GCD(3x+y, x). As we can see from the properties, GCD(3x+y,x) =GCD(y,x). But we still don't know x or y. (1) and (2) combined give us all info we need. Indeed, from (2) GCD(z,x)=GCD(y,x) and from (1) GCD(x,y)=5. So, the answer is C.

9. (1) or (2) alone is insufficient to answer the stem question. From (1) and (2) we have n = 2k + 3 and 2k - 4 = 7m. From the second equation, 2k=7m+4, so, n =7m+4 +3 = 7m + 7. Evidently, n is divisible by 7. С is the answer. 10. (1): x can be 12, 24, 36, 48, etc. Sometimes it will be divisible by 36, sometimes - no. (2): x can be 9, 18, 27, 36, 45, etc. Sometimes it will be divisible by 36, sometimes - no. (1) or (2) alone are not sufficient. Combined together (1) and (2) imply that x is divisible by LCM(9, 12)=36. С is the answer. 11. The maximum power of 2 that is presented among {2, 3, 4, 5, 6, 7, 8, 9} is 3. The maximum power

101

of 3 is 2, the maximum power of 5 is 1, the maximum power of 7 is 1. Thus, 23×32×51×71 = 8×9×5×7 = 2,520. С is the answer. GoGMAT Problem Explanation

102

Division with Remainder GoGMAT, Session 1 At first, let's have a look at the new property: Dividend  =  Divisor  Ă—  Quotient  +  Remainder. Example: 28=8Ă—3+4,  so  8  is  divisor  (we  divide 28 by 8), 3 is quotient (8 fits in 28 three times) and 4 is remainder (the part of 28 that is not divisible by 8). let’s  divide  145  by  6:

In other words, 145 contains 6 twenty four times and one more unit: 145 = 6 Ă— 24 + 1 If a number is not divisible by đ?&#x;?đ??¤ , the remainder when the number is divided by đ?&#x;?đ??¤ is the same as the remainder when the last k digits form the integer divided by đ?&#x;?đ??¤ . Examples: The number 2,346 is not divisible by 4, since the number formed by its last two digits, 46, is not divisible by 4. When 2,346 is divided by 4, the remainder is 2, since when 46 is divided by 4 the remainder is 2. The number 172,045 is not divisible by 8, since the number formed by its last three digits, 45, is not divisible by 8. When 172,045 is divided by 8, the remainder is 5, since when 45 is divided by 8 the remainder is 5. If a number is not divisible by đ?&#x;‘đ??¤ , the remainder when the number is divided by đ?&#x;‘đ??¤ is the same as the remainder when the sum of its digits is divided by đ?&#x;‘đ??¤ , k = 1,2. Examples: The number 3,248 is not divisible by 3, since the sum of its digits, 17, is not divisible by 3. When 3,248 is divided by 3, the remainder is 2, since when 17 is divided by 3 the remainder is 2. The number 172,345 is not divisible by 9, since the sum of its digits, 22, is not divisible by 9. When 172,345 is divided by 9, the remainder is 4, since when 22 is divided by 9 the remainder is 4. Example: “The  remainder  is  1  when  7  is  divided  by  3â€?  means  7  =  3  â‹… 2 + 1 A general expression of the integer X that leaves remainder 2 when divided by 4 is X = 4q + 2 Giving  q  different  integer  values:  0,  1,  2,‌  one  can  find  all  such  numbers:  2,  6,  10,  â€Ś Beginning with 2 every fourth number leaves remainder 2 when divided by 4. Thus, 2 is the least such number.

103

1. (OG) If x is an integer and y = 3x + 2, which of the following CANNOT be a divisor of y? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

2. (OG) When N is divided by T, the quotient is S and the remainder is V. Which of the following expressions is equal to N? (A) S×T (B) S+V (C) S×T+V (D) T× (S+V) (E) T× (S-V) 3. (GC) If 4 students were added to a dance class, would the teacher be able to divide her students evenly into a dance team (or teams) of 8? (1) If 12 students were added, the teacher could put everyone in teams of 8 without any leftovers (2) The number of students in the class is currently not divisible by 8 Classical type of question on the GMAT: The integer X leaves remainder 2 when divided by 4 and remainder 1 when divided by 3. One of three types of tasks will follow: 1) Find the least possible X. 2) Find one general rule for X. 3) Find the number of such integers on the interval 100...150. Explanation: First of all, interpret the statement of the problem as follows: X = 4a + 2 and X = 3b + 1, a, b - non-negative integers. Note: Be sure to use different variables for quotients in different rules. 1) Write out the equation 4a + 2 = 3b + 1 and express b through a: 4a + 1 b= . 3 We choose to express b, because its coefficient is less than the one before a. Then try different whole values (0, 1, 2,…) for a until b becomes the integer. In our case: for a=0, b=1/3 – not integer, for a=1, b=5/3 – not integer, for a=2, b=3 – integer. Thus, a = 2 and b = 3 are the least possible values of a and b to be integers simultaneously. Finally, substitute a = 2 in the rule for X (or b = 3, which yields the same X): X = 4×a + 2 = 4 × 2 + 2 = 3 × 3 + 1 = 10 Ten is the least possible X, that leaves remainder 2 when divided by 4 and remainder 1 when divided by 3.

104

X

= 10.

2)

To obtain the general rule for X use the following formula: đ??— = đ??‹đ??‚đ??Œ(đ???đ?&#x;? , đ???đ?&#x;? ) Ă— đ??œ + đ??— đ??Śđ??˘đ??§ where d , d – the divisors, X - the least possible X, c is any non-negative integer. In our example: LCM(3, 4) = 12  and  X = 10. Hence, X = 12c + 10. 3)  Let’s  find  out,  how  many  integers  with  the  required properties there are between 100 and 150: 100 < x < 150, 100 < 12c + 10 < 150, 90 < 12c < 140, 7 < c < 11 . Since c is an integer, there are 4 possibilities: c= 8, c=9, c=10 or c= 11, each of them will give us unique value of x. Thus the answer is 4.

Sample problems: 4. If x and y are positive integers, what is the remainder when 3y + 4x is divided by 10? (1) x = 25. (2) y = 1. 5. If number x is positive and x leaves the remainder 1 when divided by 3 and the remainder 3 when divided by 7, how many such numbers are there between 206 and 258? (A) 0 (B) 1 (C) 2 (D) 3 (E) Cannot be determined from the information given. 6. (GC) An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of eggs she could have had? (A) 60 (B) 81 (C) 241

105

(D) 301 (E) 361 7.

(OG) What is the units digit of (13)4(17)2(29)3? 9 (A) 7 (B) 5 (C) 3 (D) 1 (E)

8. (PR) Find the last digit of the resulting number: 135 (A) 0 (B) 2 (C) 5 (D) 7 (E) Cannot be determined

Ă— 237

Ă— 562

.

106

GoGMAT Problem

107

Answers and explanations 1. Since y = 3x + 2, y is not divisible by 3. Thus, y is not divisible by any multiple of 3, including 6. A is wrong since y = 8 = 3×2 + 2 and y is divisible by 4. There are similar examples for B, D, and E. C is the answer. 2. Using empirical rule N = T×quotient + remainder = T×S + V, one can easily get the answer, C. 3. Let S be the number of students in the class. Then (1) implies that S + 12 is divisible by 8. Then S + 12 = 8×q, where q is an integer. Then S =8q -12, S + 4 = (8q - 12) + 4 = 8q - 8 = 8(q - 1). Thus, S + 4 is divisible by 8 and groups of 8 students could be formed without any leftovers. (1) is sufficient to answer the stem. (2) states that S is not a multiple of 8. It is not sufficient since if S = 4 one group of eight dancers could be formed with new 4 students, while if S = 5 this is impossible. A is the answer. 4. The remainder of division by 10 is the last digit. The last digit of powers of 3 is repeated every 4 powers. For example, the last digit of 32 is 9 and the last digit of 36 is 9. As a coefficient 4 stands before X, the value of X does not influence the last digit of the whole expression, but Y does influence. So we need to find out the value of Y. This information is given in the second statement. 5. We can write down x=3a+1 and x=7b+3, a, b - non-negative integers. At first find x using the algorithm: 3a+1=7b+3, a= , b = 1. Then x =7b +3 =10. Now, write down general rule: x= LCM(3, 7)×c +x , or x=21c+10, where c is non-negative integer. Next step: substitute new rule in the interval: 206< 21c+10< 258 196 <21c< 248 9.333...< c< 11.8... As c is an integer, it can have two values: c=10 or c=11. Then x can be: x=21×10+10 =220 or x=21×11+10=242. The answer is C. 6. The easiest way to solve is to look at multiples of 7 among answers: only 301 is divisible by 7. The answer is D. 7. To find the units digit of product we need to know only last digits of factors: (_3)4(_7)2(_9)3=(_1)(_9)(_9)=_1. The answer is E. 8. To find the lasts digit of product we need to know only last digits of factors: ( _ _5)324( _ _7)143( _ _2)287. You can notice that we have a product of 2 and 5 in our expression. It will always give the last digit of 0 (2×5 =10). If we multiply a number of the form ...0 by any integer, the last digit of the result would be 0. The answer is A.

108

GoGMAT Problem Explanation

109

1.

2.

3.

Home assignment I. Factorization.

Number x is known to be divisible by the cube of an even prime number. Which of the following can be the value of x? (A) 6 (B) 12 (C) 24 (D) 35 (E) 50 Which of the following has the greatest number of prime factors? (A) 75 (B) 120 (C) 144 (D) 210 (E) 750 What is the number of distinct positive factors of the product of 484 and 124? (A) 3 (B) 4 (C) 7 (D) 8 (E) 30 4.

If n and p are different positive prime numbers, which of the integers n4, p3 and np has (have) exactly 4 positive divisors? (A) n4 only (B) p3 only (C) np only (D) n4 and np (E) p3 and np

5. If m = pxqy, where p and q are different positive prime numbers. Is m a square of integer? (1) xy is an odd number (2) x + y is Đ°n even number. 6. If 5x2 has two different prime factors, at most how many different prime factors does x have? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

110

7. If m and n are positive integer, and 1800m = n3, what is the least possible value of m? (A) 2 (B) 3 (C) 15 (D) 30 (E) 45 8. If n = 4p , where p is a prime number greater than 2, how many different positive even divisors does n have, including n ? (A) Two (B) Three (C) Four (D) Six (E) Eight 9. If đ?‘Ľ is divisible by 576, then the maximum number x must be divisible by is (A) 6 (B) 12 (C) 16 (D) 24 (E) 48 10. What is the largest positive integer n for which 20! is a multiple of 2n ? (A) 10 (B) 15 (C) 17 (D) 18 (E) 20 11. How many prime factors does positive integer n have? n has only a prime factor. (1) 5

(2)

3n2 has two different prime factors.

12. If 5400mn = k4, where m, n, and k are positive integers, what is the least possible value of m + n? (A) 11 (B) 18 (C) 20 (D) 25 (E) 33

111

II. LCM, GCD 1. (GC) a and b are positive integers. What is the value of b? (1) a=5 (2) The lowest common multiple of a and b is equal to the greatest common divisor of a and b 2. Comet A is seen near the Earth every 12 years while comet B every 20 years. If both comets were observed in 1979, for how many years do we have to wait to see the two comets together again? (Assume now is 2004) (A) 20 (B) 16 (C) 25 (D) 35 (E) 32 3. Integer m has 4 different prime factors and n has 3 different prime factors. If m and n has the greatest common factor of 15, how many different prime factors does mn have? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 4. If M and N are positive integers, is M divisible by N? (1) Greatest common divisor of M and N is 1 (2) Least common multiple of M and N is 210 5. Deborah and Mike visit the university library at regular intervals every 3 and 4 days respectively. If both of them were in the library on Monday, what day of the week will it be when they meet in the library again? (A) Thursday (B) Wednesday (C) Saturday (D) Friday (E) Sunday 6. (GC) The GCD of two numbers is 20 and the LCM is 840. One of the numbers is 120. What is the other number? (A)

140

(B)

120

(C)

40

(D)

20

(E)

10

112

7. Is x divisible by 48? (1) x is divisible by 8 (2) x is divisible by 6 8. What is the greatest common divisor of positive integers x and y? (1) x and y share exactly one common factor (2) x and y are both prime numbers 9. If x is a multiple of 3, is xy divisible by 48? (1) x is divisible by 8 (2) y is divisible by 8 10. (PR) If x is a sum of all even integers on the interval 13...65 and y is their number, what is the greatest common divisor of x and y? (A) 1 (B) 13 (C) 26 (D) 52 (E) 1014 11. If đ?‘€ = 14 ∙ 21 , and đ?‘ = 49 6 , then what is the number of all distinct positive factors of the greatest common factor of M and N? (A) 70 (B) 80 (C) 90 (D) 3 (E) 55 12. (GC) What is the greatest common divisor of a, b, and 3a+23b? (1) a = 4 (2) The greatest common divisor of a, b, and 23a + 3b is 4 13. (GC) What is the greatest common divisor of a + b and 4? (1) Greatest common divisor of a and 4 is 2 (2) Greatest common divisor of b and 4 is 2 14. (GC) What is the greatest common divisor of a, b, and c? (1) greatest common divisor of a and b is 3 (2) greatest common divisor of b and Â Ń Â is  4

113

15. (OG) If x and у are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y? (1) x = 12u, where u is an integer (2) у = 12z, where z is an integer 16. (GC) What is the greatest possible common divisor of two different positive integers which are less than 144? (A)

143

(B)

71

(C)

72

(D)

13

(E)

11

Division with remainder 1. If m = 7x + 9, and n = 7y + 8, what is the remainder when mn is divided by 7? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 2. Is x divisible by y? (1) x leave the remainder 0 when divided by 6. (2) у is multiple of 3. 3. When positive integer n is divided by 7, the remainder is 2. What is the value of n? (1) n is less than 20. (2) When n is divided by 8, the remainder is 0. 4.

What is the remainder when positive integer n is divided by 6? (1) When n is divided by 4, the remainder is 3. (2) When n is divided by 12, the remainder is 3.

5. When positive integer a is divided by 7, the remainder is 2; when positive integer b is divided by 7, the remainder is 3; when positive integer с is divided by 7, the remainder is 4, What is the remainder when a + b + c is divided by 7? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

114

6. (GC) What is the least positive number which leaves a remainder of 1 when divided by 3, 2 when divided by 4 and 3 when divided by 5? (A) 60 (B) 58 (C) 118 (D) 12 (E) 68 7.

Is integer x divisible by 3? (1) When x is divided by 5, the remainder is 1. (2) When x is divided by 15, the remainder is 1.

8. (OG) If a is a positive integer, and if the units' digit of a2 is 9, and the units' digit of (a + l)2 is 4, what is the units' digit of (a + 2)2? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 9.

(GC) Find the least positive number, which leaves, if divided by 3, 5, and 12, a remainder of 2. (A) 62 (B) 42 (C) 12 (D) 2 (E) 122

10. What is the sum of digits of number 1028 – 28? (A) 227 (B) 228 (C) 236 (D) 237 (E) 243 11. (GC) A certain number when divided by 8 and 11 leaves remainders of 3 and 7 respectively. What will be remainder when the number is divided by the product of 8 and 11? (A) 3 (B) 7 (C) 51 (D) 10 (E) 0 12. When n is divided by 13, the remainder is 2 and the quotient is k. When n is divided by 17, the remainder is 2. What is the remainder when k is divided by 17? (A) 0 (B) 2 (C) 4 (D) 13 (E) 15

115

13. (GC) How many numbers less than 500 exist such that when any of these is divided by 7 the remainder is 3 and when any of these is divided by 3 the remainder is 1? (A) 21 (B) 22 (C) 23 (D) 24 (E) 25 14. n is an integer between 80 and 200, inclusive. When n is divided by 8, the remainder is 4. What is the value of n? (1) n is divisible by 9. (2) n is divisible by 12.

116

Factorization 1 2 3 4 5 6 7 8 9 10 11 12

C D A E (Pl) A B C C (Pl) D D E D LCM and GCD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

C D B E C A E A C C B B C C B B (AP) Division with remainder

1 2 3 4 5 6 7 8 9 10 11 12 13, 14

B (Pl) E C B C (Pl) B (AP) B A D (AP) E C A D, E

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Test № 3. Prime factorization, LCM, GCD, division with remainder 1. If n is an integer greater than 10 and less than 40, what is the value of n? (1) When n is divided by 5, the remainder is 3. (2) When n is divided by 3, the remainder is 2. 2. If m, n, and k are positive integers, is mnk = 4? (1) mn = 4. (2) nk = 4. 3. For positive integer x, y, and z, is y > z? (1) When x is divided by 4, the remainder is y. (2) When x is divided by 6, the remainder is z. 4. If the product of the integers from 1 to n is divisible by 490, what is the least possible value of n? (A) 7 (B) 14 (C) 21 (D) 28 (E) 35 If n = 24×3×5, how many factors does n have? (A) 20 (B) 15 (C) 9 (D) 6 (E) 4 6. S is the sum of integers from 20 to 40, inclusive. How many different prime factors does S have? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

5.

7. When positive integer n is divided by 7, the remainder is 2. What is the value of n? (1) n is less than 20. (2) When n is divided by 8, the remainder is 0. 8. If n is an even integer and k is an odd integer, which of the following CANNOT be an integer? (A) (B) (C) (D) (E)

n k n k 1 n 1 k k n n 2 k 1

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9. What is the sum of the remainders when n positive integers were divided by 2? (1) The sum of n numbers is an even number. (2) Exactly eight numbers are odd. 10. How many positive integers less than 30 have no common prime factor with 30? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 11. Find the greatest common factor of 50 and 370. (A) 5 (B) 10 (C) 185 (D) 1850 (E) 18500 12. The numbers in which of the following pairs do NOT have Đ° pair of distinct prime divisors in common? (A) 10 and 20 (B) 12 and 18 (C) 24 and 32 (D) 21 and 63 (E) 22 and 88 13. The average (arithmetic mean) of five consecutive integers is an odd number. Which of the following must be true? I. The largest of the integers is even. II. The sum of the integers is odd. III. The difference between the largest and smallest of the integers is an even number. (A) I only (B) II only (C) III only (D) I and II (E) II and III 14. (1) (2)

If n is a positive integer between 30 and 60, inclusive, what is the value of n? When n is divided by 4, the remainder is 1. When n is divided by 5, the remainder is 2.

15. (1) (2)

If x denotes the remainder when 5n2 + 1 is divided by 2, what is the value of x? n is an even number. n is divisible by 5.

119

16. When positive integer n is divided by 7, the remainder is 1. If n is less than 50, what is the value of n? (1) When n is divided by 9, the remainder is 7. (2) When n is divided by 4, the remainder is 3.

Answers: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

E E E B A D C D B D B C E E A A

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Lesson  â„–  4.  Algebra  The set of all allowable inputs for a function is called the domain of the function.

Sometimes it is convenient to use letters to denote numbers . This makes algebraic expressions different from arithmetic ones. For example, we can rewrite the expression 3Ă—5 −

3Ă—2+4 5

đ?‘ŽĂ—đ?‘? −

đ?‘ŽĂ—đ?‘?+4 , 5

As

where a 3, b 5, c 2 , which makes it more general. We have learnt a lot about numbers, sets of numbers and their properties in previous lessons. In this paragraph we are going to deal with algebraic forms and methods that are applied in algebraic math problems.

Functions. Substitution. Symbolism. GoGMAT, Session 7 Defined functions are common on the GMAT, and most students struggle with them. Yet once you get used to them, defined functions can be some of the easiest problems on the test. In these problems, you are given a symbol and a mathematical expression or description that defines the symbol. For any algebraic equation you can calculate its value for given variables. This operation is called substitution. For example, substituting x=1 and y=2 into x + 2xy − 3y + 14, we get 1 + 2 ∗ 1 ∗ 2 − 3 ∗ 2 + 14 = 1 + 4 − 24 + 14 = −5. Symbolism is some unknown new operation: đ?‘Ľ#đ?‘Ś  = đ?‘Ľ + 2đ?‘Ś, đ?‘Ľđ?‘‡đ?‘Ś  = đ?‘Ľ − 4đ?‘Ľđ?‘Ś + 5, etc. 1. If ∀ is defined by the expression ∀đ?‘Ž = đ?‘Ž + 2đ?‘Ž , what is the value of a) ∀3 b) ∀đ?‘Ľ c) ∀∀đ?‘Ž 2. If ∅ is defined by the expression đ?‘Ľâˆ…đ?‘Ś = đ?‘Ľ + đ?‘Ľ Ă— đ?‘Ś, what is the value of a) 2∅3 b) 1∅(2∅3) c) (1∅2)∅3 d)

∅

121

Any function is defined for a certain set of numbers, called the domain of the function. For example, the domain of the function f(x) =3x+5 is the set of all real numbers the domain of the function g(a)=√a + 1 is đ?‘Ž ≼ −1, as under-root expression must be nonnegative. The domain of any function can be arbitrarily specified, as in the function defined by “đ?‘“(đ?‘Ľ)  = 4đ?‘Ľ − 1  đ?‘“đ?‘œđ?‘&#x;  0 < đ?‘Ľ < 3â€?.  Without  such  a  restriction,  the  domain is assumed to be all values of x that result in a real number when substituted into the function. To find the domain you need to check the following: For a fraction , its denominator must be nonzero, đ?‘? ≠0. For x , x must be nonzero, đ?‘Ľ ≠0. For √x, x must be nonnegative, đ?‘Ľ ≼ 0. For 0 ,  đ?‘Ľ must be positive, đ?‘Ľ > 0. Sample problems: 3. Find  â™Ľâ™Ľ 2   if   â™Ľđ?‘Ž = 2đ?‘Ž − 3đ?‘Ž + 1. (A) 3 (B) 6 (C) 10 (D) ♼  2   (E) ♼♼  1 4.

(MG) The operation # is defined in the following way for any two numbers: p # q = (p – q)*(q – p) If p # q = –1, then which of the following are true? I. p could equal 5 and q could equal 4 II. p could equal 4 and q could equal 5 III. p could equal 1 and q could equal –1 IV. p could equal –1 and q could equal 1 (A) I and II only (B) I and III only (C) II and IV only (D) III and IV only (E) I, II, III, IV

5.

(MH, E) What is the value of 3đ?‘Ľ  − 1.8đ?‘Ľ  + 0.3 for đ?‘Ľ  = 0.6? (A) -0.3 (B) 0 (C) 0.3 (D) 1.08 (E) 2.46

122

6.

(OG, E) The symbol @ represents one of the following operations: addition, subtraction, multiplication, or division. What is the value of 15@14? (1) 0@1 =1 (2) 1@0 =1

7. (GC, M) If P# = P/(P - 1), what is the value of P##? (A) P/(P - 1) (B) 1/P (C) P (D) 2 - P (E) P - 1

GoGMAT Problem

123

Answers and explanations 1. If ∀đ?‘Ž = đ?‘Ž + 2đ?‘Ž , then

a) ∀3 = 3 + 2 Ă— 3 = 3 + 18 = 21, b) ∀đ?‘Ľ = đ?‘Ľ + 2(đ?‘Ľ ) = đ?‘Ľ + 2đ?‘Ľ , c) ∀∀đ?‘Ž = ∀(đ?‘Ž + 2đ?‘Ž ) = đ?‘Ž + 2đ?‘Ž + 2 Ă— (đ?‘Ž + 2đ?‘Ž ) = đ?‘Ž + 2đ?‘Ž + 2 Ă— (đ?‘Ž + 4đ?‘Ž + +4đ?‘Ž ) = đ?‘Ž + 2đ?‘Ž + 2đ?‘Ž + 8đ?‘Ž + 8đ?‘Ž = đ?‘Ž + 4đ?‘Ž + 8đ?‘Ž + 8đ?‘Ž 2. If đ?‘Ľâˆ…đ?‘Ś = đ?‘Ľ + đ?‘Ľ Ă— đ?‘Ś, then a) 2∅3 = 2 + 2 Ă— 3 = 8 b) First calculate the expression in the brackets: 2∅3 = 8. Now 1∅(2∅3) = 1∅8 = 1 + 1 Ă— 8 = 9. c) Again, first calculate the expression in the brackets: 1∅2 = 1 + 1 Ă— 2 = 3. Now (1∅2)∅3 = 3∅3 = 3 + 3 Ă— 3 = 12. d) 3.

4.

5. 6.

7.

∅ = + Ă— = + 1.

Lets’  find  â™Ľ2.  The  definition  â™Ľđ?‘Ž = 2đ?‘Ž − 3đ?‘Ž + 1 gives us ♼2 = 2 Ă— 2 − 3 Ă— 2 + 1 = 3. Now, ♼♼2 = ♼3 = 2 Ă— 3 − 3 Ă— 3 + 1 = 10. The answer is (C). The best way to solve this is to plug the values into the equation: (5 – 4) * (4 – 5) = –1 (4 – 5) * (5 – 4) = –1 (1 – –1) * (–1 – 1) = –4 (–1 –1) * (1 – –1) = –4 Statements I and II give the stated value, –1. The correct answer is (A). Just substitute x=0.6 in the equation: 3(0.6*0.6) -1.8*0.6 + 0.3 = 1.08 - 1.08 + 0.3 = 0.3. We get (C). Using statement 1 we can see that the only operation giving 1 as result is addition. So, @=+ and 15+14 can be found. Sufficient. In statement 2, both 1+0=1 and 1 – 0=1, so, @ can be both division and addition. Insufficient. The answer is A. At  first  let’s  find  P#. If P# = P/(P - 1)  then  let’s  take  this  value  and  substitute  it  instead  of  P  to  find  out  P##.  Here our function P/(P - 1) becomes an argument. We will get P P P P − 1 = P − 1 = P − 1 = P. P P−P+1 1 − 1 P−1 P−1 P−1 The answer is (C).

.

124

GoGMAT Problem Explanation

125

Simplifying algebraic expressions GoGMAT, Session 5 Expressions like đ?‘Ľ  − 4đ?‘Ľ  + 15 are called polynomials. The degree of polynomial is the highest power of variable. For example, x +  5x − 10  is called a second degree (or quadratic) polynomial in On the GMAT x since the highest power of x is 2. it is often đ?‘Ś − 4 is a first degree (or linear) polynomial in y since the highest necessary to power of y is 1. factor algebraic expressions.

Factor the following polynomials: 1. đ?‘Ľ đ?‘Ś − đ?‘Ľđ?‘Ś 2. đ?‘š đ?‘› − đ?‘› đ?‘š 3. 4 − 36đ?‘Ž 4. (đ?‘Ľ + 1) − 25 5. 3đ?‘š + 3đ?‘› − 6đ?‘šđ?‘› 6. 3đ?‘Ž + 3 − đ?‘›đ?‘Ž − đ?‘› 7. 3đ?‘Ľ(đ?‘Ž + đ?‘?) + đ?‘Ś(đ?‘? + đ?‘Ž)

For example, 7đ?‘Ž  +  5đ?‘Ž = (7  +  5)đ?‘Ž = 12đ?‘Ž. 6đ?‘Ľđ?‘Ś  −  3đ?‘Ś = 3đ?‘Ś(2đ?‘Ľ − 1) 3đ?‘Ľđ?‘Ś − đ?‘Ś + 3đ?‘Ľđ?‘§ − đ?‘§ = = đ?‘Ś(3đ?‘Ľ − 1) + đ?‘§(3đ?‘Ľ − 1) = = (3đ?‘Ľ − 1)(đ?‘Ś + đ?‘§)

Here are some useful rules for simplifying algebraic expressions:

  (đ?‘Ľ + đ?‘Ś)(đ?‘§ + đ?‘¤) = đ?‘Ľđ?‘§ + đ?‘Ľđ?‘¤ + đ?‘Śđ?‘§ + đ?‘Śđ?‘¤ đ?‘Ľđ?‘Ś đ?‘Ś = , đ?‘–đ?‘“  đ?‘Ľ ≠0 đ?‘Ľđ?‘§ đ?‘§ đ?‘Ž − đ?‘? = (đ?‘Ž − đ?‘?)(đ?‘Ž + đ?‘?) (đ?‘Ž + đ?‘?) = đ?‘Ž + 2đ?‘Žđ?‘? + đ?‘?   (đ?‘Ž − đ?‘?) = đ?‘Ž − 2đ?‘Žđ?‘? + đ?‘? Sample problems:

1. (MH) If (đ?‘Ž − đ?‘?) =  64  and đ?‘Žđ?‘?  =  3, find  đ?‘Ž + đ?‘? . (A) 61 (B) 67 (C) 70 (D) 58 (E) 69 2. (PR) What is the value of Â

?

(1) m − n = 40 (2) m + n = 10

126

3. (OG) What is the value of đ?‘Ž − đ?‘? ? (1)  đ?‘Ž − đ?‘? = 16 (2) đ?‘Ž + đ?‘?  = 8

GoGMAT Problem

127

Answers and explanations 1. (a − b) = a −  2ab +  b  which is equal to 64. a −  2ab +  b = 64 and  a + b = 64 + 2ab Since ab = 3, 2ab = 6, and therefore  a + b = 64 + 6, or 70. The correct answer is (C). 2. Here, remembering that the difference of two squares can always be rewritten as the product of the sum and the difference of the two variables is key: đ?‘š − đ?‘›  =  (đ?‘š − đ?‘›) Ă— (đ?‘š + đ?‘›). Then it is easy to see that the piece of the puzzle you are looking for is the value of đ?‘š + đ?‘›. Let’s  look  at  the  statement  1  first.  We  need  to  find  the  value  of  đ?‘š + đ?‘›, but it does not give it to us. It’s  insufficient.  So,  the  answer  will  be  B,  C  or  E. Statement  2  gives  us  the  needed  value,  it’s  sufficient,  so  the  correct  answer  here  is  (B). 3. a -b = (a -b )(  a +b ). Clearly,  (1)  or  (2)  alone  is  not  sufficient  to  answer  the  stem  question,  because  they  don’t  give  us  the  opportunity to find the value of a or b themselves. Therefore, the answer must be (C) or (E). Given (1) and (2) combined, we have a-b =(a -b )/(a + b) = 2. Having this, we can make a system of two linear equations and find both a and b. Thus, the right answer is (C). Check your factorization of polynomials: 1. đ?‘Ľ đ?‘Ś − đ?‘Ľđ?‘Ś = đ?‘Ľđ?‘Ś(đ?‘Ľ − đ?‘Ś) 2. đ?‘š đ?‘› − đ?‘› đ?‘š = đ?‘š đ?‘› (đ?‘š − đ?‘›) 3. 4 − 36đ?‘Ž = (2 − 6đ?‘Ž)(2 + 6đ?‘Ž) 4. (đ?‘Ľ + 1) − 25 = (đ?‘Ľ + 1 − 5)(đ?‘Ľ + 1 + 5) = (đ?‘Ľ − 4)(đ?‘Ľ + 6) 5. 3đ?‘š + 3đ?‘› − 6đ?‘šđ?‘› = 3(đ?‘š − 2đ?‘šđ?‘› + đ?‘› ) = 3(đ?‘š − đ?‘›) 6. 3đ?‘Ľ(đ?‘Ž + đ?‘?) + đ?‘Ś(đ?‘? + đ?‘Ž) = (3đ?‘Ľ + đ?‘Ś)(đ?‘Ž + đ?‘?) 7. 3đ?‘Ž + 3 − đ?‘›đ?‘Ž − đ?‘› = 3(đ?‘Ž + 1) − đ?‘›(đ?‘Ž + 1) = (3 − đ?‘›)(đ?‘Ž + 1)

128

GoGMAT Problem Explanation

129

Linear equations. Systems of linear equations. GoGMAT, Session 5 The main task of algebra is to solve equations involving algebraic expressions. Some examples of such equations are: 3đ?‘Ľ + 5  =  4 − 2đ?‘Ľ - a linear equation with one unknown 2đ?‘Ľ  â€“  đ?‘Ś  = 3đ?‘Ľ + 4đ?‘Ś − 1 - a linear equation with two unknowns |đ?‘Ľ − 1|  = 5 – equation with the absolute value   x + 2y  = 1   2x − y  = 3

- system of linear equations

The solutions of an equation with one or more unknowns are those values that make the equation true,  or  â€œsatisfy  the  equation,â€?  when  they  are  substituted  for  the  unknowns  of  the  equation.  An  equation may have no solution or one or more solutions. If two or more equations are to be solved together, the solutions must satisfy all of the equations simultaneously. To solve a linear equation with one unknown (that is, to find the value of the unknown that satisfies the equation), the unknown should be isolated on one side of the equation. This can be done by performing the same mathematical operations on both sides of the equation. Remember that if the same number is added to or subtracted from both sides of the equation, this does not change the equality; likewise, multiplying or dividing both sides by the same nonzero number does not change the equality. For example,  let’s  solve  such  equation:

3đ?‘Ľ + 1 Â = Â 10 - multiply each side by 2 3đ?‘Ľ Â = Â 9 - subtract 1 from each side đ?‘Ľ = 3 - divide each side by 3. đ?‘Ľ Â = Â 3 is the solution.

(3đ?‘Ľ + 1) Â = Â 5 2

One linear equations with two unknowns, in general, has infinite number of solutions. For example, equation đ?‘Ľ + đ?‘Ś = 2, can be rewritten as đ?‘Ľ = 2 − đ?‘Ś. As y can be any number, we have infinite number of pairs (đ?‘Ľ, đ?‘Ś) that suit this equation. BUT If there is additional information about the variables (they are integers, positive integers, digits, etc.) one equation with two variables can have single solution. For example, if A and B are positive integers and A(B-1) = 1 we can find both A and B. If the product of two positive integers equals 1 then both these digits equal 1 => A = 1 and B-1 = 1, so, B = 2. The answer is A = 1, B = 2. Another example: There is only one pair of positive integers, satisfying the equation 32đ?‘Ľ + 15đ?‘Ś = 109. Variable đ?‘Ś can be expressed in terms of x as đ?‘Ś =

. Since đ?‘Ś must be positive, the only possible values of

130

đ?‘Ľ are 1, 2 and 3. And only for đ?‘Ľ = 2 is the value of đ?‘Ś an integer. Therefore, the single solution is đ?‘Ľ = 2, đ?‘Ś = 3. To solve a system of linear equations of two unknowns you have to remember some rules: Here is a system of two linear equations with two unknowns:   ax + by  = c  dx + ey = f x, y – variables, a, b, c, d, e, f- constants. Such system can have zero, one or infinite number of solutions. 1) The last infinite number case means that the two equations are in reality just one equation written twice. For example, consider the following system of equations: 2 đ?‘Ľ+ đ?‘Ś=5 3 6đ?‘Ľ − 30 = −4đ?‘Ś At first glance the equations look as different as chalk and cheese. But if we rewrite the second equation so that all the variables are on the left and the constant is on the right we might notice that the coefficients at đ?‘Ľ and đ?‘Ś as well as the numbers on the right are proportional: 2 đ?‘Ľ+ đ?‘Ś=5 3 6đ?‘Ľ + 4đ?‘Ś = 30 1 2â „3 5 = = 6 4 30 Therefore, if we divide the second equation by 6 (or multiply the first by 4, or multiply the first by 4 and divide the second by 2, etc.), the two of them will become indistinguishable from one another. 2) The zero number case occurs when the coefficients at the variables in two equations are proportional, but the numbers on the right are not. For example, if we take the same system as before, but change 30 for 31, we would obtain that 2 đ?‘Ľ+ đ?‘Ś=5 3 2 31 đ?‘Ľ+ đ?‘Ś= 3 6 which is impossible. 3) In all other cases, that is when the coefficients at đ?‘Ľ and đ?‘Ś are not proportional, the system of equations has exactly one solution. There are several methods of solving two linear equations with two variables. The first method is to express one of the unknowns in terms of the other using one of the equations, substitute the expression into the remaining equation to obtain an equation with one variable. Example 1:  2x −  3y  = 3 x + y  = 4 Let’s  express y from the second equation: đ?‘Ś  = 4 −  đ?‘Ľ 2đ?‘Ľ −  3(4 − đ?‘Ľ)  = 3  - substitute in the second equation

131

2đ?‘Ľ −  12 +  3đ?‘Ľ  = 3 ⇒ 5đ?‘Ľ  =  15 ⇒ đ?‘Ľ = 3. Now we can find y: đ?‘Ś  = 4  âˆ’ đ?‘Ľ  = 4 −  3  = 1. So, the pair (3, 1) is the solution of this system. The second method is to add first and second equations This method works when coefficients at one of the variables in the equations are opposite to each other, as in Example 2  2x −  3y  = 1 4x + 3y  = 5 (2đ?‘Ľ + 4đ?‘Ľ) + (−3đ?‘Ś + 3đ?‘Ś) = (1 + 5) ⇒ 6đ?‘Ľ = 6 ⇒ đ?‘Ľ = 1 Now, to find y, we can substitute x in any of the initial equations, for example the first one: 2 − 3đ?‘Ś  = 1 ⇒ −3đ?‘Ś  = 1 − 2  = −1. We’ve  got  solution:  1,

.

The third method can be used if we want to find y at first (or the first two are not working). Example 3  2x −  3y  = 1 3x − 4y  = 5 If we express đ?‘Ś in terms of đ?‘Ľ or the other way around would bring about unwanted fractional coefficients, and if we try adding the equations, it would have no positive effect – both variables would still be at large. To solve this system, we have to multiple the first equation by 3, the second by 2 6x − 9y  = 3   6x − 8y  = 10   subtract second equation from the first (6đ?‘Ľ − 6đ?‘Ľ)  + (−9đ?‘Ś + 8đ?‘Ś)  = (3 − 10) −đ?‘Ś  = −7 ⇒ đ?‘Ś =  7. If we want to find x as well, we can substitute this value of y in the first or second equation. 1. (OG, M) x =0.rstu, where r, s, t, u are nonzero digits. What is the value of x? (1) đ?‘&#x;  = 2đ?‘  = 3đ?‘Ą  = 6đ?‘˘ (2) đ?‘&#x;đ?‘  = đ?‘Ąđ?‘˘ 2. ( MH, E) If đ?‘Ľ + đ?‘Ś  = 36, what is the value of xy? (1) đ?‘Ś − đ?‘Ľ = 14 (2) đ?‘Ś  = 2đ?‘Ľ + 3 3. Martha bought several pencils. If each pencil was either a 23-cent pencil or a 21-cent pencil, how many 23-cent pencils did Martha buy? (1) Martha bought a total of 6 pencils. (2) The total value of pencils Martha bought was 130 cent.

132

GoGMAT Problem

133

Answers and explanations 1. To find x we need to find all the digits r, s, t, u. From the first statement we get a system with three equations and four unknowns: r  = 2s    r  = 3t  r  = 6u We have to remember that r, s, t, u are nonzero digits – integers from 1 till 9. Look at the third equation: r =6u. If u =1, r=6. If u =2, r = 12, but it cannot be so, because r is the digit! Clearly, if u=3,4,5,  etc.,  r  won’t  be  a  digit  as  well.  So,  the  only  possible  value  of  u  is  1.  And  we  have  such  solution: r=6, s=3, t=2, u=1. So, x=0.6321. Statement 1 is sufficient. Statement 2 alone is not sufficient. So, the answer is (A). 2. Statement (1) alone is sufficient: add the two equations together and you get 2y = 50, so y = 25 and x = 11. (2) alone is sufficient, because if you plug the equation into the question, you get x + 2x + 3 = 36, or 3x = 33, so x = 11 and y = 25. The answer is D. 3. Statement (1) provides no information about the types of pencils Martha bought. Statement (2) gives us one equation with two variables: 23đ?‘Ľ  +  21đ?‘Ś = 130. And still it is sufficient. Check, that there is the only solution here: (4, 2). The answer is B. GoGMAT Problem Explanation

134

Absolute Value. Equations and Inequalities with Absolute Value. GoGMAT, Session 5 Speaking plainly, the absolute value (or modulus) |đ?‘Ľ| of a real number đ?‘Ľ is đ?‘Ľâ€˛đ?‘ numerical value without regard to its sign. For example, |3| = 3, | − 2| = 2, |0| = 0, | − 3.4| = 3.4.

|-3|= 3 -3

|5|= 5

0

5

Absolute value of đ?‘Ľ is the distance between đ?‘Ľ and 0 on the number axis. 1)

x

x x

if x 0 if x 0

2) The main properties are: |đ?‘Ľ| ≼ 0 √đ?‘Ľ = |đ?‘Ľ| |0| = 0 |−đ?‘Ľ| = |đ?‘Ľ| |đ?‘Ľ| + |đ?‘Ś| ≼ |đ?‘Ľ + đ?‘Ś| How to solve with absolute values: Let’s  consider the following example: |đ?‘Ľ − 2| = 6 This equation means, that the value of đ?‘Ľ − 2 is either 6 or -6: đ?‘Ľâˆ’2=6 đ?‘Ľ − 2 = −6 Therefore, there are two solutions, đ?‘Ľ = 8  and  đ?‘Ľ = −4 There is also a geometrical approach to this problem. The equation |đ?‘Ľ − 2| = 6 means that đ?‘Ľ is 6 units away from 2, so, to find the value of đ?‘Ľ  , we should make 6 steps to the right and 6 steps to the left from 2, 6 6 -4 thus coming to the same two answers.

2

8

135

Inequalities with absolute values. Consider the following examples: |đ?‘Ľ| < 5  â&#x;ş  −5 < đ?‘Ľ < 5

-5

0

5

|đ?‘Ľ| > 5  â&#x;ş  đ?‘Ľ < −5  đ?‘œđ?‘&#x;  đ?‘Ľ > 5 -5

Sample problems:

0

5

1. | − 4|  (| − 20| − |5|) = (A) –100 (B) –60 (C) 60 (D) 75 (E) 100 2. What is the value of x? (1) │đ?‘Ľ + 2│ ≤  4 (2) đ?‘Ľ =  36 3.

-8

4

On the number line, the shaded interval is the graph of which of the following inequalities? (A) |đ?‘Ľ| ≤ 4 (B) |đ?‘Ľ| ≤ 8 (C) |đ?‘Ľ − 2| ≤ 4 (D) |đ?‘Ľ − 2| ≤ 6 (E) |đ?‘Ľ + 2| ≤ 6

4.

(GC) If |2 − đ?‘Ľ| + |đ?‘Ľ − 5|  = 7 , find all the values of x. (A) {0,1,7} (B) {0,7} (C) {1,7} (D) {0,2,4} (E) {7}

136

GoGMAT Problem

Answers and explanations 1. |−4|(|−20| − |5|) = 4 Ă— (20 − 5) = 60. The answer is C. 2. From statement (1) we know, that│đ?‘Ľ + 2│ ≤  4, or −4 ≤ đ?‘Ľ + 2 ≤  4,  −6 ≤ đ?‘Ľ ≤  2. This information is not sufficient to determine x. Statement (2) provides us with the information, that x is either 6 or -6, which is still not sufficient. Considering both statements together we obtain, that đ?‘Ľ = −6. The answer is C. 3. First find the middle number of the line segment: = −2. The distance between -2 and the ends of the segment is 6, which is expressed in the inequality |đ?‘Ľ − (−2)| ≤ 6, đ?‘œđ?‘&#x;  |đ?‘Ľ + 2| ≤ 6. The answer is E. 4. One of the ways to solve the problem is to consider all the possibilities: when the expressions (2 − đ?‘Ľ)  and (đ?‘Ľ − 5) are both positive, both negative or have different signs. But as you have the answer choices, it will be much easier to plug them in. If đ?‘Ľ = 0, then |2 − 0| + |0 − 5| = 7đ?‘œđ?‘&#x;  |2| + |−5| = 7, which is true. The answer is A, B or D.

If đ?‘Ľ = 7, then |2 − 7| + |7 − 5| = 7đ?‘œđ?‘&#x;  |−5| + |2| = 7. This is also true. The answer is A or B. If đ?‘Ľ = 1, then |2 − 1| + |1 − 5| = 7đ?‘œđ?‘&#x;  |1| + |−4| ≠7. The answer is B.

137

GoGMAT Problem Explanation

138

Quadratic Equations GoGMAT, Session 6

The general form of the quadratic equation is: đ??šđ??ą đ?&#x;? + đ??›đ??ą + đ??œ = đ?&#x;Ž, where a, b, and c are real numbers, and đ?‘Ž ≠0.

For example: x + 6x + 5 = 0 2x − 7x + 19 = 0 x − 10x = 0 Quadratic equation may have one, two or no solutions. Number of solutions is essential to solve data sufficiency problems.

Two ways to solve the quadratic equation: The universal rule for solving quadratic equations is:

The Discriminant đ?‘Žđ?‘Ľ + đ?‘?đ?‘Ľ + đ?‘Ľ = 0 đ??ˇ = đ?‘?  − 4đ?‘Žđ?‘? −đ?‘? Âą √đ??ˇ đ?‘Ľ, = 2đ?‘Ž

The number D, called the discriminant can tell you, how many roots an equation has: If đ??ˇ < 0, there is no solution. If đ??ˇ Â = 0, there is only one root. If đ??ˇ > 0, there are two roots

Most quadratic equations invite you to use Viet method, which is applicable to equation with a=1: The Viet Theorem Practice: The Viet theorem 1. đ?‘Ľ + 3đ?‘Ľ + 2 = 0 7. đ?‘Ľ + 8đ?‘Ľ − 9 = 0 đ?‘Ľ + đ?‘?đ?‘Ľ + đ?‘? = 0 2. đ?‘Ľ − 15đ?‘Ľ + 14 = 0 8. đ?‘Ľ + 9đ?‘Ľ + 20 = 0 đ?‘Ľ + đ?‘Ľ = −đ?‘? 3. đ?‘Ľ − 19đ?‘Ľ + 18 = 0 9. đ?‘Ľ − 15đ?‘Ľ + 36 = 0 đ?‘Ľ Ă—đ?‘Ľ =đ?‘? 4. đ?‘Ľ + 8đ?‘Ľ + 7 = 0 10. đ?‘Ľ + 5đ?‘Ľ − 14 = 0 5. đ?‘Ľ + 3đ?‘Ľ − 4 = 0 6. đ?‘Ľ − 12đ?‘Ľ − 13 = 0

11. đ?‘Ľ − 7đ?‘Ľ − 30 = 0

Remember, that roots of quadratic equation help to simplify the expression: đ?‘Žđ?‘Ľ + đ?‘?đ?‘Ľ + đ?‘? = đ?‘Ž(đ?‘Ľ − đ?‘Ľ )(đ?‘Ľ − đ?‘Ľ ) Factoring Practice: 1. 2. 3. 4. 5. 6. 7. 8.

đ?‘Ľ − 11đ?‘Ľ + 24 đ?‘Ľ − 8đ?‘Ľ + 15 đ?‘Ľ + 7đ?‘Ľ + 12 đ?‘Ľ + 3đ?‘Ľ − 10 – đ?‘Ľ + 16đ?‘Ľ − 15 – đ?‘Ľ − 8đ?‘Ľ + 9 – đ?‘Ľ + 5đ?‘Ľ − 6 – đ?‘Ľ + 7đ?‘Ľ − 12

139

Geometric interpretation. The graph of the quadratic function đ?‘Ś = đ?‘Žđ?‘Ľ + đ?‘?đ?‘Ľ + đ?‘? is a parabola. The direction of the branches depends on the sign of a: If đ?‘Ž > 0, the branches are in upward direction If đ?‘Ž < 0, the branches are in downward direction

Now determine the points of intersection with axis (y-intercept and x-intercept). Coefficient c is y-intercept. Consider the following examples:

Roots of the equation are the x-intercepts. For example, consider the two functions and find roots of respective equations: đ?‘“ (đ?‘Ľ) = đ?‘Ľ − 9đ?‘Ľ + 14, đ?‘Ľ = 7, đ?‘Ľ = 2;   đ?‘“ (đ?‘Ľ) = đ?‘Ľ − 3đ?‘Ľ − 10, đ?‘Ľ =  −2;  đ?‘Ľ = 5. Here are their graphs. Pay attention to the x-intercepts.

140

Finally,  let’s  learn  to  find  the  vertex  of  the  parabola.  The  vertex  is  the  minimum  if  the branches are upwards and maximum if they are downwards. The x-coordinate of the vertex is defined by the formula: đ?‘Ľ =

.

To find the y-coordinate simply plug đ?‘Ľ into the function: đ?‘“ = đ?‘“(đ?‘Ľ ). Example: find the vertexes of the parabolas đ?‘“(đ?‘Ľ) =  −  2đ?‘Ľ + 4đ?‘Ľ + 2, đ?‘“(đ?‘Ľ) = 2đ?‘Ľ − 8đ?‘Ľ + 3 Check yourself with the graph:

1. In the xy-plane, a parabola intersects with y-axis at point (0, y). Is y < 0? (1) The vertex of parabola is (2,-5). (2) The parabola intersects with x-axis at points (-2, 0) and (6, 0). 2. (OG, E) x > 2  đ?‘Žđ?‘›đ?‘‘  đ?‘Ľ − 4x + 3 = 0. Find x. (A) 3 (B) 4 (C) 1 (D) 2 (E) 5

141

3. (GC, M) What is X? (1) đ?‘‹  +  24  =  8  Ă—  (đ?‘‹  +  1) (2) đ?‘‹ −  6đ?‘‹  + 5  = 0 4. (OG, M) If  đ?‘Ą − 8 is a factor of đ?‘Ą  âˆ’ đ?‘˜đ?‘Ą  âˆ’ 48, then đ?‘˜  = (A) -6 (B) -2 (C) 2 (D) 6 (E) 14

GoGMAT Problem

142

Answers and explanations 1. The 1st statement is not sufficient, as the vertex tells us nothing about the interceptions with y-axis. The 2nd statement leaves 2 variants:

So  it’s  not  sufficient. Both statements together provide us with the vertex and the x-intercepts, which uniquely define the parabola. The answer is C. 2. Let’s  solve  this  equation  with  the  Discriminant. D=  4Ă—4  -4Ă—3Ă—1=16-12=4>0, 2 roots. đ?‘Ľ = đ?‘Ľ =

√ Ă— √ Ă—

= = 1, or = = 3.

X has to be greater then 2, so, x=3. The answer is A. Alternative solution: Viet theorem. Ă— +Ă— =4 Ă— Ă— =3. The roots are 1 and 3, because their sum = 4 and their product =3. The answer is A. 3. Statement (1) simplifies to X2 - 8X +16 =0 or (X - 4)2 = 0 from where X = 4. Sufficient. Statement (2) gives us X2 - 6X +5 =0 or X =5, X=1. We cannot find the only value of X. Insufficient. Answer: (A). 4. If t-8 is a factor of t - kt -48, then this polynomial can be expressed as t - kt -48 = (t-8)Ă—(t-a), where a is some number. According to Viet theorem, 8 and a will be roots of the equation t - kt - 48=0 Thus, substituting t=8 in t -kt -48,  we’ll  get  0: 64 -8k-48=0 8k =16, k=2. Answer (C).

143

The Viet theorem practice: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

-2;-1 1;14 1;18 -7;-1 -4;1 -1;13 -9;1 -5;-4 3;12 -7;2 -3;10

Factoring practice: 1. 2. 3. 4. 5. 6. 7. 8.

ЁЭСе тИТ 11ЁЭСе + 24 = (ЁЭСе тИТ 3)(ЁЭСе тИТ 8) ЁЭСе тИТ 8ЁЭСе + 15 = (ЁЭСе тИТ 3)(ЁЭСе тИТ 5) ЁЭСе + 7ЁЭСе + 12 = (ЁЭСе + 4)(ЁЭСе + 3) ЁЭСе + 3ЁЭСе тИТ 10 = (ЁЭСе + 5)(ЁЭСе тИТ 2) тАУ ЁЭСе + 16ЁЭСе тИТ 15 = тИТ(ЁЭСе тИТ 1)(ЁЭСе тИТ 15) тАУ ЁЭСе тИТ 8ЁЭСе + 9 = тИТ(ЁЭСе + 9)(ЁЭСе тИТ 1) тАУ ЁЭСе + 5ЁЭСе тИТ 6 = тИТ(ЁЭСе тИТ 2)(ЁЭСе тИТ 3) тАУ ЁЭСе + 7ЁЭСе тИТ 12 = тИТ(ЁЭСе тИТ 3)(ЁЭСе тИТ 4)

GoGMAT Problem Explanation

144

Exponential equations GoGMAT, Session 6 The general form of an exponential equation with equal bases is:

Be careful when seeing a variable in the base of an

đ?‘Ž =đ?‘Ž Consider the following cases: 1) If đ?‘Ž = 1, then đ?‘Ľ and đ?‘Ś can take on any values. 2) If đ?‘Ž = 0, then đ?‘Ľ and đ?‘Ś are any positive numbers 3) If đ?‘Ž ≠1  and  đ?‘Ž ≠0, then đ?‘Ž =đ?‘Ž â&#x;şđ?‘Ľ=đ?‘Ś

exponent in DS problems – there  might  be  a  â€œđ?‘Ž = 1â€?   or  â€œđ?‘Ž = 0â€?  trap!

4) Equation đ?‘Ž = đ?‘? , where đ?‘Ž ≠đ?‘?, đ?‘Ľ and đ?‘Ś are integers, has I. Single solution đ?‘Ľ = đ?‘Ś = 0 if đ?‘Ž is not an integer power of đ?‘? For example, equation (−3) = 217 gives us 2đ?‘š − 1 = 15đ?‘Ľ = 0, or đ?‘š = 1/2 and đ?‘Ľ = 0. II. Infinite number of solution, if đ?‘Ž = đ?‘? for some integer đ?‘›. For example, equation 3 = 9 gives us đ?‘Ś = 30đ?‘Ľ. 1. Consider 2

= đ?‘› + 1 where n is an integer number. What is a value of m?

(1) 3

= 5  , where k is an integer. (2)  đ?‘› − đ?‘š ≤ 1

2. (OG, M) If (−2) (A)1 (B)2 (C)3 (D)4 (E)6

=2

, m –integer, then m=

3. (OG, M) If (7 ) =7, then n= (A)1/3 (B)2/3 (C)4/3 (D)5/3 (E)6/3 4. If đ?‘Ž = đ?‘? , where đ?‘Ž, đ?‘?, đ?‘Ľ and đ?‘Ś are positive integers, is x divisible by y? (1) đ?‘Ž < đ?‘? (2) đ?‘Ś is an odd number

145

GoGMAT Problem

146

Answers and explanations 1. Let’s  take  Statement  (1).  The  power  of  number  3  could  be  equal  to  the  power of number 5 only in one case: when 5n = 3k = 0. So,  2 = n + 1 = 1, and 3m  =  0. Statement (1) is sufficient to answer the question. Statement (2) is not insufficient because for n=0 we obtain m=0, but for n=1 m=1/3. The answer is (A). 2. At first look at (−2) . 2m is an even integer, so, (−2) =2 : 2 =2 , equalizing the powers of left and right sides we get 2m =9-m, 3m =9, m=3. The answer is (C). 3. (7 ) = 7 , we get: 7 = 7 , so, 3n/4 =1, 3n =4, n=4/3. Answer (C). 4. Statement (1) leaves two options: either đ?‘? = đ?‘Ž for some positive integer đ?‘›, in which case đ?‘Ľ = đ?‘›đ?‘Ś, or both đ?‘Ľ = đ?‘Ś = 0. In the latter case đ?‘Ľ is not divisible by đ?‘Ś, so, statement (1) is not sufficient. Statement (2), when considered together with (1) excludes the possibility of đ?‘Ľ = đ?‘Ś = 0, therefore đ?‘Ľ = đ?‘›đ?‘Ś ≠0 and đ?‘Ľ is divisible by đ?‘Ś. The answer is C. GoGMAT Problem Explanation

147

Inequalities. Exponential inequalities. GoGMAT, Session 7 An inequality is a statement that uses one of the following symbols: ≠ not equal to; > greater than; ≥ greater than or equal to;; < less than; ≤ less than or equal to. Some examples of inequalities are: 3x - 4y + 5 < 1 - 7x, y ≥ 3, 4a – b ≤ 0, etc. Very often in the problem you will need to find not the solution of the inequality but the interval that contains this solution. For example, for x > 3 the answer on the question “What must be true for all values of x?” could be x ≥ 1. Solving a linear inequality with one unknown is similar to solving an equation; the unknown is isolated on one side of the inequality. As in solving an equation, the same number can be added to or subtracted from both sides of the inequality, or both sides of an inequality can be multiplied or divided by a positive number without changing the truth of the inequality. However, multiplying or dividing an inequality by a negative number reverses the order of the inequality. For example, 6 > 2, but (−1)×6 <(−1)×2.

1) 2) 3) 4) 5) 6) 7)

Such rules are useful: If a > b, for any c a+c > b+c if a > b and c > 0 then ac > bc if a > b and c < 0 then ac < bc if a > b and b > c then a >c if a > b and c > d then a+c > b+d if a, b, c, d > 0 and a > b and c > d then ac > bd. if a > b > 0 then 1/a < 1/b 1. If d > a and b < a, which of the following CANNOT be true? (A) d + b = 14 (B) d – b = 7 (C) d – b = 1 (D) a – d = 9 (E) a + d = 9 2. If 0< ab < ac, is a negative? (1) c < 0 (2) b > c

148

4. Is |đ?‘Ľ − 1| < 3? a) (1) |đ?‘Ľ + 1| > 4        (2)  |đ?‘Ľ − 2| > 7;

b) (1) |đ?‘Ľ − 2| < 1.5  (2) |đ?‘Ľ − 3| < 4;

c) (1) |đ?‘Ľ + 1| < 3 (2) |đ?‘Ľ + 2| > 3

Solving an exponential inequality is very alike with solving an exponential equation: to solve an exponential inequality with the same bases of power: a <a look at the base a. If it is in the interval (0 ,1), write down the same inequality with the powers of left and right sides, but change the sign: đ?‘Ľ > đ?‘Ś. If it is in the interval (1,  +∞),  just  write  down  the  same  inequality  with  the  powers  of  left and right sides: đ?‘Ľ < đ?‘Ś. 5. (MG, E) If 4 – x > 5, then (A) x > 1 (B) x > –1 (C) x < 1 (D) x < –1 (E) x = –1 6. (GC, M) If x2 - 100 < 300, how many integers x satisfy this condition? (A) 42 (B) 39 (C) 38 (D) 37 (E) 19 7.

(PR, M) Is mn <m/n? (1) mn is a positive integer. (2) n is a negative number.

8.

(OG, M) What is the smallest integer n for which 25 > 5 ? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 9. Is � > � ? (1)  � < 1 (2) � ≠0

149

GoGMAT Problem

150

Answers and explanations 1. If d > a then 0 > a – d. So, (D) is FALSE. 2. While c < 0 and ac > 0 the only possible for a is a < 0. Statement (1) is sufficient. If b>c then b-c>0. If ac>ab than ab-ac>0. So, a(b-c)>0. While b-c >0 the only possible for a is a>0. The answer is (D) 3. The  question  can  be  rephrased  as  â€œis  đ?‘Ľ between -2 and 4?â€?. a) Statement (1) gives that đ?‘Ľ  is either greater than 3 or less than -5. Which is not sufficient, for it allows both situations: when đ?‘Ľ belongs to the interval in question (for example, đ?‘Ľ = 3.5) and when it does not (for example, đ?‘Ľ = −7) -5

-2

3

4

Statement (2) means that đ?‘Ľ is either greater than 9 or less than -5. None of the values from these two regions belong to the interval (−2, 4), therefore, (2) is sufficient. The answer is (B). -5

-2

4

9

b) Statement (1) means that đ?‘Ľ  is between 0.5 and 3.5. Which is sufficient, for any đ?‘Ľ from the interval (0.5, 3.5) belongs to the containing interval(−2,4). So, (1) is sufficient.

-2

0.5

3.5

4

Statement (2) means that đ?‘Ľ is between -1 and 7, which is not sufficient to determine whether đ?‘Ľ belongs to the interval (−2,4). -2

-1

4

7

So, the answer is (A). c) Statement (1) means that đ?‘Ľ is between -4 and 2, which is not sufficient to determine whether đ?‘Ľ belongs to the interval (−2,4). -4

-2

2

4

Similarly, statement (2), giving that đ?‘Ľ is either smaller than -5 or larger than 1, is not sufficient. -5

-2

1

4

9

But two statements together give that đ?‘Ľ is  between  1  and  2,  thus,  the  answer  is  Š. -2

1

2

4

4. 4 – x > 5 –x > 1 Divide by –1 and change the inequality sign: x < –1 The correct answer is (D).

151

5. First of all the inequality we have to solve is đ?‘Ľ < 400 or |đ?‘Ľ| < √400 . Then −20 < đ?‘Ľ < 20. This gives  us  19  Ă—  2  +  1  =  39  integers  (we  have  to  add  1  for  zero).  Answer  (B). 6. Statement 1 is not sufficient, because we can take m=-5, n=-6  and  we  will  get  30>  5/6,  answer  â€œnoâ€?.  Or we can take m=-50, n=-0.1  and  we  will  get  5<500,  answer  â€œyesâ€?. Statement 2 does even less, since we do not know something about m. Together, the statements still do not tell us what pair to take: m=-5, n=-6 or m=-50, n=-0.1, so the correct answer is (E). 7. 25 = (5 ) = 5 . We get 5 > 5 , so, we can write down the same inequality with the powers of left and right sides: 2n>12, n>6. The smallest integer n that suits this inequality is n=7. Correct answer is (B). 8. Statement  (1)  is  â€œnearly  sufficientâ€?  â€“ for any value of đ?‘Ľ which is less than 1 and not zero it is true that đ?‘Ľ > đ?‘Ľ . But for đ?‘Ľ = 0 it is not, therefore (1) is not sufficient. Statement (2) excludes the unpleasant exception. The answer is (C).

GoGMAT Problem Explanation

152

Home assignment. 1. (OG E) If x = -1, then -(x4 + x3 + x2 + x) = (A) -10 (B) -4 (C) 0 (D) 4 (E) 10 2. (GC M) What is 987 × 987? (A) 974,169 (B) 974,219 (C) 974,549 (D) 975,019 (E) 975,369 3. (OG E) Which of the following equations is NOT equivalent to 25x2 = y2 - 4? (A) (B) (C) (D) (E)

25x2 + 4= y2 75x2=3y2-12 25x2=(y-2)(y+2) 5x=y-2 x2=(y2-4)/25

4. (OG M) If x ≠0, what is the value of (xp/xq)4? (1) p = q (2) x = 3 5. (MG M) The square root of 800 is between which of the following integers? (A) 21 and 25 (B) 24 and 28 (C) 27 and 31 (D) 30 and 34 (E) 33 and 37 6. (MG E) If point X is on line segment AB, all of the following may be true EXCEPT (A) AX = XB (B) AX > XB (C) AX < XB (D) AB > XB (E) AX + XB < AB 7. (OG M) If m/n = 5/3, what is the value of m + n? (1) m > 0 (2) 2m + n = 26

153

8. (OG E) If b < 2, and 2x - 3b = 0, which of the following must be true? (A) x >-3 (B) x > 2 (C) x =3 (D) x<3 (E) x >3 9. (OG E) If 4x + 3y = -2 and 3x + 6 = 0, what is the value of y? (A) -10/3 (B) -2 (C) -2/3 (D) 2/3 (E) 2 10. (OG E) If x*y = xy- 2(x + y) for all integers x and y, then 2*(-3) = (A) -16 (B) -11 (C) -4 (D) 4 (E) 16 11. (OG E) If a, b, and c are nonzero numbers and a + b = c, which of the following is equal to 1? (A) (a-b)/c (B) (a-c)/b (C) (b-c)/a (D) (b-a)/c (E) (c-b)/a 12. (MG M) If x*y =(4x – 3y)(x+y) then 5*6= (A) 1 (B) 11 (C) 15 (D) 22 (E) 34 13. (MH) What is the value of the greater of two numbers if one of the numbers is three times the smaller number? (1) One of the numbers is 12. (2) The sum of the two numbers is 16. 14. (MH H) Of the following, which is most nearly equal to but not greater than √15? (A) 3.7 (B) 3.8 (C) 3.9 (D) 4.0 (E) 4.1

154

15. (OG H)If x increases from 165 to 166, which of the following must increase? I. 2x- 5 II. 1 - 1/x III. 1/(x2 - x) (A) I only (B) III only (C) I and II (D) I and III (E) II and III 16. (GC H) A perfect number is defined as one for which the sum of all the unique factors less the number itself equals to the number. For instance, 6 is a perfect number, because the factors of 6 (apart from 6 itself) are 1, 2 and 3, and 1 + 2 + 3 = 6. Which of the following is a perfect number? (A) 12 (B) 28 (C) 15 (D) 13 (E) 67 17. (MG H) Every letter in the alphabet has a number value that is equal to its place in the alphabet; thus, the letter A has a value of 1 and C a value of 3. The number value of a word is obtained by adding up the value of the letters in the word and then multiplying that sum by the length of the word. The word “DFGH” would have a number value of (A) 22 (B) 44 (C) 66 (D) 100 (E) 108 18. (MG M) A perfect number is one which is equal to the sum of all its positive factors that are less than the number itself. Which of the following is a perfect number? (A) 1 (B) 4 (C) 6 (D) 8 (E) 10 19. (OG E) If x and y are different integers and x2 = xy, which of the following must be true? I. x=0 II. y=0 III. x=-y (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III

155

20. (GC H) If x is a number such that -2  â‰¤  x  â‰¤  2,  which of the following has the greatest possible absolute value? (A) 3x - 1 (B) x2 + 1 (C) 3 - x (D) x - 3 (E) x2 - x 21. Is  x2  â€“  5x  +  6  =  |  x  â€“  2  |  Ă—  |  x  â€“  3  |? (1) (2)

x > 2. x > 3.

22. (OG M) If °  represents one of the operations +, -, and Ă—, is k°(n  + m) = (k°n)  + (k°m)  for all numbers k, n,and m? (1) k°n  is not equal to n°k  for some n numbers k (2) °  represents subtraction 23. (OG M) If the operation °  is defined for all a and b by the equation a °  b = a2b/3, then 2 °  (3 °  -1) = (A) 4 (B) 2 (C) -4/3 (D) -2 (E) -4 24. (GC M) Is integer R positive? (1) R3 = R (2) |R| = R 25. (OG M) Is x = 0? (1)  x  Ă—  y  =  x (2) x + y = y 26. (GC M) If x is a positive integer, is √đ?‘Ľ > 2.5x - 5? (1) x < 2 (2) x is a prime number 27. (GC H) How many values of x satisfy this equation? A  Ă—  x  +  A  =  B (1) A does not equal 0 (2) B does not equal 0 28. (GC E) What is X? (1) X2 - 1 = X + 1 (2) X + 3 is a prime number

156

29. (GC M) Is 4t3 - 2t2 - 8t + 16 divisible by t2? (1) t > 1 (2) t is an even prime number 30. (GC E) If 3  +  √đ?‘Ľ  −  1 = 4, what is the value of x? (A) 101 (B) 122 (C) 157 (D) 170 (E) 197 31. (GC M) If a, b, and c are positive and a2 + c2 = 202, what is the value of b - a - c? (1) b2 + c2 = 225 (2) a2 + b2 = 265 32.

(MG M) The trinomial x + x – 20 is exactly divisible by (A) x – 5 (B) x + 4 (C) x – 10 (D) x – 4 (E) x – 2 33. Is đ?‘Žđ?‘? = 1? (1) đ?‘Žđ?‘?đ?‘Ž = đ?‘Ž (2) đ?‘?đ?‘Žđ?‘? = đ?‘? 34. (GC M) What is the value of the following expression: 6x2 + 9y2? (1) x = 2 (2) 6y2 + 4x2 = 22 35. (GC M) If x2 - 100 < 300, how many integers x satisfy this condition? (A) 42 (B) 39 (C) 38 (D) 37 (E) 19 36. (GC M) Is (X3)  ×  (Y2)  ×  (Z2) > 0? (1)  X  ×  Y  >  0 (2)  X  ×  Z  >  0 37. (GC H) X, Y, and Z are positive integers. Is (X-Y)  ×  (Y-Z)  ×  (X-Z) > 0? (1) X2 +  Y  ×  Z  =  X  ×  Y  +  X  ×  Z (2)  X  ×  Y  - Y2 =  X  ×  Z  - Y  ×  Z

157

38. (MG M) If x < y, 2x = A, and 2y = B, then (A) (B) (C) (D) (E)

A=B A<B A>B A<x B<y

39. (MG M) If x and y are positive integers such that (x + y) < 10, then which of the following must be true? (A) x < 8 (B) x > 3 (C) x > y (D) x + y = 5 (E) x – y ≤ 7 40. If xy + z = z, is |x-y|>0? (1) x ≠ 0 (2) y = 0 41. (PR M) If –2 < a < 11 and 3 < b < 12, then which of the following is NOT true? (A) 1 < a + b < 23 (B) -14 < a – b < 8 (C) -7 < b – a < 14 (D) 1 < b + a < 23 (E) -24 < ab < 132

42. (OG H) Is

Ă— (đ?‘š  +  đ?‘›  +  đ?‘˜ )  =  đ?‘Ľđ?‘š  +  đ?‘Śđ?‘›  +  đ?‘§đ?‘˜?

(1) = (2)

=

43. Is A positive? (1) đ?‘Ľ − 2đ?‘Ľ + đ??´ is positive for all x. (2) đ??´đ?‘Ľ + 1 is positive for all x. 44. (MH M) Is y between 1 and 2, exclusive? (1) y   is more than y (2) y is between 1 and 2, exclusive 45. (GC H) Which of the following describes value of x2 - x when x is between 0 and 1? (A) 1 > x2 - x > 0 (B) 1 > x2 - x ≼  -1/4 (C) 0 > x2 - x ≼  -1/4 (D) 1 > x2 - x > -1 (E) 0 > x2 - x ≼ -1

158

46. (GC M) Is (đ?‘Ľ − 5) = 5 − đ?‘Ľ? (1) −đ?‘Ľ|đ?‘Ľ|  >  0 (2) 5  âˆ’  đ?‘Ľ  >  0 47. (GC M) $Y$ = (YY)/(Y  Ă—  Y  Ă—  2).  What  is  the  last  digit of $($4$)$? (A) 8 (B) 6 (C) 4 (D) 2 (E) 1 48. If m, n, Ń€, and q are positive integers, and mŃ€ = nq, what is value of pq? (1) m2 = n (2) pp = 4. 49. (MH H) If 5a = 9b = 15c, what is the value of a + b + c? (1) 3c – a = 5c – 3b (2) 6cb = 10a 50. (GC H) What is (đ?‘Ľ  âˆ’  6  Ă—  đ?‘Ľ  +  9)  +  âˆš2  âˆ’  đ?‘Ľ  +  (đ?‘Ľ  âˆ’  3) if each term in this expression is well defined? (A)

(2  −  đ?‘Ľ)

(B) 2  ×  đ?‘Ľ  −  6  +  √2  −  đ?‘Ľ  (C) √2  −  đ?‘Ľ  +  (đ?‘Ľ  −  3) (D)  2  ×  đ?‘Ľ  −  6  +  √đ?‘Ľ  −  2 (E) đ?‘Ľ  +  √đ?‘Ľ  −  2 # of question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Topic of question (type) Your answer ALGEBRA

Correct Answer C A (LD) D A C E B D E C E D B B C B

159

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

D C A A B D E E B A A E B D C D E B B E D B E A C C A E C D D E E A

160

Test â„–  4. Algebra. 1. If x < y, which of the following must be true? I. |đ?‘Ľ|  < |đ?‘Ś|. II. x – y < 0. III. x2 – y2 < 0. (A) I only (B) II only (C) III only (D) I and II only (E) II and III only 2. If x and Ńƒ are integers, what is the value of (x – y)4? (1) The product of x and Ńƒ is 7. (2) The sum of x and Ńƒ is – 8. 3. If x2 + 12x + 20 < 0, how many integer values can x be? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 4. If x2 – y2 < 0, which of the following must be true? I. x + y < 0 II. x – y < 0 III. x2 – y3 < 0 (A) None (B) I only (C) II only (D) III only (E) I and III only 5. If Ńƒ is a positive integer and x is a negative integer, is yx < l? (1) Ńƒ  is greater than 1. (2) x > – 5. 6. If 0 < đ?‘› < đ?‘š < 1, đ?‘š − đ?‘› must be less than which of the following expressions? If x and Ńƒ are x

positive, is y greater than 1? (1) xy > 1 (2) x – Ńƒ > 0.

161

8. Is x = y? (1) x2 + y2 = 2xy. (2) 4

= 1.

9. If the product of x and у is – 15, what is the sum of x and y? (1) xy2 = – 75. (2) x – y = – 8. 10. How many positive integers, from 2 to 100, inclusive, are not divisible by odd integers greater than 1? (A) 5 (B) 6 (C) 8 (D) 10 (E) 50 11. On four successive days, a farmer picks exactly twice as many apples each day as on the previous day. If in the course of the four days he picks a total of 12,000 apples, how many apples does he pick on the second of the four days? (A) 800 (B) 1,000 (C) 1,600 (D) 2,000 (E) 6,000 12. If

for any none-zero integer N, then, N* – (N – l)*=

(A) (B)

1 N

2

(C) (D)

1 N(N 1)

(E)

1 N 1

13. If equation ах2 + bх + с = 0 have two distinct roots, which of the following must be true? I. b>0 II. ac > 0 III. ac < 0 (A) None (B) I only (C) II only (D) III only (E) I and II only

162

14. The function f is defined for any integer n that f(n) is the product of all positive even numbers less than or equal to n, for example, f(10) = 2×4×6×8×10. So, what is the greatest prime factor of f(24)? (A) (B) (C) (D) (E)

3 5 7 11 13

15. If 0 < x < l, which of the following is the greatest? (A) x1/2 (B) x + 1 (C) 2x (D) x2 (E) x2 + 1 16. Which of the following CANNOT be the sum of two prime numbers? (A) 19 (B) 45 (C) 68 (D) 79 (E) 88 17. Is x divisible by y? (1) x is multiple of 6. (2) у is multiple of 3. x2

18. If x is not equal to 1, is x 1 x ? (1) x > 0. (2) x < 2. 19. If n is an integer, is 30 the factor of n? n

(1) 45 is a factor of 2 . (2) 30 is a factor of n2. 20. If x is a positive integer, what is the value of x? (1) x3 = 4x. (2) x2 – 4x = – 4.

163

21. If m and n are integers, is km + n > 0? (1) k < 0. (2) km – n < 0. 22. If xy2 > 0, which of the following must be true? I. x>0 II. y>0 III. yx2 > 0 (A) I only (B) II only (C) III only (D) I and II only (E) I and III only 23. If x and у are none negative integer, what is the value of xy? (1) 7x = 13y. (2) 3x = 9y. Answers: 1. B 2. A 3. B 4. A 5. A 6. D 7. B 8. D 9. A 10. B 11. C 12. A 13. A 14. D 15. B 16. D 17. E 18. E 19. D 20. D 21. B 22. A 23. A

164

Lesson № 5. Word Problems Many problems on GMAT contain a lot of text. Such tasks are called Word Problems. There are several types of them: Word translations, Percentage Problems, Overlapping Sets problems, Rate Problems, Mixture Problems, Work Problems. Note some general tips for solving word problems on GMAT; 1. Identify what kind of word problem you're up against. 2. Translate English into algebra as you read the problem for the first time, sentence by sentence. You cannot afford reading the problem twice. 3. Use convenient variables, which remind you of what they denote, for example, h for hardback books, p for paperback. 4. Draw a figure or a table. A nice visualization can make the solution much quicker. 5. Read the question carefully. The GMAT problems makers love to trick those who do not pay careful attention to what is being asked, and provide you with all the answers you can obtain as a result of such carelessness. On the first steps of your preparation the following common translations may be helpful: Verbal expressions Equals, is, was, will has, be, costs, the same amount, adds up to Times, of, product of, twice as much as, thrice, double, triple, half Divided by, per, out of, each, ratio of Plus, added to, sum, combined, and, more than, a total of, increased by Minus, subtracted from, less than, decreased by, difference between What, how much, how many

Arithmetic symbols = × ÷ + A variable

Now let’s have a close look to each type of Word Problems.

165

Word translations GoGMAT, Session 8 At first we will talk about Word Problems that don’t use special formulae. These problems can take a big part of your test time, but their solution is not so hard. You just have to write down and to solve the system of linear equations or one linear/quadratic equation. 1. (GC) Aunt Marge is giving candy to each of her nephews and nieces. She has 20 candy pieces and she gives candy to children according to her wish. Thus, Robert gets 2 more candy pieces than Kate. Bill gets 6 less than Mary. Mary, in turn, gets 2 more candy than Robert. Kate gets 2 more candy than Bill. How many candy pieces does Kate get? (A) 2 (B) 4 (C) 6 (D) 8 (E) 10 2. (OG) Bill can buy 3 pairs of jeans and 2 shirts for a price of $69 or he can buy 2 pairs of jeans and 3 shirts for the price of $66. How much does one shirt cost? (A) $10 (B) $12 (C) $13.20 (D) $15 (E) $16.80 3. (OG) Jan lives x floors above the ground floor of a high-rise building. It takes her 30 seconds per floor to walk down and 2 seconds per floor to ride the elevator. If it takes Jan the same amount of time to walk down the steps to the ground floor as to wait for the elevator for 7 minutes and ride down, then x equals (A) 10 (B) 12 (C) 13 (D) 14 (E) 15 4. (OG) Salesperson A's compensation for any week is $360 plus 6 percent of the portion of A's total sales above $1000 for that week. Salesperson B's compensation for any week is 8 percent of B's total sales for that week. For what amount of total weekly sales would both salespeople earn the same compensation? (A) $21,000 (B) $18,000 (C) $15,000 (D) $4,500 (E) $4,000

166

5. The number n of units of its product that Company X is scheduled to produce in month t of its next fiscal year is given by the formula 900 đ?‘›= , 1 + đ?‘?2 where c is a constant and t is a positive integer between 1 and 6, inclusive. What is the number of units of its product that Company X is scheduled to produce in month 6 of its next fiscal year? (1) Company X is scheduled to produce 180 units of its product in month 1 of its next fiscal year. (2) Company X is scheduled to produce 300 units of its product in month 2 of its next fiscal year.

GoGMAT Problem

167

Answers and explanations 1. Denote the number of candy given to Robert for r. Similarly pick initial letter of each name as name of variable (convenience saves time), so other variables are b, m, and k. Read the problem line by line and write the equations: r=k+2 b=m-6 m=r+2 k=2+b Let us find the values of b ,m, and k in terms of r: from equation 1: k = r – 2; from equation 4: b = k - 2 = (r - 2) - 2 = r - 4 ; from equation 3: m = r + 2. Since the sum of all these r + b + m + k = 20, substituting each value for expression with r, we obtain r + r - 4 + r + 2 + r - 2 = 20, so, r = 6. Then k=r-2=6-2=4 So the answer is B. 2. Let the price of shirt be s and of jeans – j, then we can construct system of equations: 3j + 2s = 69; 2j + 3s = 66. It is wise to subtract the second from the first: j - s = 3 or j = s + 3. Then plug j in either of the equations and solve for s. s = $12, so the answer is B. 3. If it takes Jan 30 seconds per floor to walk down and she lives x floors above the ground floor then total time needed to walk down the steps to the ground floor is 30x seconds. As for elevator, it will take 2x seconds to ride down to the ground floor. If it takes Jan the same amount of time to walk down the steps to the ground floor as to wait for the elevator for 7 minutes (= 420 seconds) and ride down, we can write down such an equation: 30x =2x +420; 28x = 420; x= 15. The answer is E. 4. A’s  compensation  for  any  week  can  be  written  as  360  +  0.06(t-1000), where t is amount of total  sales.  B’s  compensation is 0.08t. To find for what amount of total weekly sales would both salespeople earn the same compensation we have to solve the equation: 360 + 0.06(t-1000) =0.08t; 360 +0.06t -60 -0.08t =0; 0.02t =300; t=15 000. The answer is C. 5. Given the formula đ?‘›=

900 , 1 + đ?‘?2

180 =

900 , 1 + đ?‘?2

Determine the value of n when t = 6. (1) Given that n = 180 when t = 1, then

168

This equation can be solved for a unique value of c. Then, by substituting this value for c and 6 for t into 900 đ?‘›= , 1 + đ?‘?2 The value of n can be determined; SUFFICIENT. (2) Given that n = 300 when t = 2, then 900 , 1 + đ?‘?2 This equation can be solved for a unique value of c. Then, by substituting this value for c and 6 for t into 900 đ?‘›= , 1 + đ?‘?2 the value of n can be determined; SUFFICIENT. 300 =

The answer is D.

169

GoGMAT Problem Explanation

170

Rate problems GoGMAT, Session 8 For successful solving rate problems you need to remember the following rules: 1) The distance that an object travels is equal to the product of the average speed at which it travels and the amount of time it takes to travel that distance that is đ?‘šđ?’‚đ?’•đ?’† Ă— đ?‘ťđ?’Šđ?’Žđ?’† = đ?‘Ťđ?’Šđ?’”đ?’•đ?’‚đ?’?đ?’„đ?’†. 2) To determine the average rate at which an object travels, divide the total distance traveled by the total amount of traveling time. Average speed = Â

đ??­đ??¨đ??­đ??šđ??Ľ  đ???đ??˘đ??Źđ??­đ??šđ??§đ??œđ??ž đ??­đ??¨đ??­đ??šđ??Ľ  đ??­đ??˘đ??Śđ??ž

.

3) Relative speed of oncoming object is sum of both speeds: đ?‘šđ?’†đ?’?đ?’‚đ?’•đ?’Šđ?’—đ?’†  đ?’”đ?’‘đ?’†đ?’†đ?’…  (đ?’?đ?’?đ?’„đ?’?đ?’Žđ?’Šđ?’?đ?’ˆ)  =  đ?’”đ?’‘đ?’†đ?’†đ?’…  (đ?’?đ?’ƒđ?’‹đ?’†đ?’„đ?’•   đ?&#x;?)  +  đ?’”đ?’‘đ?’†đ?’†đ?’…  (đ?’?đ?’ƒđ?’‹đ?’†đ?’„đ?’•  đ?&#x;?) Â

Ra m/h

A

Rb m/h

D miles

B

Example: Two cyclists ride towards each other with their constant respective rates Ra and Rb as shown on the figure above. If they started simultaneously and the distance between A and B equals D, then how long will it take them to meet? As we know from the rule above, the relative (oncoming) speed đ?‘šđ?’“ equals to sum of the individual speeds. So, here we have đ?‘šđ?’“  =  đ?‘šđ?’‚  +  đ?‘šđ?’ƒ . Now, the time the cyclists need to reach each other is given by the general formula: đ?‘Ť đ?‘Ť đ?’•= = đ?‘šđ?’“ đ?‘šđ?’‚ + đ?‘šđ?’‚ 4) Relative speed of two objects moving in the same direction is the difference between their individual speeds: đ?‘šđ?’†đ?’?đ?’‚đ?’•đ?’Šđ?’—đ?’†  đ?’”đ?’‘đ?’†đ?’†đ?’…  (đ?’?đ?’—đ?’†đ?’“đ?’•đ?’‚đ?’Œđ?’Šđ?’?đ?’ˆ) =  đ?’”đ?’‘đ?’†đ?’†đ?’…  (đ?’‡đ?’‚đ?’”đ?’•đ?’†đ?’“  đ?’?đ?’ƒđ?’‹đ?’†đ?’„đ?’•  )  −  đ?’”đ?’‘đ?’†đ?’†đ?’…  (đ?’”đ?’?đ?’?đ?’˜đ?’†đ?’“  đ?’?đ?’ƒđ?’‹đ?’†đ?’„đ?’•)

171

Rb m/h

Ra m/h

A

D miles

B

Example: Two cyclists ride in the same direction with their constant respective rates Ra and đ?‘šđ?’ƒ , where đ?‘šđ?’‚ > đ?‘šđ?’ƒ . If they started simultaneously and the distance between A and B equals D, then how long will it take the faster cyclist to catch up the slower one? As we know from the rule above, the relative speed đ?‘šđ?’“ equals to difference of the individual speeds. So, here we have đ?‘šđ?’“  =  đ?‘šđ?’‚ −  đ?‘šđ?’ƒ . Now, the time the cyclists need to reach each other is given by the general formula: đ?‘Ť đ?‘Ť đ?’•= = đ?‘šđ?’“ đ?‘šđ?’‚ − đ?‘šđ?’‚ 5) Note the following points about the upstream/downstream motion: Upstream Downstream

against the stream same direction as the stream

Downstream đ??ľđ?‘œđ?‘Žđ?‘Ąâ€™đ?‘  đ?‘ đ?‘?đ?‘’đ?‘’đ?‘‘  =  đ?‘†đ?‘?đ?‘’đ?‘’đ?‘‘  đ?‘œđ?‘“  đ?‘Ąâ„Žđ?‘’  đ?‘?đ?‘œđ?‘Žđ?‘Ą  đ?‘–đ?‘›  đ?‘ đ?‘Ąđ?‘–đ?‘™đ?‘™  đ?‘¤đ?‘Žđ?‘Ąđ?‘’đ?‘&#x;  +  đ?‘†đ?‘?đ?‘’đ?‘’đ?‘‘  đ?‘œđ?‘“  đ?‘Ąâ„Žđ?‘’  đ?‘ đ?‘Ąđ?‘&#x;đ?‘’đ?‘Žđ?‘š Upstream đ??ľđ?‘œđ?‘Žđ?‘Ąâ€™đ?‘  đ?‘ đ?‘?đ?‘’đ?‘’đ?‘‘  =  đ?‘†đ?‘?đ?‘’đ?‘’đ?‘‘  đ?‘œđ?‘“  đ?‘Ąâ„Žđ?‘’  đ?‘?đ?‘œđ?‘Žđ?‘Ą  đ?‘–đ?‘›  đ?‘ đ?‘Ąđ?‘–đ?‘™đ?‘™  đ?‘¤đ?‘Žđ?‘Ąđ?‘’đ?‘&#x;  âˆ’  đ?‘†đ?‘?đ?‘’đ?‘’đ?‘‘  đ?‘œđ?‘“  đ?‘Ąâ„Žđ?‘’  đ?‘ đ?‘Ąđ?‘&#x;đ?‘’đ?‘Žđ?‘š Let’s  denote  for  shortness đ?‘˘ = đ?‘ đ?‘?đ?‘’đ?‘’đ?‘‘  đ?‘œđ?‘“  đ?‘Ąâ„Žđ?‘’  đ?‘?đ?‘œđ?‘Žđ?‘Ą  đ?‘–đ?‘›  đ?‘ đ?‘Ąđ?‘–đ?‘™đ?‘™  đ?‘¤đ?‘Žđ?‘Ąđ?‘’đ?‘&#x;, đ?‘Ł = đ?‘ đ?‘?đ?‘’đ?‘’đ?‘‘  đ?‘œđ?‘“  đ?‘Ąâ„Žđ?‘’  đ?‘ đ?‘Ąđ?‘&#x;đ?‘’đ?‘Žđ?‘š. So, we can rewrite đ?‘Ł = đ?‘˘ + đ?‘Ł, đ?‘Ł = đ?‘˘ − đ?‘Ł.

172

Now  it’s  easy  to  see,  that  if  both   đ?‘Ł Speed in still water

and đ?‘Ł

are given, then

�=

1 (đ?‘Ł 2

+đ?‘Ł

)

đ?‘Ł=

1 (đ?‘Ł 2

−đ?‘Ł

)

Speed of the stream

6) Units conversion on GMAT. The only conversions you need to know are the units of time 1 minute = 60 seconds, 1 hour = 60 minutes, 1 day = 24 hours Other units are used in GMAT problems, but if you need to convert them, all the necessary information about units relations will be provided in the problem. Otherwise, it would be unfair to people who are not familiar with some region-specific units. Hint: when expressing units, treat them as factors, for example: 1  đ?‘šđ?‘–đ?‘™đ?‘’  =  1760  đ?‘Śđ?‘Žđ?‘&#x;đ?‘‘đ?‘     1  Ă—  đ?‘šđ?‘–đ?‘™đ?‘’  =  1760  Ă—  đ?‘Śđ?‘Žđ?‘&#x;đ?‘‘đ?‘  1 Ă— đ?‘šđ?‘–đ?‘™đ?‘’    đ?‘Śđ?‘Žđ?‘&#x;đ?‘‘  =   1760 Let’s  practice: (1) Convert 5 km per hour into meters per second (2) Convert 100 meters per minute into km per hour (1) (2)

Â

= Â

Ă—

Â

Ă—

=

  Â

= =

Ă—

Â

Ă—

Â

Â

Ă— Â

= =

đ?‘š/đ?‘ Â

= 6  đ?‘˜đ?‘š/â„Ž

Now  let’s  apply  this  methodology  to  some  rate  problems. 1. (OG) A hiker walked for 2 days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day? (A) (Đ’)

2 3

(ĐĄ)

4

(D)

5

(E)

6

173

2.

(GC) A passenger sitting near the window in a train moving at 40 km/h noticed that it took 3 seconds for the oncoming train to pass by. What was the speed of the oncoming train if the length of the oncoming train was 75 meters? (A) 50 km/h (B) 52 km/h (C) 56 km/h (D) 60 km/h (E) 70 km/h 3. (OG) One hour after Yolanda started walking from X to Y, a distance of 45 miles, Bob started walking along the same road from Y to X. If Yolanda's walking rate was 3 miles per hour and Bob's was 4 miles per hour, how many miles had Bob walked when they met? (A) (B) (С) (D) (E)

24 23 22 21 19.5

4. (GC) A man can row 4½ miles per hour in still water and he finds that it takes him twice as long to row a certain distance upstream as to row the same distance downstream. What is the rate of the river stream in miles per hour? (A) 1½ (B) 2 (C) 2½ (D) 3 (E) 3½

5. (GC) A hiker walking at a constant rate of 4 miles per hour is passed by a cyclist travelling in the same direction along the same path at a constant rate of 20 miles per hour. The cyclist stops and waits for the hiker 5 min after passing her while the hiker continues walking at her constant rate. How many minutes must the cyclist wait until the hiker catches up? (A) (B) (C) (D) (E)

6 2/3 15 20 25 26 2/3

6. If an object travels 100 feet in 2 seconds, what is the object’s approximate speed in miles per hour? (Note: 1 mile = 5280 feet) (A) (B)

3.4 3.8

174

(C) (D) (E)

34 38 340

7. At  a  certain  instant  in  time,  the  number  of  cars,  N,  traveling  on  a  portion  of  a  certain  highway  can  be  estimated  by  the  formula  đ?‘ =

20đ??żđ?‘‘ Â 600 + đ?‘

where  L  is  the  number  of  lanes  in  the  same  direction,  d  is  the  length  of  the  portion  of  the  highway,  in  feet,  and  s  is  the  average  speed  of  the  cars,  in  miles  per  hour.  Based  on  the  formula,  what  is  the  estimated  number  of  cars  traveling  on  a   mile  portion  of  the  highway  if  the  highway  has  2  lanes  in  the  same  direction  and  the  average  speed  of  the  cars  is  40  miles  per  hour?  (5,280  feet  =  1  mile) (A) 155 (B) 96 (C) 80 (D) 48 (E) 24

GoGMAT Problem

175

Answers and explanations 1. If the time spent walking on the first day is t and the time spent walking on the second day is t then t + t = 18. We know that on the second day the hiker walked 2 hours longer, so, t = t + 2.  Solving the system of these two equations we get that walker spent 8 hours walking on the first day and 10 hours on the second day. If his speed on the first day was v1 and the speed on the second day was v2 , the distance he walked a total is d =  t v + t v or 64 = 8v1 + 10v2 and, as on second day his average speed was 1 mile per hour greater, v1 = v2 – 1. So, we get 64=8(v2 – 1) +10 v2. Thus, v1= 3. The answer is B. 2. Here we have 2 objects – train 1 and train 2. Speed of train 1 is 40 km/h. Relative speed of two trains is equal to 75m/3s=25 m/s. It can be expressed as 90 km/h, because 1 m/s = 3.6 km/h. Since train 2 is oncoming, its speed is the difference between relative speed and the speed of the train 1: Train 2 speed= 90 – 40 = 50 km/h. Here we used formula: Relative speed (oncoming)= speed (object 1) + speed (object 2) The answer is A. 3. If t denotes time that Bob spent walking, then (t+1) is the time that Yolanda spent walking. Distance which Yolanda covered is 3(t + 1), and distance which Bob covered is 4t. The total distance then is: 4t + 3(t + 1) = 45. So, t = 6 and Bob walked 6x4 = 24 miles. Therefore, A is the answer. 4. Let’s  draw  a  table.  If  v  is  the  speed  of  the  stream,  then Rate

Distance D

Time

đ??ˇ 4.5 − đ?‘Ł đ??ˇ Downstream D 4.5  +  đ?‘Ł 4.5 + đ?‘Ł From the problem we know the relationship between the time upstream and the time downstream: Upstream

4.5  –  �

đ??ˇ 2đ??ˇ = 4.5 − đ?‘Ł 4.5 + đ?‘Ł 1 2 = 4.5 − đ?‘Ł 4.5 + đ?‘Ł 9 − 2đ?‘Ł = 4.5 + đ?‘Ł đ?‘Ł = 1.5. The answer is A.

176

5. As the cyclist and the hiker move in one direction, they move away at a relative rate of 8. 20  đ?‘š/â„Ž  âˆ’  4  đ?‘š/â„Ž  =  16  đ?‘š/â„Ž. In  5  minutes  or  1/12  hour  after  they  met,  the  distance  between  the  hiker  and  the  cyclist  will  be đ??ˇ=

Ă—

â„Ž=

đ?‘š = đ?‘š.

The  time  that  the  hiker  takes  to  cover  4/3  miles  is đ?‘‡= đ?‘šá

= â„Ž = 20  đ?‘šđ?‘–đ?‘›đ?‘˘đ?‘Ąđ?‘’đ?‘ .

The  answer  is  C. 6. The rate of the object in feet per second equals  Â

= 50 Â đ?‘“đ?‘Ą/đ?‘ đ?‘’đ?‘?.

Now convert it to miles per hour: 50  đ?‘“đ?‘Ą 50 Ă— 60 Ă— 60  đ?‘š 50 Ă— 60 Ă— 6  đ?‘š 50 Ă— 60  đ?‘š 50 Ă— 15  đ?‘š 750  đ?‘š = = = = = ≈ 34  đ?‘š/â„Ž đ?‘ 5280  ℎ 528  ℎ 88  ℎ 22  ℎ 22  ℎ The answer is C.

7. Substitute đ??ż = 2, đ?‘‘ = (5,280), đ?‘Žđ?‘›đ?‘‘  đ?‘ = 40 into the given formula and calculate the value for N: 1 20(2)( )(5,280) 20(5,280) 20(5,280) 2(528) 528 2 đ?‘ = = = = = = 48. 600 + 40 600 + 1,600 2,200 22 11

The answer is D.

177

GoGMAT Problem Explanation

178

Work problems GoGMAT, Session 8 Work problems on GMAT usually describe a situation when a number of people (machines, pumps, etc.) work together. Individual rates of each person (machine, pump) are given. We often need to find the time necessary for the group of workers (machines) to complete a certain task. There  is  only  one  basic  concept  for  solving  such  problems.  Let’s  divide  it  in  several  steps. 1) Calculate how much work each person/machine does in one unit of time (could be days, hours, minutes, etc). The work per one unit of time is called rate and can be found using the formula: đ?‘šđ?’‚đ?’•đ?’†  =

đ?’•đ?’?đ?’•đ?’‚đ?’?  đ?’˜đ?’?đ?’“đ?’Œ  đ?’…đ?’?đ?’?đ?’† . đ?’•đ?’?đ?’•đ?’‚đ?’?  đ?’•đ?’Šđ?’Žđ?’†

2) Add up the amount of work done by each person/machine in that one unit of time. For example, if two persons are working at a certain job together with respective constant rates đ?‘… and đ?‘… , their work rate đ?‘… can be expressed as the sum of their individual rates: đ?‘šđ?’•đ?’?đ?’ˆđ?’†đ?’•đ?’‰đ?’†đ?’“ = đ?‘šđ?&#x;? + đ?‘šđ?&#x;? . In general, if n workers/machines are working together with respective constant rates đ?‘šđ?&#x;? , đ?‘šđ?&#x;? , ‌ , đ?‘šđ?’? , the rate of their collective work equals đ?‘šđ?’•đ?’?đ?’ˆđ?’†đ?’•đ?’‰đ?’†đ?’“ = đ?‘šđ?&#x;? + đ?‘šđ?&#x;? + â‹Ż + đ?‘šđ?’? . Note: In problems dealing with water pumps, which not only pour the water in, but also out, to find the rate of filling, we need to subtract these rates: đ?‘šđ?’‡đ?’Šđ?’?đ?’?đ?’Šđ?’?đ?’ˆ = đ?‘šđ?’Šđ?’?đ?’•đ?’? − đ?‘šđ?’?đ?’–đ?’• . 3) Calculate total amount of time necessary for work to be completed when all workers/machines are working together. đ?’•đ?’?đ?’•đ?’‚đ?’?  đ?’˜đ?’?đ?’“đ?’Œ  đ?’…đ?’?đ?’?đ?’† đ?‘ťđ?’Šđ?’Žđ?’†  = . đ?’“đ?’‚đ?’•đ?’†đ?’•đ?’?đ?’ˆđ?’†đ?’•đ?’‰đ?’†đ?’“ 1.

(OG) It takes one tube 6 hours to fill the dish with water. If it takes 4 hours for another pipe to do the same job, how much time it would take for both pipes to fill the same dish? (A) 1 (B) 4.3 (C) 5/12 (D) 2 (E) 2.4

179

2. (GC) If Samson is filling a bathtub with COLD water, it will take him 6 minutes and 40 seconds, and if he fills it with HOT water, it will take him 8 minutes. If draining the tub takes 13 minutes 20 seconds, how many minutes will it take to fill up the bath tub with both HOT and COLD water running while the plug is out, so the water is constantly draining? (A) 4.75 (B) 5 (C) 8.6 (D) 12 (E) 16 3.

Twenty five men can reap a field in 20 days, each working with the same uniform rate. How many days after all of them start working together should 15 men leave the field, if the other 10 men finish the remaining part of the work in 37.5 days? (A) 5 (B) 10 (C) 7 (D) 7.5 (E) 8

GoGMAT Problem

180

Answers and explanations 1. Using our formula, we get 1/T = 1/4 + 1/6 = (3+2)/12 =5/12, Therefore, T = 12/5 = 2.4 hours, thus, E is the answer. 2. Let’s count the rate for each variant: COLD: 400 seconds to fill the whole tub or 1/400 tub a second; HOT: 480 seconds to fill the whole tub or 1/480 tub a second; Drain: 800 seconds to drain the whole tub or 1/800 tub a second. We get water inflow/outflow per second: 1/400 + 1/480 - 1/800 = 12/4800 + 10/4800 - 6/4800 = 16/4800 = 1/300 = 1/T T =300. So, it takes 300 seconds, or 5 minutes. The answer is B. 3. Let’s consider the concept of man-hours (or, to be more exact, man-days)in this problem. Twenty five man can do the total work in 20 days, so to reap the whole field we need 25 men × 20 days = 500 man-days. Let x denote the number of days they worked together. During these x days they will complete 25 men × x days = 25x man-days. After 15 men left, the remaining 10 men finished the work in 37.5 days, resulting in 10 men × 37.5 days = 375 man-days. As a result, the whole work is completed. So, 500 = 375 + 25x 125 = 25x x = 5. The answer is A.

181

GoGMAT Problem Explanation

182

Mixture Problems GoGMAT, Session 8 In mixture problems, substances with different characteristics are combined, and it is necessary to determine the characteristics of the resulting mixture. Concentration of the substance in some mixture can be determined as: đ??Żđ??¨đ??Ľđ??Žđ??Śđ??ž  đ??¨đ??&#x;  đ??­đ??Ąđ??ž  đ??Źđ??Žđ??›đ??Źđ??­đ??šđ??§đ??œđ??ž

Concentration of the substance = đ??­đ??¨đ??­đ??šđ??Ľ  đ??Żđ??¨đ??Ľđ??Žđ??Śđ??ž  đ??¨đ??&#x;  đ??­đ??Ąđ??ž  đ??Śđ??˘đ??ąđ??­đ??Žđ??Ťđ??ž. The main property for solving mixture problems is: đ??ąđ?&#x;? Ă—đ??œđ?&#x;? đ??ąđ?&#x;? Ă—đ??œđ?&#x;? đ??ąđ?&#x;? đ??ąđ?&#x;?

=C

Where: x is the volume of the first mixture, x  is the volume of the second mixture, c is concentration of the first substance, c is concentration of the second substance, C is concentration of the mixture. Let’s  consider  some  typical  situations. Assume, that we have 200 grams of 20% solution of salt.

50 grams of pure salt are

120 grams of pure water are

300 grams of 30% solution of

added to the initial

added to the initial solution.

salt are added to the initial

solution. Find the new

Find the new concentration.

solution. Find the new

concentration.

x = 200,

x = 200,

x = 120,

x = 50,

c = 20% = 0.2,

c = 20% = 0.2, c = 100% = 1. đ?&#x;?đ?&#x;Žđ?&#x;ŽĂ—đ?&#x;Ž.đ?&#x;? đ?&#x;“đ?&#x;ŽĂ—đ?&#x;? đ?&#x;?đ?&#x;Žđ?&#x;Ž đ?&#x;“đ?&#x;Ž

concentration.

x = 200,

đ?&#x;—

= đ?&#x;?đ?&#x;“ = đ?&#x;Ž. đ?&#x;‘đ?&#x;” =

= 36%

c = 20% = 0.2,

c = 0% = 0. đ?&#x;?đ?&#x;Žđ?&#x;ŽĂ—đ?&#x;Ž.đ?&#x;? đ?&#x;?đ?&#x;?đ?&#x;ŽĂ—đ?&#x;Ž đ?&#x;?đ?&#x;Žđ?&#x;Ž đ?&#x;?đ?&#x;?đ?&#x;Ž

x = 300,

đ?&#x;?

= đ?&#x;– = đ?&#x;Ž. đ?&#x;?đ?&#x;?đ?&#x;“ =

=12.5%

c = 30% = 0.3. đ?&#x;?đ?&#x;Žđ?&#x;ŽĂ—đ?&#x;Ž.đ?&#x;? đ?&#x;‘đ?&#x;Žđ?&#x;ŽĂ—đ?&#x;Ž.đ?&#x;‘ đ?&#x;?đ?&#x;Žđ?&#x;Ž đ?&#x;‘đ?&#x;Žđ?&#x;Ž

đ?&#x;?đ?&#x;‘

= đ?&#x;“đ?&#x;Ž = đ?&#x;Ž. đ?&#x;?đ?&#x;”=

=26%

Remember that mixture formula can be used not only for chemical substances. See the following examples.

183

1. (OG) For an agricultural experiment, 300 seeds were planted in one plot and 200 were planted in a second plot. If exactly 25 percent of the seeds in the first plot germinated and exactly 35 percent of the seeds in the second plot germinated, what percent of the total number of seeds germinated? (A) 12% (B) 26% (C) 29% (D) 30% (E) 60% 2. (GC) If 12 ounces of strong solution of vinegar is mixed with 50 ounces of water to form three percent vinegar, what was the original strength of the vinegar solution? (A) 19.3 (B) 17 (C) 16.67 (D) 15.5 (E) 12.5 3. (OG) A club sold an average of 92 raffle tickets per member. Among the female members, the average number sold was 84, and among the male members, the average number sold was 96. What was the ratio of the number of male members to the number of female members in the club? (A) 1:1 (B) 1:2 (C) 1:3 (D) 2:1 (E) 3:1

184

GoGMAT Problem

185

Answers and explanations 1. If   the  number  of  seeds  in  the  first  plot  is  the  volume  of  the  first  substance:  x = 300, the  number  of  seeds  in  the  first  plot  is  the  volume  of  the  second  substance:  x = 200, Percent of germinated seeds is the concentration: c = 25, c = 35,  And the  consentration  of  the  mixture is the percent of the total number of seeds germinated, we can write down such an equation: Ă—

C=

Ă— Â ( Ă—

= C; Ă—

)

=

=

= 29 Â %.

So, the answer is C. 2. 50 = Â x ounces of water that have 0% = c of vinegar and 12 = Â x ounces of strong solution have c % of vinegar. The resulting mixture has 3% = C of vinegar. Using our formula we get: 50 Ă— 0 + 12 Ă— c =3 50 + 12 Solving this equation we will get c = 15.5% The answer is D.

3. Let the number of female members be f and the number of male members be m. Then we can write down such equation: 84f + 96m = 92 f+m Multiplying both parts by (đ?‘“ + đ?‘š), we get: 84đ?‘“ Â + 96đ?‘š Â = 92 Â (đ?‘“ + đ?‘š); 84đ?‘“ Â + 96đ?‘š Â = Â 92đ?‘“ Â + 92đ?‘š; 4đ?‘š Â = 8đ?‘“; đ?‘š Â = 2đ?‘“. We need to find the the ratio of the number of male members to the number of female members in the club, or simply m/f. From the last equation we can find m/f =2/1. The answer is D.

186

GoGMAT Problem Explanation

187

Percentage and Profit Problems GoGMAT, Session 9 We came across percentage problems at the beginning of our course in Arithmetic. In this chapter we will sum everything up, review the tips helpful for percentage word problems and define two new terms: simple interest and compound interest. 1) Convert percent into decimals: 5% 0.05 0.25% 0.0025 2) If some quantity X is increased by p%, multiply it by (1 + If some quantity X is decreased by p%, multiply it by (1 −

); );

For example, X X

Increased by 10%

1.1X

X

Increased by 25% 1.25X

X

decreased by 10%

0.9X

decreased by 25% 0.75X

This approach is especially convenient to organize multiple percent changes. Assume, that X is increased by 40%, then increased by 25%, and finally decreased by 60 percent: đ?‘‹

+ 40%

1.4đ?‘‹

+ 25%

1.4đ?‘‹ Ă— 1.25

-60%

1.4đ?‘‹ Ă— 1.25 Ă— 0.4 = 0.7đ?‘‹.

In other words, this chain of percent changes is equivalent to a 30% decrease. Note: If some quantity X is increased by, for example, 10% and then decreased by 10%, it does not equal to X: đ?‘‹

+ 10%

1.1đ?‘‹

-10%

1.1đ?‘‹ Ă— 0.9 = 0.99đ?‘‹

3) To find the percent of increase or decrease of two quantities, first find their difference and then divide it by the initial quantity (the one you compare with).

For example, (1) 320 is what percent greater than 250? (2) 400 is what percent less than 640? đ?‘ƒđ?‘’đ?‘&#x;đ?‘?đ?‘’đ?‘›đ?‘Ą  đ?‘œđ?‘“  đ?‘–đ?‘›đ?‘?đ?‘&#x;đ?‘’đ?‘Žđ?‘ đ?‘’ =

=

đ?‘ƒđ?‘’đ?‘&#x;đ?‘?đ?‘’đ?‘›đ?‘Ą  đ?‘œđ?‘“  đ?‘‘đ?‘’đ?‘?đ?‘&#x;đ?‘’đ?‘Žđ?‘ đ?‘’ =

= 28%. = = 37.5%.

188

4) There are two types of interests on GMAT: simple and compound. If the interest is simple, then each period the percent is calculated of the initial sum invested. If the interest is compound, then each period the percent is calculated of the previous period result. Assume that you have invested $100 at 10% annually. Here is how the amount will grow depending on the type of interest: 10% simple annual interest Year 0

$100

Year 1

$110

Year 2

$120

+$10 +$10

10% compound annual interest Year 0

$100

Year 1

$110

Year 2

$121

Formula:

Formula:

� = �(1 + ��)

� = �(1 + �)

Where

Where

R is the resulting sum

R is the resulting sum

S is the initial sum

S is the initial sum

đ?‘– is interest as a decimal

đ?‘– is interest as a decimal

đ?‘Ą is the number of periods

đ?‘Ą is the number of periods

Ă—1.1 Ă—1.1

1. (OG) The price of a certain television set is discounted by 10 percent, and the reduced price is then discounted by 10 percent. This series of successive discounts is equivalent to a single discount of (A) 20% (B) 19% (C) 18% (D) 11% (E) 10% 2. A merchant purchased a jacket for $60 and then determined a selling price that equaled the purchase price of the jacket plus 25 percent of the selling price. During a sale, the merchant discounted the selling price by 20 percent and sold the jacket. What was the  merchant’s  gross  profit on this sale? (A)  $0

189

(B) $3 (C) $4 (D) $12 (E) $15

Problems testing graphics interpretation are quite rare on the quantitative part of GMAT. Still take 2 minutes to solve the following problem:

3.

Of the following, which is closest to the increase from 1975 to 1980 in the amount received by the processor in producing 6 ounces of frozen orange juice? (A) $0.03 (B) $0.05 (C) $0.06 (D) $0.08 (E) $0.13 4. On a certain date, Pat invested $ 10,000 at x percent annual interest, compounded annually. If the total value of the investment plus interest at the end of 12 years will be $ 40,000, in how many years, the total value of the investment plus interest will increase to $ 80,000? (A) 15 (B) 16 (C) 18 (D) 20 (E) 24 5. Mike invested a total of $ 5,000 at 5 % simple annual interest rate for n years. Is n > 4? (1) At the end of n years, Mike’s investment plus interest was more than $ 5, 500. (2) At the end of n years, Mike’s investment plus interest was less than $ 6,500.

190

6. Leona bought a 1-year, $10,000 certificate of deposit that paid interest at an annual rate of 8 percent compounded semiannually. What was the total amount of interest paid on this certificate at maturity? (A) $10,464 (B) $ 864 (C) $ 816 (D) $ 800 (E) $ 480 GoGMAT Â Problem

191

Answers and explanations 1. If the price of the television is p, after first reducing it becomes 0.9đ?‘? and after second reducing: 0.9đ?‘? Ă— 0.9  đ?‘? = 0.81đ?‘?. So, such series of successive discounts is equivalent to a single discount of  đ?‘? − 0.81đ?‘?  = 0.19đ?‘? or simply 19%. The  answer  is  B. 2. The selling price is the purchase price $60 plus 25% of the selling price. Let s denote the selling price, then the equation can be written in the following way: đ?‘ = 60 + 0.25đ?‘ 0.75đ?‘ = 60 4 đ?‘ = 60 Ă— 3 đ?‘ = 80. After  the  selling  price  was  reduced  by  20%,  it  became  $80  Ă—  0.8  =  $64. Gross profit is the difference between total income and total cost. 64 − 60 = 4. The answer is C. 3. The amount received by the processor was 0.317  Ă—  0.3  in  1975 0.18  Ă—  0.7  in  1980. The  increase  was:  0.18  Ă—  0.7  -  0.317  Ă—  0.3  =  0.3  (0.42  -  0.317)  =  0.3  Ă—  0.103  =  0.0309. The  answer  is  A. 4. According to the formula of compound interest, at the end of the 12th year the total value of the investment will be: đ?‘Ľ 40,000 = 10,000(1 + ) 100 If  t  is  the  number  of  years  necessary  for  the  sum  to  increase  to  $80,000,  then  80,000 = 10,000(1 +

đ?‘Ľ ) 100

Now,  we  have  the  system  of  equations.  Firstly,  simplify  them: 4 = (1 + 8 = (1 +

đ?‘Ľ ) 100

đ?‘Ľ )      (∗) 100

Finding  the  square  root  from  both  parts  of  the  first  equation,  we  get:

192

2 = (1 +

đ?‘Ľ ) . 100

To  get  8  in  the  left  part  of  this  equation,  we  need  to  cube  both  its  parts: 2 = ( 1+ 8= 1+

đ?‘Ľ ) 100

đ?‘Ľ 100

     (∗∗)

Comparing  (*)  with  (**)  we  see,  that  t  =  18. Answer  C. 5. Mike’s  total  income in n years can be calculated using the formula of simple interest. From statement 1 we know, that 5,000(1 + 0.05đ?‘›) > 5,500 5,000 + 250đ?‘› > 5,500 đ?‘› > 2. But  we  do  not  know,  whether  n  is  greater  than  4  or  not.  Not  sufficient. Statement  2  gives  us  the  following  inequality: 5,000(1 + 0.05đ?‘›) < 6,500 đ?‘› < 6. Still  we  do  not  know  if  n  is  greater  than  4. Combining  the  information  from  both  statements  we  obtain,  that  n  is  between  2  and  6,  but  it  can  be  3,  4,  or  5,  which  means,  that  n  can  be  less,  equal  of  greater  than  4. The  answer  is  E. 6. Using the formula đ?‘… = đ?‘†(1 + đ?‘–) , where R is the amount of money after t times the compounding occurs (2 half-year periods), S is the initial amount invested (10,000$), i is the interest rate per period (0.08/2 = 0.04), the given information can be expressed as follows and solved for R: đ?‘… = (10,000)(1 + 0.04) đ?‘… = (10,000)(1.04) đ?‘… = (10,000)(1.0816) đ?‘… = 10,816 So, the amount of interest is 10,816 – 10,000 = 816.

193

The answer is C. GoGMAT Problem Explanation

194

Overlapping Sets Problems GoGMAT, Session 8 In mathematics a set is a collection of numbers or other objects. The objects are called the elements of the set. If S is a set having a finite number of elements, then the number of elements is denoted by |S|. Such a set is often defined by listing its elements. For example, S= {–5, 0, 1} is a set with |S| = 3. The order in which the elements are listed in a set does not matter; thus {–5, 0, 1} = {0, 1, –5}. If all the elements of a set S are also elements of a set T, then S is a subset of T. For example, S = {–5, 0, 1} is a subset of T = {–5, 0, 1, 4, 10}. For any two sets A and B, the union of A and B is the set of all elements that are in A or in B or in both. The intersection of A and B is the set of all elements that are both in A and in B. The union is denoted by âˆŞ . The intersection is denoted by ∊. For example, if A = {3, 4} and B = {4, 5, 6}, then đ??´ âˆŞ đ??ľ =  {3, 4, 5, 6}, and đ??´ ∊ đ??ľ = {4}. Two sets that have no elements in common are said to be disjoint or mutually exclusive. The relationship between sets is often illustrated with a Eulerian circles (or Venn diagram) in which sets are represented by regions in a plane.

A

đ?‘¨âˆŠđ?‘Š

B

This diagram illustrates a fact about any two finite sets A and B: the number of elements in their union equals the sum of their individual numbers of elements minus the number of elements in their intersection (because the latter are counted twice in the sum);

|đ??´ âˆŞ đ??ľ| = |đ??´| + |đ??ľ| − |đ??´ ∊ đ??ľ|

1. đ??´ = {1, 2, 3, 4, 5}, đ??ľ = {2, 4, 6, 8}. đ??´ ∊ đ??ľ  đ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘™đ?‘ A. {1, 2, 3, 4, 5, 6, 7, 8} B. {2, 4} C. {1, 2, 3, 4, 5, 6, 8, 10} D. {9} E. {1, 2, 6, 8}

195

2. đ??ś = {đ?‘Ž, đ?‘?, đ?‘?, đ?‘‘}. đ??ˇ = {3, 4, đ?‘?}. đ??ś âˆŞ đ??ˇ  đ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘™đ?‘ A. {a, b, c, d, 3, 4} B. {b} C. {3, 4} D. {b, d, 4} E. {a, c, 3, 4} 3. đ??´ = {1, 2, 3}. đ??ľ = {2, 3, 4}. đ??ś = {3, 4, 5}. (đ??´ ∊ đ??ľ) ∊ đ??ś  đ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘™đ?‘ A. {1, 2, 3, 4, 5} B. {1, 3, 5} C. {2, 3, 4} D. {1} E. {3} Word problems that involve two or more given sets of data that partially intersect with each other are called Overlapping Sets problems. In this chapter we consider two approaches to such problems and both of them involve visualization. 1) Eulerian circles – help to solve problems with two or three overlapping sets. 2) Tables – are good for dealing with two sets problems. Let’s  now  see  how  they work.

196

Eulerian circles for two or three sets. 4. In a class of 30 students 15 students are taking math class and 25 are taking literature class. If three students take neither math nor literature class, how many students are taking both math and literature class? (A) 5 (B) 7 (C) 10 (D) 13 (E) 15 5. (OG) Of the 65 cars on a car lot, 45 have air-conditioning, 30 have power windows, and 12 have both air conditioning and power windows. How many of the cars on the lot have neither air-conditioning nor power windows? (A) 2 (B) 8 (C) 10 (D) 15 (E) 18

The formula for three overlapping sets:

A

B

x y

w C

z

Assume that there are three sets of objects: A, B and C. x, y and z represent the numbers of objects belonging to exactly two of the sets, w objects belong to exactly three – A, B and C simultaneously.

|đ??´ âˆŞ đ??ľ âˆŞ đ??ś| = |đ??´| + |đ??ľ| + |đ??ś| − đ?‘Ľ − đ?‘Ś − đ?‘§ − 2đ?‘¤ “Exactly  twoâ€?

6. If the 600 residents of Clermontville, 35% watch the television show Island Survival, 40% watch Lovelost Lawyers and 50% watch Medical Emergency. If all residents watch at least one of these three shows and 18% watch exactly 2 of these shows, then how many Clermontville residents watch all of the shows? (A) 150 (B) 108 (C) 42

197

(D) 21 (E) -21

7. In a group of 68 students, each student is registered for at least one of three classes – History, Math and English. Twenty five students are registered for History, twenty five students are registered for Math, and thirty four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes? (A)

13

(B)

10

(C)

9

(D)

7

(E)

5

2. Tables for two set Overlapping Sets problems. Another approach to more complex Overlapping Sets problems is organizing a table. It’s extremely useful and time-saving for problems, where one set of objects is divided into overlapping groups by two different properties. But it may take some time to master. Let’s consider the following example: 8. Out of a total of 1,000 employees at a certain corporation, 52 percent are female and 40 percent of these females work in research. If 60 percent of the total number of employees work in research, how many male employees do NOT work in research? (A) 520 (B) 480 (C) 392 (D) 208 (E) 88 The set of all workers is divided into groups by two characteristics: 1. By gender, 2. By a form of activity. Let’s start organizing a table. When it is empty, it looks the following way: Male

female

Total

Work in research Do not work in research Total

198

Reading sentence by sentence, fill in the information in this order: a. out of a total of 1,000 employees at a certain corporation, b. 52 percent are female c. and 40 percent of these females work in research. as soon as percent of some quantity appears in the problem, put this quantity into appropriate Total cell. Note: finding percentage of the wrong quantity is a very widely spread mistake in this problem type. So, we must find 40 percent of females, not of all the employees, and the cell female – total is already filled up. Now we can fill up the cell female – work in research. d. If 60 percent of the total number of employees work in research, e. how many male employees do NOT work in research? Male

female

Total

Work in research Do not work in e. ??? research

c. 0.4×0.52×1000

d. 0.6×1000

total

b. 0.52×1000

a. 1000

Now make simple calculations. The table works in a way, that if two of the three cells of one row or line are known, then the third one can be easily calculated: Male 600-208=392

Work in research Do not work in ???= 480research 392=88 total 1000-520=480

female 208

Total 600

520

1000

So, the correct answer is E. 9. Of a group of people surveyed in a political poll, 60 percent said that they would vote for candidate R. Of those who said they would vote for R. 90 percent actually voted for R. And of those who did not say that they would vote for R. 5 percent actually voted for R. What percent of the group voted for R? (A) 56% (B) 59% (C) 62% (D) 65% (E) 74%

199

GoGMAT Problem

200

Answers and explanations 1. B 2. A 3. E

literature

math both

neither 4. Step 1. From the problem we understand, that only 30-3 = 27 students participate in at least one of the two classes. Step 2. On the other hand, we know that 15 take math and 25 take literature. Summing up the two quantities we get 15+25=40. Now, why is this sum greater than the total number of students attending at least one of the two classes, which is 27? The answer is because some of the students were considered twice while summing 15 and 25. These are the students participating in both activities. Step 3. So, to eliminate the desired quantity, we need to subtract 40 – 27 = 13. Generally, the equation for this problem looks the following way: 30 = 15 + 25 − đ?‘?đ?‘œđ?‘Ąâ„Ž + 3, 27 = 40 − đ?‘?đ?‘œđ?‘Ąâ„Ž, đ??ľđ?‘œđ?‘Ąâ„Ž = 13. The correct answer is D. 5. Let’s  draw  the  diagram  for  this  problem. 65

Air Cond. 45

both

Power win. 30

According to the formula, 65 = 45 + 30 − 12 + "đ?‘›đ?‘’đ?‘–đ?‘Ąâ„Žđ?‘’đ?‘&#x;", 65 = 63 + "đ?‘›đ?‘’đ?‘–đ?‘Ąâ„Žđ?‘’đ?‘&#x;",

12

2 = "đ?‘›đ?‘’đ?‘–đ?‘Ąâ„Žđ?‘’đ?‘&#x;". Neither

The correct answer is A.

6. According to the formula for three overlapping sets,

đ??´ âˆŞ đ??ľ âˆŞ đ??ś = đ??´ + đ??ľ + đ??ś − đ?‘Ľ − đ?‘Ś − đ?‘§ − 2đ?‘¤ “Exactly  twoâ€?

201

Let’s  first  operate  with  percentages.  There  are  three  overlapping  sets  in  this  problem: đ??´ âˆŞ đ??ľ âˆŞ đ??ś = 100, A = 35%, B = 40%, C = 50%, 18% people watch exactly two shows, so đ?‘Ľ + đ?‘Ś + đ?‘§ = 18. Substituting the given values, we obtain the following equation: 100 = 35 + 40 + 50 − 18 − 2đ?‘¤, 100 = 107 − 2đ?‘¤, 2đ?‘¤ = 7, đ?‘¤ = 3.5. So, 3.5% of the residents watch all three shows, which is 3.5 Ă— 600 = 21. The answer is D. 7. In terms of the formula for three overlapping sets, đ??´ âˆŞ đ??ľ âˆŞ đ??ś = 68, A = 25, B = 25, C = 34, 3 students are registered for all three classes, so w =3. Finally, the equation is 68 = 25 + 25 + 34 − “đ?‘’đ?‘Ľđ?‘Žđ?‘?đ?‘Ąđ?‘™đ?‘Ś  đ?‘Ąđ?‘¤đ?‘œâ€? − 6. 68 = 78 − “đ?‘’đ?‘Ľđ?‘Žđ?‘?đ?‘Ąđ?‘™đ?‘Ś  đ?‘Ąđ?‘¤đ?‘œâ€?. “đ?‘’đ?‘Ľđ?‘Žđ?‘?đ?‘Ąđ?‘™đ?‘Ś  đ?‘Ąđ?‘¤đ?‘œâ€? = 10. The answer is B. 8. See explanation in the lesson. 9. Let’s  organize  a  table. Said  that  they  would  vote Actually  voted 0.9Ă—0.6T Did  not  vote Total The answer is A.

0.6T

Said  that  they  would  not  vote 0.05Ă—0.4T

Total

T Â - Â 0.6T Â = Â 0.4T

T

0.9Ă—0.6T+0.05Ă—0.4T Â = Â 0.56T

202

GoGMAT Problem Explanation

203

Home assignment Word translations 1. (OG) Marion rented a car for $18.00 plus $0.10 per mile driven. Craig rented a car for $25.00 plus $0.05 per mile driven. If each drove d miles and each was charged the same amount for the rental, then d equals (A) 100 (B) 120 (C) 135 (D) 140 (E) 150 2. (PR) A tourist has travelers’ checks in $20 and $100 denominations. How many $20 checks are there? (1) If half of the $20 checks are spent, the remaining amount is $520. (2) The total value of the checks is $740 3. (GC) Mary bought some kiwis, bananas, and lemons at the grocery store in proportion of 1:4:7 accordingly. How many lemons did Mary buy? (1) The total number of fruits Mary bought is 24. (2) Mary bought 8 bananas. 4. (OG) In a weight-lifting competition, the total weight of Joe's two lifts was 750 pounds. If twice the weight of his first lift was 300 pounds more than the weight of his second lift, what was the weight, in pounds, of his first lift? (A) 225 (B) 275 (C) 325 (D) 350 (E) 400 5. (OG) Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in 5 years? (A) 9 (B) 19 (C) 21 (D) 23 (E) 33 6. (OG) A sum of $200,000 from a certain estate was divided among a spouse and three children. How much of the estate did the youngest child receive? (1) The spouse received 1/2 of the sum from the estate, and the oldest child received 1/4 of the remainder (2) Each of the two younger children received $12,500 more than the oldest child and $62,500 less than the spouse

204

7. (OG) If the Lincoln Library's total expenditure for books, periodicals, and newspapers last year was $35,000, how much of the expenditure was for books? (1) The expenditure for newspaper was 40 percent greater than the expenditure for periodicals (2) The total of the expenditure for periodicals and newspapers was 25 percent less than the expenditure for books 8. (OG) A certain bakery sells rye bread in 16-ounce loaves and 24-ounce loaves, and all loaves of the same size sell for the same price per loaf regardless of the number of loaves purchased. What is the price of a 24-ounce loaf of rye bread in this bakery? (1) The total price of a 16-ounce loaf and a 24-ounce loaf of this bread is $2.40 (2) The total price of two 16-ounce loaves and one 24-ounce loaf of this bread is $3.40 9. (OG) In a certain company the ratio of the number of managers to the number of production-line workers is 5 to 72. If 8 additional production-line workers were to be hired, the ratio of the number of managers to the number of production-line workers would be 5 to 74, How many managers does the company have? (A) 5 (B) 10 (C) 15 (D) 20 (E) 25 10. (OG) The sum of the ages of Doris and Fred is y years. If Doris is 12 years older than Fred, how many years will Fred be y years from now, in terms of y? (A) y-6 (B) 2y-6 (C) y/2-6 (D) 3y/2-6 (E) 5y/2-6 11. (OG) In a certain furniture store, each week Nancy earns a salary of $240 plus a 5 percent of the amount of her total sales that exceeds $800 for the week. If Nancy earned a total $450 one week, what were her total sales that week? (A) $2200 (B) $3450 (C) $4200 (D) $4250 (E) $5000 12. (OG) What is the number of female employees in Company X? (1) If Company X were to hire 14 more workers and all of these workers were females, the ratio of the number of male employees to the number of female employees would then be 16 to 9 (2) Company X has 105 more male employees than female employees

205

13. (OG) A swim club that sold only individual and family memberships charged $300 for an individual membership. If the club's total revenue from memberships was $480,000, what was the charge for a family membership? (1) The revenue from individual memberships was 1/4 of the total revenue from memberships (2) The club sold 1.5 times as many family memberships as individual memberships 14. (OG) Currently there are 50 picture books on each shelf in the children's section of a library. If these books were to be placed on smaller shelves with 30 picture books on each shelf, how many of the smaller shelves would be needed to hold all these books? (1)The number of smaller shelves needed is 6 more than the current number of shelves (2)Currently there are 9 shelves in the children's section 15. (OG) If Sara's age is exactly twice Bill's age, what is Sara's age? (1) Four years ago, Sara's age was exactly 3 times Bill's age (2) Eight years from now, Sara's age will be exactly 1.5 times Bill's age 16. (OG) If Aaron, Lee, and Tony have a total of $36, how much money does Tony have? (1) Tony has twice as much money as Lee and 1/3 as much as Aaron (2) The sum of the amounts of money that Tony and Lee have is half the amount that Aaron has 17. (OG) A bookstore that sells used books sells each of its paperback books for a certain price and each of its hardcover books for a certain price. If Joe, Marina, and Paul bought books in this store, how much did Maria pay for 1 paperback book and 1 hardcover book? (1) Joe bought 2 paperback books and 3 hardcover books for $12.50 (2) Paul bought 4 paperback books and 6 hardcover books for $25.00 18. The temperature of a certain cup of coffee 10 minutes after it was poured was 120 degree Fahrenheit. If the temperature F of the coffee t minutes after it was poured can be determined by the formula đ??š = 120(2 ) + 60, where F is in degrees Fahrenheit and a is a constant, then the temperature of the coffee 30 minutes after it was poured was how many degrees Fahrenheit? (A) 65 (B) 75 (C) 80 (D) 85 (E) 90 19. (GC) Aunt Marge is giving candy to each of her nephews and nieces. She has 20 candy pieces and she gives candy to children according to her wish. Thus, Robert gets 2 more candy pieces than Kate. Bill gets 6 less than Mary. Mary, in turn, gets 2 more candy than Robert. Kate gets 2 more candy than Bill. How many candy pieces does Kate get? (A) 2 (B) 4

206

(C) 6 (D) 8 (E) 10 20. What is sum of cost of one apple, one orange and one pear, in dollars? (1) The cost of one apple, two oranges, and three pears is 7 dollars. (2) The cost of one apple, three oranges, and five pears is 12 dollars. 21. (GC) One fisherman was telling his friends that he caught a fish that had a 60 foot long head. It also had a tail that was the size of the fish's head and a half of its body, and the body was half the size of the whole fish. What is the length of this fish? (A) 120 (C) 240 (E) 480 (B) 200 (D) 400

207

Rate problems 1.

(OG) How many miles long is the route from Houghton to Callahan? (1) It will take 1 hour less time to travel the entire route at an average rate of 55 miles per hour than at an average rate of 50 miles per hour (2) It will take 11 hours to travel the first half of the route at an average rate of 25 miles per hour

2.

(OG) If it took Mike 3 hours to ride 162 kilometers, what was his average speed, meters per second? (A) 1.5 (B) 5 (C) 5.4 (D) 15 (E) 54

3.

Andrew and Stephen drive on the highway in the same direction at respective rates of 72 and 80 kmh. If Stephen is 4 km behind Andrew, by how much does he have to increase his speed to catch up with Andrew in 20 minutes? 12. (A) 1 km/h (B) 2 km/h (C) 3 km/h (D) 4 km/h (E) 5 km/h 13. (GC) Maureen runs a 26-mile marathon in 5.4 hours. If she runs the first half of the race in 2.8 hours, what is her average rate, in miles per hour, for the second half of the race? (A) 10

4.

(B) 9 (C) 7 (D) 6 (E) 5 5.

6.

(MH) A hiker walked for 3 days. She walked 18 miles on the first day, walking 3 miles per hour. On the second day she walked for one less hour but she walked one mile per hour faster than on the first day. On the third day she walked the same number of hours as on the first day, but at the same speed as on the second day. How many miles in total did she walk? 14. (A) 60 15. (B) 58 16. (C) 62 17. (D) 24 18. (E) 44 19. A boy can row a boat at a constant rate of 5 miles per hour in still water. In a river with the constant current rate, he rows upstream for 15 minutes and then rows downstream, returning to his starting point in another 12 minutes. What is the rate of the current, in miles per hour?

208

7.

8.

9.

(A)  5/9 (D)  25 (B)  5 (E)  45 (C)  9 20. (MH) The time it took car P to travel 600 miles was 2 hours less than the time it took car R to travel  the  same  distance.  If  car  P’s  average  speed  was  10  miles  per  hour  greater  than  that  of  car  R,  what  was  car  R’s  average  speed,  in  miles  per  hour? 21. (A)  30 22. (B)  40 23. (C)  50 24. (D)  60 25. (E)  70 26. (GoGMAT) A fly and a train are moving towards each other along a straight rail track. If the train remained stationary, the fly would reach the train in 4 hours. If the fly remained stationary, the train would hit it in 3 hours. Both the fly and the train started moving at the same time towards each other. After one hour and 20 minutes of flight, the fly will stop and remain stationary at its new position. The train will continue to move towards the fly at its usual speed. How long after the journey began will the train hit the fly? (A)

 â„Žđ?‘œđ?‘˘đ?‘&#x;

(B)

 â„Žđ?‘œđ?‘˘đ?‘&#x;

(C)

 â„Žđ?‘œđ?‘˘đ?‘&#x;

(D)

 â„Žđ?‘œđ?‘˘đ?‘&#x;

(E)

2  â„Žđ?‘œđ?‘˘đ?‘&#x;đ?‘

(GC) A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 level, and 1/3 downhill. If he covers the uphill part of the route at the rate of 16 miles per hour and the level part at the rate of 24 miles per hour, what rate in miles per hour would he have to travel the downhill part of the route in order to average 24 miles per hour for the entire route? (A) 32 (B) 36 (C) 40 (D) 44 (E) 48

10. (GP) If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did. How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day? (A) 100 (B) 120 (C) 140 (D) 150 (E) 160

209

Work Problems 1. (MG) Mr. Bridges can wash his car in 15 minutes, while his son Dave takes twice as long to do the same job. If they work together, how many minutes will the job take them? (A) 5 (B) 7.5 (C) 8 (D) 10 (E) 22.5 2. (GC) Bill takes 20 days to empty a 50 - liter barrel of beer and, together with Henry, he can empty it in 14 days, how many days will it take Henry to finish a barrel of beer by himself? (A) 17 (B) 33 (C) 46 (D) 47 (E) 58 3. (OG) Hoses X and Y simultaneously fill an empty swimming pool that has a capacity of 50,000 liters. If the flow in these hoses is independent, how many hours will it take to fill the pool? (1) Hose X alone would take 28 hours to fill the pool. (2) Hose Y alone would take 36 hours to fill the pool. 4. (MH) If a copier makes 3 copies every 4 seconds, then continues at this rate, how many minutes will it take to make 9,000 copies? (A) 3,000 (B) 200 (C) 120 (D) 100 (E) 60 5. Matt and Peter can do together a certain work in 20 days. After they have worked together for 12 days Matt stops and Peter completes the remaining work in 10 days. In how many days can Peter complete the work separately? (A) 26days (B) 27days (C) 23days (D) 25days (E) 24 days 6.

(OG) Tree machines, individually, can do a certain job in 4, 5, and 6 hours, respectively. What is the greatest part of the job that can be done in one hour by two of the machines working together at their respective rates? (A) 11/30 (B) 9/20 (C) 3/5 (D) 11/15 (E) 5/6

210

7. (GC) Water flows into an empty tank of 54 liters via 12 small pipes. The rate of each pipe is 1 liter per hour. However, water flows out of the tank via several big pipes at the rate of 1.5 liter per hour. If after 12 hours, the tank is completely full, how many big pipes are there? (A) 2.5 (B) 3 (C) 5 (D) 6 (E) 7 8. (MH) An employee at a company was given the task of making a large number of copies. He spent the first 45 minutes making copies at a constant rate on copier A, but copier A broke down before the task was completed. He then spent the next 30 minutes finishing the task on copier B, which also produced copies at a constant rate. How many total minutes would the task have taken had copier A not broken down? (1) Copier B produces 10 copies per minute. (2) Copier A produced twice as many copies in its first 5 minutes of operation as copier B produced in its first 15 minutes. 9. (GC) In a TV factory 9 persons can assemble 10 tv sets in 20 days of 7 1/2 working hours. How long will it take 12 persons to assemble 20 tv sets working 6 hours per day, if 2 persons in the latter case do as much work as 3 men in the former? 27. (A) 10 28. (B) 12.5 29. (C) 20 30. (D) 25 31. (E) 50 10. (GC) It takes 6 technicians 10 hours to build and program a new server from Direct Computer, each working at the same uniform rate. If six technicians start to build the server at 11:00 AM, and one technician per hour is added beginning at 5:00 PM, at what time will the server be complete? (A) 6:40 PM (B) 7:00 PM (C) 7:20 PM (D) 7:40 PM (E) 8:00 PM

Mixture Problems 1. (OG) If 3 pounds of dried apricots that cost x dollars per pound are mixed with 2 pounds of prunes that cost y dollars per pound, what is the cost, in dollars, per pound of the mixture? (A) (3x+2y)/5 (B) (3x+2y)/(x+y) (C) (3x+2y)/(xy) (D) 5(3x+2y) (E) 3x+2y

211

2. (GC) In what proportion must sugar at $3.1 per pound be mixed with sugar at $3.6 per pound, so that the mixture costs $3.25 per pound? (A) 1:2 (B) 3:2 (C) 7:3 (D) 3:7 (E) 2:3 3. (OG) A $500 investment and a $1,500 investment have a combined yearly return of 8.5 percent of the total of the two investments. If the $500 investment has a yearly return of 7 percent, what percent yearly return does the $1,500 investment have? (A) 9% (B) 10% (C) 10.625% (D) 11% (E) 12% 4. (GC) Committee X and Committee Y, which have no common members, will combine to form Committee Z. Does Committee X have more members than Committee Y? (1) The average (arithmetic mean) age of the members of Committee X is 25.7 years and the average age of the members of Committee Y is 29.3 years (2) The average (arithmetic mean) age of the members of Committee Z will be 26.6 years 5. (OG) During a certain season, a team won 80 percent of its first 100 games and 50 percent of its remaining games. If the team won 70 percent of its games for the entire season, what was the total number of games the team played? (A) 180 (B) 170 (C) 156 (D) 150 (E) 105 6. (OG) A rabbit on a controlled diet is fed daily 300 grams of a mixture of two foods, food X and food Y. Food X contains 10 percent protein and food Y contains 15 percent protein. If the rabbit's diet provides exactly 38 grams of protein daily, how many grams of food X are in the mixture? (A) 100 (B) 140 (C) 150 (D) 160 (E) 200

212

7. (OG) On Jane's credit card account, the average daily balance for a 30-day billing cycle is the average of the daily balances at the end of each of the 30 days. At the beginning of a certain 30day billing cycle, Jane's credit card account had a balance of $600. Jane made a payment of $300 on the account during the billing cycle. If no other amounts were added to or subtracted from the account during the billing cycle, what was the average daily balance on Jane's account for the billing cycle? (1)Jane's payment was credited on the 21st day of the billing cycle (2)The average daily balance through the 25th day of the billing cycle was $540 8. (GC) If 200 lbs. of a mixture is 80 percent husk and 20 percent sand. Then how much husk should be extracted in order to have 75 percent concentration of husk? (A) ¼ (B) 20/3 (C) ½ (D) 40 (E) 60 9. (MG) How many ounces of pure acid must be added to 20 ounces of a solution that is 5% acid to strengthen it to a solution that is 24% acid? (A) 5 (B) 5.5 (C) 6 (D) 7.5 (E) 10 10. (GC) How must a grocer mix 4 types of peanuts worth 54 c, 72 c, $1.2 and $1.44 per pound so as to obtain a mixture at 96 cents per pound? (A) 8:4:4:7 (B) 24:12:12:50 (C) 4:8:7:4 (D) 16:42:28:10 (E) Cannot be uniquely determined

Percentage and profit problems 1. (GC) If today the price of an item is $3,600, what was the price of the item exactly 2 years ago? (1) The price of the item increased by 10 percent per year during this 2-year period (2) Today the price of the item is 1.21 times its price exactly 2 years ago 2. (OG) Tickets for all but 100 seats in a 10,000-seat stadium were sold. Of the tickets sold, 20 percent were sold at half of the price and the remaining tickets were sold at the full price of $2. What was the total revenue from ticket sales? (A) $15,840 (B) $17,820 (C) $18,000 (D) $19,800 (E) $21,780

213

3. (OG) Increasing the original price of an article by 15 percent and then increasing the new price by 15 percent is equivalent to increasing the original price by (A) 32.25% (B) 31.00% (C) 30.25% (D) 30.00% (E) 22.50% 4. (OG) A certain manufacturer produces items which the production costs consist of annual fixed costs totalling $130,000 and variable costs averaging $8 per item. If the manufacturer's selling price per item is $15, how many items must the manufacturer produce and sell to earn an annual profit of $150,000? (A) 2,858 (B) 18,667 (C) 21,429 (D) 35,000 (E) 40,000 5. (OG) An author received $0.80 in royalties for each of the first 100,000 copies of her book sold, and $0.60 in royalties for each additional copy sold. If she received a total of $260,000 in royalties, how many copies of her book were sold? (A) 130,000 (B) 300,000 (C) 380,000 (D) 400,000 (E) 420,000 6. The market value of a certain machine decreased by 30 percent of its purchase price each year. If the machine was purchased in 1982 for its market value of $8,000, what was its market value two years later? (A) $8,000 (B) $5,600 (C) $3,200 (D) $2,400 (E) $800 7. On January, 1, 2002, Michael invested a certain amount of money at r percent interest rate, compounded annually. What is the value of r? (1) At the end of 2004, the investment plus the total interest was $ 5,955. (2) At the end of 2006, the investment plus the total interest was $ 6,690.

214

Revenue (in billions of dollars)

15 11,3 10,0

10 8,0 6,0 5

0 1993

1994

1995

1996

8. The graph above shows the combined revenue, in billions of dollars, of a chain of food stores for each year over a four-year period. In 1994 a certain store’s revenue accounted for 2.0 percent of the combined revenue for that year, and in 1995 the same store accounted for 2.3 percent of the combined revenue for that year. What was the approximate percent increase in revenue for this store from 1994 to 1995 ? (A) 0.3% (B) 15.0% (C) 25.0% (D) 30.4% (E) 43.8% 9. (OG) Henry purchased 3 items during a sale. He received a 20 percent discount off the regular price of the most expensive item and 10 percent discount off the regular price of each of the other 2 items. Was the total amount of the 3 discounts greater than 15 percent of the sum of the regular prices of the 3 items? (1) The regular price of the most expensive item was $50, and the regular price of the next most expensive item was $20 (2) The regular price of the least expensive item was $15 10. (GC) $20,000 was deposited into bank, with interest compounded quarterly. What is compound interest? (1) The interest in the second quarter is 1.1 times more than the first quarter. (2) The interest in the second quarter is 200 dollars more than the first quarter. 11. (GC) A filter decreases concentration of harmful substances in the water by 50%. How many filters are required to clean 200 liters of water containing 0.5 liters of harmful substances if the water that contains less than 0.1 percent of harmful substances is considered clean? (A) 8 (B) 6 (C) 5 (D) 4 (E) 2

215

12. Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w ? w (A) 1 1.08 w (B) 1.08 1.16 w (C) 1.16 1.24 w (D) 1.08 (1.08) 2 w (E) 2 (1.08) (1.08) 3 13. (GC) The price of a certain commodity increased at a rate of X% per year between 2000 and 2004. If the price was A dollars in 2001 and B dollars in 2003, what was the price in 2002 in terms of A and B? (A) √(A × B) (B) B × √(B/A) (C) B × √A (D)B × A/√B (E) B × A3/2 14. (GC) If the circus were to sell all of its 220 tickets for this month's performance at their usual price, the revenue from sales would be 10% greater than that collected last month. However, the circus raised the ticket price by 5% and sold only 200 tickets as a result. What percent less was last month's revenue than that of this month? (A) 2 (B) 4 (C) 110/20 (D) 100/21 (E) 8/4 15. One-fifth of the light switches produced by a certain factory are defective. Four-fifths of the defective switches are rejected and

of the non-defective switches are rejected by mistake. If all

the switches not rejected are sold, what percent of the switches sold by the factory are defective? (A) 4% (B) 5% (C) 6.25% (D) 11% (E) 16%

216

16. John deposited $10,000 to open a new savings account that earned 4 percent annual interest, compounded quarterly. If there were no other transactions in the account, what was the amount of money in John’s account 6 months after the account was opened? (A) $10,100 (B) $10,101 (C) $10,200 (D) $10,201 (E) $10,400

Overlapping Sets Problems The  questions  1  â€“  2  refer  to  the  following  sets: A  =  {1,  3,  2,  5}.  B  =  {2,  4,  6}.  C  =  {1,  3,  5}. 1.  (đ??´ âˆŞ đ??ľ) ∊ đ??ś equals (A) {1,  2,  3} (B) {2,  4,  5} (C) {1,  2,  5} (D) {1,  3,  5} (E) {3,  4,  5} 2.  (đ??´ ∊ đ??ľ) âˆŞ đ??ś  equals (A) {1,  2,  3,  5} (B) {4} (C) {2,  4} (D) {1,  3,  5} (E) {1,  2,  3,  4,  5} 3.  A  set  consists  of  12  numbers,  all  are  even  or  multiple  of  5.  If  5  numbers  are  even  and  8  numbers  are  multiple  of  5,  how  many  numbers  is  multiple  of  10? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5 4.  In  Company  X,  30  percent  of  the  employees  live  over  ten  miles  from  work  and  60  percent  of  the  employees  who  live  over  ten  miles  from  work  are  in  car  pools.  If  40  percent  of  the  employees  of  Company  X  are  in  car  pools,  what  percent  of  the  employees  of  Company  X  live  ten  miles  or  less  from  work  and  are  in  car  pools? (A) 12% (B) 20% (C) 22% (D) 28% (E) 32%

217

5. On a certain road, 10 percent of the motorists exceed the posted speed limit and receive speeding tickets, but 20 percent of the motorists who exceed the posted speed limit do not receive speeding tickets. What percent of the motorists on that road exceed the posted speed limit? (A) 10.5% (B) 12.5% (C) 15% (D) 22% (E) 30% 6. One night 18 percent of the female officers on a police force were on duty. If 180 officers were on duty that night and half of these were female officers, how many female officers were on the police force? (A) 90 (B) 180 (C) 270 (D) 500 (E) 1,000 7. Of the high school students who participated in a recent survey, 110 had taken a course in biology and 230 had taken a course in history. How many students participated in the survey? (1) Of the students who participated in the survey, 120 had taken neither a course in biology nor a course in history. (2) Of the students who participated in the survey, 60 had taken both a course in biology and a course in history. 8. Of the n students in a certain class, x took a course in biology and у took a course in history, and z took neither of the courses. If there were students who took both two courses, in terms of n, x, y, and z, what fraction of the students took both two courses? (A)

1

x

y n

z

x

y n

z

1

x

y

(B) (C)

z n n

n x

(D)

y z n

n x

(E)

y

z

n

9. Twenty-four dogs are in a kennel. Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Two of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and long hair but are not black. If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails?

218

(A) (B) (C) (D) (E)

1 2 3 5 6

10. All trainees in a certain aviator training program must take both a written test and a flight test. If 70 percent of the trainees passed the written test, and 80 percent of the trainees passed the flight test, what percent of the trainees passed both tests? (1) 10 percent of the trainees did not pass either test. (2) 20 percent of the trainees passed only the flight test. 11. How many of the boys in a group of 100 children have brown hair? (1) Of the children in the group, 60 percent have brown hair. (2) Of the children in the group, 40 are boys. 12. A group of people participate in some curriculums, 20 of them practice Yoga, 10 study

cooking, 12 study weaving, 3 of them study cooking only, 4 of them study cooking and yoga only, 2 of them participate all curriculums. How many people study cooking and weaving only? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

219

# of question

Topic of question (type) Your answer WORD PROBLEMS Word Translations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Correct Answer

D C D D D B B C D D E C C D D A E B B C E Rate Problems

1 2 3 4 5 6 7 8 9 10

D D D E C A C (AP) E E D Work Problems

1 2 3 4 5 6 7 8

D D C B D B C B

220

9 10

D E Mixtures Problems

1 2 3 4 5 6 7 8 9 10

A C A C D B D D A E Percentage and Profit

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

D B A E D C C E A D E D A D B D Overlapping Sets Problems

1 2 3 4 5 6 7 8 9 10 11 12

D A B C (Pl) B D C B C D E A

221

Test № 5. Word problems 1. Of the 200 students in a college, 125 learned Spanish and 81 learned French. How many students learn neither Spanish nor French? (1) 20 percent of the students who learned Spanish also learned French. (2) 50 out of the students who learn French did learn Spanish. 2. Train A leaves the station traveling at 30 miles per hour. Two hours later train В leaves the same station traveling in the same direction at 40 miles per hour. How far from the station was train A overtaken by train B? (A) 180 (B) 240 (C) 270 (D) 300 (E) 320 3. Each of six students, four girls and two boys, spent an average of $60 on books. If the girls spent a total amount of $300, how much did boy X and Y spends on books? (1) X spent twice as much as Y spent on books. (2) X spent $20 more than Y on books. 4. Of the students in a certain class, 55% of the female and 35% of the male passed an exam. Did more than half of the students in the class pass the exam? (1) More than half of the students in the class are female. (2) The number of the female students is 20 more than the number of the male students. 5. Jim traveled from X to Y, and arrived at Z. If the average speed for the whole trip was 60 miles per hour, what is the average speed for the trip from Y to Z? (1) The average speed for the trip from X to Y is 55 miles per hour. (2) It took Jim 0.5 hour from Y to Z. 6. A riverboat leaves Mildura and travels upstream to Renmark at an average speed of 6 miles per hour. It returns by the same route at an average speed of 9 miles per hour. What is its average speed for the round trip, in miles per hour? (A) 7.0 (B) 7.2 (C) 7.5 (D) 7.8 (E) 8.2 7. There are only three kinds of books on a bookshelf, science book, novel book, and workbooks, in the ratio of 2 to 5 to 7. What is the total number of the books on the shelf? (1) There are less than 5 science books on the shelf. (2) The total number of novel book and workbook is 24.

222

8. If 0 < n < m < 1, m2 – n2 must be less than which of the following expressions? (A) m–n (B)

m n 2

(C) mn (D) (m + n)2 (E) (m – n)2 9. Steve took a math test that consists of basic questions and advanced questions. A correct answer to the basic question will gain 1 point and a correct answer to advanced question will gain 2 points. How many questions does the test consist of? (1) Steve answered 80% of the basic question and 30% of the advanced questions correctly, and gained 88 points. (2) There are twice as many advanced questions as the basic question. 10. A set consists of 12 numbers, all are even or multiple of 5. If 5 numbers are even and 8 numbers are multiple of 5, how many numbers is multiple of 10? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5 11. Seven students each bought some books on a book fair. Is the total number of books they bought greater than 27? (1) No student bought more than 7 books. (2) No two students bought same number of books. 12. Of the people stand in a line waiting for service, 20% have waited for more than 5 minutes and 8% have waited for more than 8 minutes. How many people have waited for more than 5 minutes and no more than 8 minutes? (1) 20 people waited for no more than 5 minutes. (2) 2 people have waited for more than 8 minutes. 13. On a certain date, Jack invested $13,000 at x percent simple annual interest and a different amount at у percent simple annual interest. If he earned same amount of interest from two investments, what amount did Jack invest at у percent simple annual interest? (1) The amount of interest earned by Jack from the investment that pay x% interest was $390. (2) The ratio of x to у is 3 to 2. 14. Maura drives to work in 40 minutes. She takes the same route to return home. If her average speed on the trip home is half as fast as her average speed on the trip to work, how much time does she spend driving on the round trip? (A) 1 hour (B) 1 hour, 15 minutes (C) 1 hour, 20 minutes

223

(D) (E)

1 hour, 40 minutes 2 hours

15. If m, n, and mn are nonzero integers, which of the following must be true? I. mn is positive II. nm is an integer III. nm is positive (A) None (B) I only (C) II only (D) III only (E) I and II only 16.What is the remainder when positive integer x is divided by 6? (1) 15x is divisible by 2. (2) 15x is divisible by 3. 17. Which of the following must be greater than x? I. 2x II. x2 III. x2 + 1 (A) I only (B) II only (C) III only (D) I and II only (E) II and III only 8

18. Working together at their respective rates, machine А, В, and С can finish a certain work in 3 hours. How many hours will it take A to finish the work independently? (1) Working together, A and В can finish the work in 4 hours. 48

(2) Working together, В and С can finish the work in 7 hours. 3

19. 4 of the employees at a global company are female. If all the employees are from American, Canadian, or Mexican, what percent of the male employees are Canadian? 4

(1) 5 of the female are American or Mexican. 1

(2) 3 of all the employees are Canadian.

20. One inlet pipe fills an empty tank in 5 hours. A second inlet pipe fills the same tank in 3 hours. If 2

both pipes are used together, how long will it take to fill 3 of the tank?

224

(A) (B) (C) (D) (E)

8 15 15 8 3 4 5 4 8 3

21. In a survey of 200 college graduates, 30 percent said they had received student loans during their college careers, and 40 percent said they had received scholarships. What percent of those surveyed said that they had received neither student loans nor scholarships during their college careers? (1) 25 percent of those surveyed said that they had received scholarships but no loans. (2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships. 1

1

22. Machine A can finish 4 of a work in 4 hours, while В can finish 6 of the work in 2 hours, and С 1

can finish 3 of the work in 8 hours. How many hours will it take for A, B, and С together to finish the work? (A) (B) (C) (D) (E)

8 3 14 3 16 3 7 8 7 3

23. John always jogs to school at a speed of 6 kilometers per hour, and walks home along the same route at a speed of 3 kilometers per hour. If he spends exactly one hour total traveling both ways, how many kilometers is his school from his home? (A) 2 (B) 3 (C) 4 (D) 6 (E) 9 24. What is the remainder when x is divided by 12? (1) When x is divided by 4, the remainder is 3. (2) When x is divided by 36, the remainder is 27.

225

25. Of the residents in Stamford, 75% have a laptop, 40% have both laptop and desktop, and 5% have none. What percent of the residents have only desktop? (A) 10 % (B) 15 % (C) 20 % (D) 25 % (E) 30 % 26. The students in a certain class each joined a club: basketball club, soccer club, or swimming club. What fraction of the students joined swimming club? (1) The number of the students who joined the basket club is three times the number of the students who joined the soccer club. (2) The number of the students who joined the swimming club is twice the number of the students who joined soccer club or basketball club. 27. During a certain day, a bookshop sold 100 used books and 50 new books. Is the revenue on new books sold greater than the revenue on used books sold? (1) The price for any new book is more than twice the price for the most expensive used book. (2) The average price for all of the new books is more than twice the average price for all of the used books. 28. On January 1, 2003, Jack invested $5,000 at r percent interest rate, compounded annually. What is the value of r? (1) The investment plus the total interest at the end of 2004 was $550 more than the amount at the end of 2003. (2) The investment plus the total interest at the end of 2004 increased by 21% from the original investment. 1

1

29. A 50-liter solution of alcohol and water is 5 percent alcohol. If 1 2 liters of alcohol and 8 2 liters of water are added to this solution, what percent of the solution produced is alcohol? 1

(A) 5 2 % (B) 6 % (C) (D) (E)

1 % 3 2 6 % 3 6

7%

30. Working at their respective constant rates, printing machine X, Y, and Z can finish a certain work in 9, 12, and 18 hours. If three machines work together to finish the work, what fraction of the work will be finished by the machine Z?

226

(A) (B) (C) (D) (E)

4 9 1 3 1 4 2 9 1 9

31. Solution X is 25 percent mercury and solution Y is 10 percent mercury. If 12 ounces of solution X are added to 13 ounces of solution Y, approximately what percent of the resulting solution will be mercury? (A) 12 % (B) 17 % (C) 21 % (D) 23 % (E) 37 % 32. What is the sum of 5 evenly spaced integers? (1) The middle integer is zero (2) Exactly 2 of the integers are negative Answers 1. D 2. B 3. D 4. E 5. E 6. B 7. B 8. D 9. C 10. B 11. B 12. D 13. B 14. E 15. A 16. E

17. C 18. B 19. C 20. D 21. D 22. C 23. A 24. B 25. C 26. B 27. D 28. D 29. D 30. D 31. B

227

Lesson  â„–  6.  Geometry. Geometry questions account for about 20% of quantitative section in GMAT test. Even though  there  aren’t  a  lot  of  these  type  of  questions,  they  can  determine  the  difference  between  a  good and an excellent score. The key to cracking tricky Geometry questions is to know the properties of different shapes, and practice a variety of questions. We have divided GMAT Geometry into six basic topics: Lines and angles Triangles GMAT Quadrilaterals (squares, rectangles, parallelograms, etc.) geometry Circles problems Simple Solids (cubes, cylinders, spheres, etc.) always involve Coordinate Geometry more than one step.

Let’s  consider  them  step  by  step.

Lines and angles. GoGMAT, Session 10 đ?‘™ P

Q

In  geometry,  the  word  â€œlineâ€?  refers  to  a  straight  line  that  extends  without  end  in  both  directions. The line above can be referred to as line đ?‘ƒđ?‘„ or line đ?‘™. The part of the line from đ?‘ƒ to đ?‘„ is called a line segment. đ?‘ƒ and đ?‘„ are the endpoints of the segment. The notation đ?‘ƒđ?‘„ is used to denote both the segment and the length of the segment. Line segments that have equal lengths are called congruent line segments. The point that divides a line segment into two congruent line segments is called the midpoint of the line segment. An angle with a measure of 90°  is called a right angle. An angle with a measure less than 90° is called an acute angle, and an angle with measure between 90° and 180° is called an obtuse angle. An angle with a measure of 180°  is called a straight angle.

When two lines intersect at a point, they form four angles, as indicated below. Each angle has a vertex at point đ??ľ, which is the point of intersection of the two lines. In the figure, angles đ??´đ??ľđ??¸ and đ??śđ??ľđ??ˇ are called opposite angles, also known as vertical angles. Angles đ??´đ??ľđ??ś and đ??¸đ??ľđ??ˇ are also C opposite angles. Opposite angles have equal measures, and angles that have equal measures are called congruent angles. đ?‘ĽÂ° đ?‘ŚÂ° Hence, opposite angles are congruent. B đ?‘ĽÂ° D A đ?‘ŚÂ° Angles đ??´đ??ľđ??ś and đ??śđ??ľđ??ˇ can be called neighboring or adjacent angles. They add up to one straight line, or 180 degrees. E

228

Hence, đ?‘Ľ + đ?‘Ś = 180. Perpendicular and parallel lines

â„“đ?“

An angle that has a measure of 90°  is a right angle. â„“đ?“ If two lines intersect at right angles, the lines are perpendicular. Lines â„“đ?“ and â„“đ?“ on the right are perpendicular, denoted by â„“đ?“ ⊼ â„“đ?“ . If two lines that are in the same plane do not â„“đ?“ đ?‘ĽÂ° đ?‘ŚÂ° intersect, the two lines are parallel. đ?‘ŚÂ° đ?‘ĽÂ° Lines â„“đ?“ and â„“đ?“ on the left are parallel, denoted ° ° â„“đ?“ đ?‘Ś đ?‘Ľ by â„“đ?“ âˆĽ â„“đ?“ . ° ° đ?‘Ś If two parallel lines are intersected by a third đ?‘Ľ line, as shown on the left, than the angle measures are related as indicated, where đ?‘Ľ + ° đ?‘Ś = 180 . 1. (OG) In the figure below, the value of đ?‘Ś is 3đ?‘Ľ 2đ?‘Ľ

(A) (B) (C) (D) (E)

6° 12° 24° 36° 42°

đ?‘Ś + 30°

The symbol of degrees ° could not be with variables. You should understand from the context what do they mean in the problem.

229

230

Answers and explanations 1. (E) Clearly, 2đ?‘Ľ and 3đ?‘Ľ are adjacent angles. Thus, 2đ?‘Ľ + 3đ?‘Ľ = 180° . Therefore, 5đ?‘Ľ = 180° and đ?‘Ľ = 36° . Angles 2đ?‘Ľ and đ?‘Ś + 30° are vertical. Thus, 2đ?‘Ľ = đ?‘Ś + 30° and đ?‘Ś = 2đ?‘Ľâ€“ 30° = 2 ∙ 36° − 30° = 42° . E is the correct answer.

GoGMAT Problem Explanation

231

Triangles A triangle is a three-sided figure that contains three interior angles. The sum of interior angles is 180 degrees.

B

The longest side is opposite the largest angle. c

A

The length of any side of a triangle must be smaller than the sum of the other two sides:

a

b

đ?‘Ž Â + đ?‘? Â > Â đ?‘? Â đ?‘? Â + đ?‘? Â > Â đ?‘Ž Â Â đ?‘Ž Â + đ?‘? Â > Â đ?‘?

C

Hint: It is enough to check only one inequality for the largest side. For example, to check whether exists triangle with sides 5,6 and 7, you need only to check is 5+6 greater than 7. 1.

(OG) In ABC below, what is đ?‘Ľ in terms of đ?‘§? B 37°

đ?‘Ľ A

�

70°

C

(A) đ?‘§ + 73° (B) đ?‘§ − 73° (C) 70° − đ?‘§ (D) đ?‘§ − 70° (E) 73° − đ?‘§ 2. (MH)  If  4  and  11  are  the  lengths  of  two  sides  of  a  triangular  region,  which  of  the  following  can  be  the  length  of  the  third  side? I. 5 II. 13 III. 15 (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III The altitude (the height) of a triangle is the segment drawn from a vertex perpendicular to the side opposite that vertex. Relative to that vertex and altitude, the opposite side is called the base. Line segment đ??ľđ??ť on the right picture is the altitude of the triangle đ??´đ??ľđ??ś.  It’s  perpendicular  to  đ??´đ??ś, so đ??´đ??ś is the base. You can draw three heights in the triangle – perpendicular to each side.

B

h

A

H

C

232

B

Line segment đ??ľđ?‘€ on the left picture is the median. It means that point đ?‘€ divides đ??´đ??ś on two equal parts: đ??´đ?‘€  = đ?‘€đ??ś. You can draw three medians in the triangle.

đ?‘š

A

B

C

M

Line segment đ??ľđ??ż denotes the bisector, which means that angles đ??´đ??ľđ??ż and đ??żđ??ľđ??ś are equal. â„“đ?“ You can draw three bisectors in the triangle. Midsegment of a triangle is a A L B line segment joining the midpoints of two sides of a triangle. The midsegment is always parallel to the third side of the M N triangle. The midsegment is always half the length of the third side. On the picture shown above the midsegment đ?‘€đ?‘ âˆĽ đ??´đ??ś and C

A

C

đ?‘€đ?‘ = đ??´đ??ś.

To calculate the area of a triangle use the formulae S=

altitude Ă— base 2

where p =

S=

p(p − a)(p − b)(p − c)

or

is the semi perimeter of the triangle.

Special types of triangles. A scalene triangle is a triangle with all sides of different lengths. A

An isosceles triangle has at least two sides of the same length. The angles opposite the two equal sides are also equal. Conversely, if two angles of a triangle have the same measure, then the sides opposite those angles have the same length. In isosceles triangle altitude traced to the base is a median and a bisector.

B

C

3. (GC) B

A

đ?‘Ľ

2đ?‘Ľ D

2đ?‘Ľ

When a question involving angles has a triangle, remember that the interior angles of a triangle add up to 180 degrees. Apply that rule in every possible way you can. C

In the triangle đ??´đ??ľđ??ś above, what is the length of side đ??ľđ??ś? (1) Line segment đ??´đ??ˇ has length 6.

233

(2) đ?‘Ľ  =  36° . An equilateral triangle has all sides of equal length, all angles equal 60 degrees. The area of an equilateral triangle is S=

A

a √3 4

60

B

where đ?‘Ž is a side of the triangle. The radius of inscribed circle đ?‘&#x; and circumscribed circle đ?‘… are r=

a 2√3

R=

and

C

a √3

4. (GC) If two triangles shown in the figure above are equilateral and the radius of the outer circle equals to 3, what is the radius of the smallest circle? (A) √3 (B) 3/2 (C) 3/4 (D) 1/√3 (E) 1/2√3 A triangle that has a right angle (90 degrees) is a right triangle. The side opposite the right angle is the hypotenuse, and the other two sides are the legs.

A

c

Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. đ?‘? =đ?‘Ž +đ?‘?

b

C

a

B

where đ?‘? is the hypotenuse, đ?‘Ž and đ?‘? are the legs. Any triangle whose sides are in the ratio đ?&#x;‘: đ?&#x;’: đ?&#x;“ is a right triangle. Such triangles that have their sides in the ratio of whole numbers are called Pythagorean Triples. In general, if đ?‘Ž, đ?‘? and đ?‘? are the lengths of the sides of a triangle and đ?‘? = đ?‘Ž + đ?‘? , then the triangle is a right triangle. đ?‘? > đ?‘Ž + đ?‘? , then the triangle is an obtuse triangle. đ?‘? < đ?‘Ž + đ?‘? , then the triangle is an acute triangle. The area of a right triangle can be calculated as follows

234

S=

leg Ă— leg 2

S=

altitude Ă— hypotenuse 2

or There  are  two  special  types  of  right  triangle.  They  come  up  so  often  that  they’re  worth  memorizing. A right isosceles triangle, which angles are 45° −  45° − 90°. A The sides of such triangle are always in the ratio đ?&#x;?: đ?&#x;?: √đ?&#x;?. 45°

A

The triangle on the left is isosceles and right, so đ??´đ??ś = đ??ľđ??ś and đ??´đ??ľ = đ??ľđ??śâˆš2.

45°

C

30°

B

A right triangle, which angles are 30° − 60° − 90°: The sides of such triangle are always in the ratio đ?&#x;?: √đ?&#x;‘: đ?&#x;?. B

A

C O

.

60°

B

C

A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle.

Conversely, if a triangle is inscribed in a circle so that one of its sides is a diameter of the circle, then the triangle is a right triangle. Q

2

P

R a

b

5. (OG) If arc đ?‘ƒđ?‘„đ?‘… above is a semicircle, what is the length of diameter đ?‘ƒđ?‘… ? (1) đ?‘Ž = 4 (2) đ?‘? = 1 6. (OG)

In the figure above, segments đ?‘…đ?‘† and đ?‘‡đ?‘ˆ represent two positions of the same ladder leaning against the side đ?‘†đ?‘‰ of a wall. The length of đ?‘‡đ?‘‰ is how much greater than the length of đ?‘…đ?‘‰? (1) The length of đ?‘‡đ?‘ˆ is 10 meters (2) The length of đ?‘…đ?‘‰ is 5 meters

235

Hint: If you know two angles of a triangle, it is sufficient to determine the shape of the triangle. If you know two angles and a side, it is sufficient to determine the triangle (you can find two other sides). Congruent and similar triangles. Triangles can be called congruent (equal) if all their respective sides are equal and all their respective angles are equal. Triangles that have the same shape but the different sizes are called similar triangles. For example, any two equilateral triangles of different sizes are similar. B E

C

A

F

D

We say that two triangles đ??´đ??ľđ??ś and đ??ˇđ??¸đ??š are similar if either of the following conditions holds: 1. Corresponding sides have lengths in the same ratio: =

=

.

2. Corresponding angles are equal: ∠đ??ľđ??´đ??ś = ∠đ??¸đ??ˇđ??š, ∠đ??´đ??ľđ??ś = ∠đ??ˇđ??¸đ??š, and ∠đ??´đ??śđ??ľ = ∠đ??ˇđ??šđ??¸. When two triangles đ??´đ??ľđ??ś  and đ??ˇđ??¸đ??š  are similar, we write ∆đ??´đ??ľđ??ś~∆đ??ˇđ??¸đ??š. Three criterions for similarity: I. (AAA) If in two triangles, the corresponding angles are equal the triangles are similar. II. (SSS) If the corresponding sides of two triangles are proportional the triangles are similar. III. (SAS) If one angle of a triangle is equal to one angle of the other triangle and the sides containing these angles are proportional, the triangles are similar. The ratio of the areas of similar triangles is equal to the ratio of the squares on their corresponding sides. For triangles đ??´đ??ľđ??ś and đ??ˇđ??¸đ??š we have

=

7. What is the value of the height, â„Ž meters, in the following diagram, if the tennis ball must be hit so that it will just pass over the net and land 6 meters away from the base of the net?

(A) 1.8

236

(B) 2.2 (C) 2.7 (D) 2.4 (E) 3.6 8. (OG) In triangle đ??´đ??ľđ??ś, point đ?‘‹ is the midpoint of side đ??´đ??ś and point đ?‘Œ is the midpoint of side đ??ľđ??ś. If point đ?‘… is the midpoint of line segment đ?‘‹đ??ś and if point đ?‘† is the midpoint of line segment đ?‘Œđ??ś, what is the area of triangular region đ?‘…đ??śđ?‘†? (1) The area of triangular region đ??´đ??ľđ?‘‹ is 32. (2) The length of one of the altitudes of triangle đ??´đ??ľđ??ś is 8.

If  you  get  stuck  on  a  geometry  problem  that  doesn’t  have  triangles  in  it,  ask  yourself:  can  the  figure in this problem be divided into triangles (especially right triangles) that I can work with? This will sometimes the key to solution.

237

GoGMAT Problem

Answers and explanations (E) Since the sum of the interior angles is 180° , đ?‘Ľ  +  (đ?‘§  +  37° )  +  70°  =  180° , đ?‘Ľ  +  đ?‘§  =  180°  â€“  70°  â€“  37°  =  73° . Therefore, đ?‘Ľ  =  73°  âˆ’  đ?‘§. E is the answer. 2. (B) Let for the triangle đ?‘Ž = 4 and đ?‘? = 11 and we are looking for đ?‘?. Since the length of any side of a triangle must be smaller than the sum of the other two sides, 4 + 11 > đ?‘? and 4 + đ?‘? > 11. Hence, the third side should be less than 15 and more than 7. 13 is the only one length of side which is possible. B is the answer. 3. (A) Triangle đ??ľđ??śđ??ˇ is isosceles (since 2 of its angles are equal). Thus, đ??ľđ??ś = đ??ľđ??ˇ. Angle đ??´đ??ˇđ??ľ is equal to 180  â€“  2đ?‘Ľ since it is adjacent to the angle đ??ľđ??ˇđ??ś. The sum of all angles of triangle đ??´đ??ľđ??ˇ is equal to 180: 180 = đ??´đ??ľđ??ˇ + 180– 2đ?‘Ľ + đ?‘Ľ, so, angle đ??´đ??ľđ??ˇ is equal to đ?‘Ľ. This implies that triangle đ??´đ??ľđ??ˇ  is isosceles with angle đ??ľđ??´đ??ˇ equal to angle đ??´đ??ľđ??ˇ. Having this, we can find that 1.

238

đ??´đ??ˇ = đ??ľđ??ˇ = đ??ľđ??ś. Statement (1) is sufficient to answer the stem question since đ??ľđ??ś = đ??´đ??ˇ = 6. Thus, the answer must be A or D. Statement (2) is insufficient to determine any length since only angles are given. Thus, A is the answer. 4. (C) For an equilateral triangle radius of inscribed circle is r = and of circumscribed circle √

is R =

√

, so, đ?‘&#x;  = đ?‘…/2. Let đ?‘… be the radius of the outer circle, đ?‘&#x; – the radius of the middle

circle and đ?‘&#x; ∗ - the radius of the smallest circle. We have đ?‘&#x; = đ?‘…/2, đ?‘&#x; ∗= đ?‘&#x;/2 = đ?‘…/4 = 3/4. Thus, the answer is C. 5. (D) Since angle đ?‘ƒđ?‘„đ?‘… isinscribed in a semicircle, it is a right angle, and â–ł đ?‘ƒđ?‘„đ?‘…is a right triangle. â–ł đ?‘ƒđ?‘„đ?‘… is divided into two right triangles by the vertical line from đ?‘„ to side đ?‘ƒđ?‘…. Let đ?‘Ľ = đ?‘ƒđ?‘„ and đ?‘Ś = đ?‘„đ?‘…. The larger right triangle has hypotenuse đ?‘Ľ, so đ?‘Ľ = 4 + đ?‘Ž ; the smaller right triangle has hypotenuse đ?‘Ś, so đ?‘Ś = 4 + đ?‘? . From â–ł đ?‘ƒđ?‘„đ?‘…, (đ?‘Ž + đ?‘?) = đ?‘Ľ + đ?‘Ś , so by substitution, (đ?‘Ž + đ?‘?) = 4 + đ?‘Ž + 4 + đ?‘? , and by simplification, đ?‘Ž + 2đ?‘Žđ?‘? + đ?‘? = 8 + đ?‘Ž + đ?‘?  or 2đ?‘Žđ?‘? = 8 or đ?‘Žđ?‘? = 4. (1) If đ?‘Ž = 4 is substituted in đ?‘Žđ?‘? = 4,then đ?‘? must be 1 and diameter đ?‘ƒđ?‘… is 5; SUFFICIENT. (2) If đ?‘? = 1 is substituted in đ?‘Žđ?‘? = 4, then đ?‘Ž must be 4 and diameter đ?‘ƒđ?‘… is 5; SUFFICIENT 6. (D) Since we know all angles of đ?‘‡đ?‘ˆđ?‘‰: 45, 45, 90 and đ?‘†đ?‘…đ?‘‰: 30, 60, 90 and also that đ?‘†đ?‘… = đ?‘‡đ?‘ˆ (as the same ladder), we need to know the length of any side of these two triangles. If we have such information we can determine the lengths of the other sides, because for these two triangles we have ratios of the sides: 1: 1: √2  and 1: √3: 2 respectively. So, having đ?‘‡đ?‘ˆ we can find đ?‘‡đ?‘‰. And as đ?‘†đ?‘… = đ?‘‡đ?‘ˆ we can find đ?‘…đ?‘‰. Similarly, having the length of đ?‘…đ?‘‰, we can determine all lengths in this triangle. And đ?‘†đ?‘… = đ?‘‡đ?‘ˆ will give us information about đ?‘‡đ?‘‰. Thus, each statement is sufficient to answer the stem question. Therefore, D is the best answer. 7. (C) We draw and label diagram as shown.

Let’s  consider  ,  âˆ†đ??´đ??ˇđ??¸ and ∆đ??´đ??ľđ??ś. Since ∠đ??´đ??ˇđ??¸ = ∠đ??´đ??ľđ??ś as right angles, and ∠đ??´ is common, ∆đ??´đ??ˇđ??¸ and ∆đ??´đ??ľđ??ś are similar. Therefore

=

, and

.

=

, and â„Ž = 2.7.

So, the height at which the ball should be hit is 2.7 meters. C is an answer. 8. (A)

239

B

Y S

A

X

R

C

As shown in the figure above, đ?‘‹ and đ?‘Œ are the midpoints of đ??´đ??ś and đ??ľđ??ś, respectively, of đ?›Ľđ??´đ??ľđ??ś, and đ?‘… and đ?‘† are the midpoints of đ?‘‹đ??ś and đ?‘Œđ??ś, respectively. đ?‘‹đ?‘Œ and đ?‘…đ?‘† are midsegments of triangles đ??´đ??ľđ??ś and đ?‘‹đ?‘Œđ??ś respectively. Thus, đ?‘…đ?‘† âˆĽ đ?‘‹đ?‘Œ âˆĽ đ??´đ??ľ and đ?‘…đ?‘† = đ?‘‹đ?‘Œ = đ??´đ??ľ. Also, đ?›Ľđ??´đ??ľđ??ś and đ?›Ľđ?‘…đ?‘†đ??ś are similar triangles, since their corresponding interior angles have the same measure (AAA criterion). Thus,

=

=

, and the area of ∆đ?‘…đ?‘†đ??ś can

be determined exactly when the value of the area of ∆đ??´đ??śB can be determined. (1) Given that the area of ∆đ??´đ??ľđ?‘‹, which is

đ??´đ?‘‹ ∙ đ?‘Žđ?‘™đ?‘Ąđ?‘–đ?‘Ąđ?‘˘đ?‘‘đ?‘’  đ?‘œđ?‘“  đ??´đ??śđ??ľ = đ?‘†

, is 32. It is

sufficient to answer the question. (2) Without knowing the length of the side to which the altitude is drawn, the area of đ?›Ľđ??´đ??śđ??ľ, cannot be determined; NOT suďŹƒcient. The correct answer is A. 9. (E) Statement  1  is  NOT  sufficient.  Let’s  look  at  the  right  triangles  and  pick  numbers:  if  area  of  ABC is 1 the sides are 1 and 2. The hypotenuse is √3, Perimeter is 3+√3. Perimeter is much greater than the area of a triangle with these values. However if sides of ABC are 10, 10, and 10√2; then perimeter is 20+10√2 and area is 50. Perimeter is much smaller than the area. Insufficient. Statement 2 is insufficient as well. All it tells us is that both triangles are similar or proportionate to each other, but nothing about their size. Combining the two statements, we still cannot determine whether the triangles have small values of their sides that yield greater perimeters or large values that yield greater area measurements. Not sufficient. Therefore E is the answer.

240

GoGMAT Problem Explanation

241

Quadrilaterals GoGMAT, Session 11 A polygon with four sides is a quadrilateral. Sum of interior angles of any quadrilateral is 360 Â degrees.

A

B

Properties of a parallelogram: Opposite sides are equal in length. Opposite angles are equal in measure. The diagonals bisect each other.

O C

D

Parallelogram is a quadrilateral with opposite sides parallel and congruent.

A

B

The area of a parallelogram is â„Ž

S=hĂ—b

, where đ?‘? is the base of the parallelogram and â„Ž is its height.

D

đ?‘?

C

1. (GC) The area of a parallelogram is 100. What is the perimeter of the parallelogram? (1) The base of the parallelogram is 10. (2) One of the angles of the parallelogram is 45 degrees Rhombus is parallelogram, in which all sides are equal. A rhombus therefore has all of the properties of a parallelogram. Properties of a rhombus: B All sides are equal in length. Opposite angles are equal in measure. The diagonals bisect each other at 90°. A C There are two ways to find the area of a rhombus: D

S=

S = altitude Ă— base

or

diagonal Ă— diagonal 2

.

2. (GC) Is quadrilateral đ??´đ??ľđ??śđ??ˇ a rhombus? (1) đ??´đ??ś is perpendicular to đ??ľđ??ˇ. (2) đ??´đ??ľ + đ??śđ??ˇ = đ??ľđ??ś + đ??´đ??ˇ. Rectangle is a parallelogram with right angles. A rectangle therefore has all of the properties of a parallelogram. Properties of a rectangle: A B Opposite sides are parallel and congruent. The diagonals bisect each other. The diagonals are congruent. C The area of a rectangle is D

242

S = length Ă— width

.

3. (MH) The diagonal of the rectangle is 13. What is the area of the rectangle? (1) The length of the rectangle is 12 (2) The width of the rectangle is 5 Square is a rectangle with all sides of equal length, or is a rhombus with right angles. Properties of a square: A B Each diagonal of a square is the perpendicular bisector of the other. Radius of inscribed circle is r = Radius of circumscribed circles is R = D

C

√

The area of a square is S=a

S= or

d 2

where đ?‘Ž is side and đ?‘‘ is diagonal of a square. 4. (GC) If the area of square A is three times the area of square B, what is the ratio of the diagonal of square B to that of square A? (A) (B) (C) 3 (D) 3 (E) 3

a

B

Trapezoid is a quadrilateral which has one pair of parallel sides. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs.

C

h

A

b

D

243

The usual way to calculate the area of a trapezoid is the average base length times altitude S=hĂ—

a+b 2

.

5. (MR) A

6 cm

B

8 cm

D

C

What is the area of the trapezoid pictured above? (1) ∠đ??´ = 120 degrees (2) The perimeter of trapezoid đ??´đ??ľđ??śđ??ˇ = 36.

244

GoGMAT Problem

Answers and explanations 1. (C) From Statement (1) we know the base and the height of the parallelogram. This height can equal the side of the parallelogram (in case of a rectangle) but it can also exceed it. Thus, Statement (1) is not sufficient. From Statement (2) nothing can be established about the length of the base of the parallelogram. In fact, the base can be arbitrarily long. Statement (1) and Statement (2) completely define the B A parallelogram: đ??ˇđ??ś = đ??´đ??ľ = đ??ľđ??ˇ = 10, the length of the sides đ??ľđ??ś and đ??´đ??ˇ can be 10 found by Pythagorean theorem. D

10

C

So, the right answer is C. 2. (E) đ??´đ??ľđ??śđ??ˇ can be a rhombus but it can also be a kite-shaped figure. The diagonals of this figure form a cross and Statement (2) holds because of the symmetry (đ??´đ??ľ = đ??ľđ??ś and đ??śđ??ˇ = đ??´đ??ˇ). The answer is E. 3. (D) Statement (1): knowing the diagonal and the length we can find the width. So, Statement (1) is sufficient. Statement (2): knowing the diagonal and the width we can find the length. So, Statement (2) is sufficient. The answer is D.

245

4. (A) The ratio of the diagonals is the same as the ratio of the sides. If the side of B is đ?‘?, then the side of A must be √3đ?‘?. The required ratio is

√

=

√

=

. We answer A.

5. (D) At  first  let’s  draw  a  picture: To calculate the area of this trapezoid we need đ??śđ??ˇ, đ??´đ??ľ 6 cm A B and đ??´đ??ť. đ??´đ??ľ = 6 by definition, so, the question is to find đ??śđ??ˇ and đ??´đ??ť. Here đ??´đ??ľđ?‘€đ??ť is a rectangle, so đ??ťđ?‘€ = 8 cm đ??´đ??ľ = 6. From statement 1 we can get that angle đ??ś equals 60 degrees. Then we can find đ??śđ??ť and đ??´đ??ť from the D C H M triangle đ??´đ??śđ??ť (as its angles are 30, 60, and 90 its sides are always in the ratio 1: √3: 2). đ??śđ??ť = đ?‘€đ??ˇ,  so,  we’ll  find  đ??śđ??ˇ as đ??śđ??ť + đ??ťđ?‘€ + đ?‘€đ??ˇ = 2đ??śđ??ť + 6. We can answer the question. Statement 1 is sufficient. From statement 2 we can get đ??śđ??ˇ = 36 − 8 − 8 − 6 = 14. đ??ťđ?‘€ = đ??´đ??ľ = 6, so, đ??śđ??ť + đ?‘€đ??ˇ = 14 − 6 = 8. These line segments are equal: đ??śđ??ť = đ?‘€đ??ˇ = 4. Using Pythagorean theorem for the triangle đ??´đ??śđ??ť we’ll  find  đ??´đ??ť. Statement 2 also is sufficient. The answer is D.

GoGMAT Problem Explanation

246

Polygons GoGMAT, Session 11 A polygon is a closed plane figure with three or more sides that are all straight. Polygons are named according to the number of sides. The names of the most common polygons are given below:

đ?‘†đ?‘˘đ?‘š  đ?‘œđ?‘“  đ?‘–đ?‘›đ?‘Ąđ?‘’đ?‘&#x;đ?‘–đ?‘œđ?‘&#x;  đ?‘Žđ?‘›đ?‘”đ?‘™đ?‘’đ?‘  =  180 Ă— (đ?‘› − 2) đ??¸đ?‘Žđ?‘?â„Ž  đ?‘Žđ?‘›đ?‘”đ?‘™đ?‘’  (đ?‘œđ?‘“  đ?‘&#x;đ?‘’đ?‘”đ?‘˘đ?‘™đ?‘Žđ?‘&#x;  đ?‘?đ?‘œđ?‘™đ?‘Śđ?‘”đ?‘œđ?‘›)  = Â

180 Ă— (đ?‘› − 2) đ?‘› ,

where đ?‘› is the number of sides. For example, for a regular heptagon the sum of interior angles is 180(7 − 2) = 900 degrees and each angle of it is

= 128  degrees. đ?‘Śâˆ˜ đ?‘Ľâˆ˜

đ?‘§âˆ˜ đ?‘Łâˆ˜

đ?‘¤âˆ˜

1. (OG) In the figure shown, what is the value of � + � + � + � + �? (A) 45 (B) 90 (C) 180 (D) 270 (E) 360

247

GoGMAT Problem

248

Explanations 1. (C) In the following figure, the center section of the star is a pentagon. The sum of the interior angles of any polygon is 180(đ?‘› − 2),where đ?‘› is the number of sides. Thus, đ?‘Ž + đ?‘? + đ?‘? + đ?‘‘ + đ?‘’ = 180(5 − 2) = 180(3) = 540, where đ?‘Ž, đ?‘?, đ?‘?, đ?‘‘, đ?‘’ are interior angles of the pentagon. Each of the interior angles of the pentagon defines a triangle with two of the angles at the points of the star. This gives the following five equations: đ?‘Ž + đ?‘Ľ + đ?‘§ = 180, đ?‘? + đ?‘Ł + đ?‘Ś = 180, đ?‘? + đ?‘Ľ + đ?‘¤ = 180, đ?‘‘ + đ?‘Ł + đ?‘§ = 180, đ?‘’ + đ?‘Ś + đ?‘¤ = 180. Summing these 5 equations gives: đ?‘Ž + đ?‘? + đ?‘? + đ?‘‘ + đ?‘’ + 2đ?‘Ł + 2đ?‘Ľ + 2đ?‘Ś + 2đ?‘§ + 2đ?‘¤ = 900 Substituting 540 for đ?‘Ž + đ?‘? + đ?‘? + đ?‘‘ + đ?‘’ gives: 540 + 2đ?‘Ł + 2đ?‘Ľ + 2đ?‘Ś + 2đ?‘§ + 2đ?‘¤ = 900 From this: đ?‘Ł  +  đ?‘Ľ  +  đ?‘Ś  +  đ?‘§  +  đ?‘¤  = 180 GoGMAT Problem Explanation

249

Circle GoGMAT, Session 11 A circle is a set of points in a plane that are all located the same distance from a fixed point (the center of the circle). The number of degrees of arc in a circle (or the number of C degrees in a complete revolution) is 360 degrees. A chord of a circle is a line segment that has its endpoints on the circle. On the right diagram đ??śđ??ˇ and đ??´đ??ľ are chords. A O A chord that passes through the center of the circle is a diameter of the circle. A radius of a circle is a segment from the center of the circle to a point on the circle. The words  â€œdiameterâ€?  and  â€œradiusâ€?  are  also  used  to  refer  to the lengths of these segments. On the right diagrams đ??´đ??ľ is a diameter, đ??´đ?‘‚ and đ?‘‚đ??ľ are radii.

D B

The circumference of a circle is the distance around the circle. If đ?‘&#x; is the radius of the circle, then the circumference is equal to đ??ś = 2đ?œ‹đ?‘&#x;

where  Ď€  is  approximately 3.14 or

,

, and the area of a circle is equal to đ?‘† = đ?œ‹đ?‘&#x;

. An inscribed angle is an angle đ??´đ??ľđ??ś formed by points đ??´, đ??ľ, and đ??ś on a circle. A central angle is an angle đ??´đ?‘‚đ??ś with endpoints đ??´ and đ??ś located on a circle and vertex đ?‘‚ located at the circle's center. C

C

O

B

A ∠đ??´đ??ľđ??ś is an inscribed angle

B

O A ∠đ??´đ?‘‚đ??ś is a central angle

C

The Central Angle Theorem states: O

the measure of inscribed angle is always half the measure of the central angle.

B

A ∠đ??´đ?‘‚đ??ś = 2∠đ??śđ??ľđ??´

250

A central angle in a circle determines an arc đ??´đ??ś. A circle sector is the portion of a circle enclosed by two radii and an arc Area of a sector is Îą C S = Ď€r 360 , where Îą is the central angle. đ?›ź An arc length is Îą l = 2Ď€R 360 A , where Îą is the central angle of the arc.

O

A line that has exactly one point in common with a circle is said to be tangent to the circle, and that common point is called the point of tangency. A radius or diameter with an endpoint at the point of tangency is perpendicular to the tangent line, and, conversely, a line that is perpendicular to a diameter at one of its endpoints is tangent to the circle at that endpoint.

B

â„“đ?“

Î&#x;

T

The line â„“đ?“ on the right is tangent to the circle and radius đ?‘‚đ?‘‡ is perpendicular to â„“đ?“ . 1. (OG)

8 feet

Î&#x;

The figure above represents a circular flower bed, with its center đ?‘‚, surrounded by a circular path that is 3 feet wide. What is the area of the path, in square feet? (A) 25đ?œ‹ (B) 38đ?œ‹ (C) 55đ?œ‹ (D) 57đ?œ‹ (E) 64đ?œ‹

251

2. (OG)

If đ?‘‚ is the center of the circle above, what fraction of the circular region is shaded? (A) (B) (C) (D) (E)

3. The circle above circumscribes square đ??´đ??ľđ??śđ??ˇ. What is the value of đ?‘¤ + đ?‘Ľ + đ?‘Ś + đ?‘§? (A) 180° (B) 135° (C) 120° (D) 100° (E) 90° For questions involving circles watch out for inscribed angles, central angles and tangent lines. Also, since the radii of a circle are always the same length, be sure to look for isosceles triangles hiding within the figure.

252

GoGMAT Problem

253

Answers and explanations 1. (D) Clearly, total area of the shaded region is the difference between areas of two circles. The area of the outer is 112 Ď€  and  the  area  of  the  inner  is  82Ď€.  So,  the  area  of  the  path  is  Ď€  Ă—  (112 82)  =  57Ď€.  Thus,  D  is  the  answer. 2. (C) Each of the vertical angles is equal to 30 degrees. Thus, two angles have a share of 60/360 = 1/6 of the entire circle. The right answer is C. 3. (E) The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. So the central angle which subtends the minor arc đ??ľđ??ś equals to twice the angle đ?‘Ľ + đ?‘§ (as inscribed angle đ?‘Ľ plus inscribed angle đ?‘§ subtend the same minor arc đ??ľđ??ś). Now, minor arc đ??ľđ??ś is of the circumference hence the central angle which subtends it equals to

= 90, so 90 = 2(� + �), and � + � = 45. Similarly 90 = 2(� + �) (for minor

arc đ??´đ??ˇ) and đ?‘¤ + đ?‘Ś = 45; đ?‘Ľ + đ?‘§ + đ?‘¤ + đ?‘Ś = 90. The answer is E.

254

GoGMAT Problem Explanation

255

Volume geometry GoGMAT, Session 11

đ?‘‘

Rectangular solid is a three-dimensional figure formed by 6 rectangular surfaces, as shown on the right. Each rectangular surface is a face. Each solid or dotted line segment is an edge, and each point at which the edges meet is a vertex. A rectangular solid has 6 faces, 12 edges, and 8 vertices. Opposite faces are parallel rectangles that have the same dimensions. Volume of a rectangular solid is

đ?‘? đ?‘Ž

đ?‘?

V = abc

, Where đ?‘Ž, đ?‘? and đ?‘? are the width, length and height of rectangular solid, respectively. Surface area of a rectangular solid is S = 2(ab + bc + ac) Volume diagonal is the line that go from a vertex of the rectangular solid, through the center of rectangular solid, to the opposite vertex. The length of the volume diagonal is đ?‘‘=

đ?‘Ž +đ?‘? +đ?‘?

Cube is the rectangular solid with equal edges. Volume of a cube is V=a Surface area of rectangular solid is

,

đ?‘‘

S = 6a The length of the volume diagonal is

đ?‘Ž đ?‘Ž

đ?‘Ž

đ?‘‘ = đ?‘Žâˆš3 1. (GC) What is the maximum straight line distance between any two points of the box 10 Ă— 8 Ă— 6? (A) 10 (B) 2√41 (C) 2√34 (D) 8 (E) 10√2

256

Sphere consisting of those points in a three-dimensional space which are equidistant from a given point called the center.

Volume of a sphere is 4 V = πR 3

,

where đ?‘… is the radius of the sphere. Surface area of a sphere is S = 4Ď€R

.

2. (PR) Find the volume of the sphere inscribed in the cube with edge 6. (A) 12Ď€ (B) 16Ď€ (C) Â 24Ď€ (D) 36Ď€ (E) 42Ď€ Cylinder is geometric shape, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The two bases are circles of the same size and altitude is perpendicular to the bases. Volume of a cylinder is â„Ž

đ?‘&#x;

V = πr h

, where đ?‘&#x; is the radius of the base and â„Ž is the altitude of the cylinder. Surface area of a cylinder is S = 2Ď€rh + 2Ď€r

3. (PR) Cylindrical tank with height 4 and radius of the base 3 is half full of water. Find the volume of this water. (A) 6  π (B) 12  đ?œ‹ (C) 18  đ?œ‹ (D) 24  đ?œ‹

257

(E) 36  đ?œ‹ A right circular cone is a cone whose axis is a line segment joining the vertex to the midpoint of the circular base. Volume of a cone is 1 V = Ď€r h 3

â„“đ?“ â„Ž

, where đ?‘&#x; is the radius of the base and â„Ž is the altitude of the cylinder. đ?‘&#x;

4. A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere? (A) √3: 1 (B) 1: 1 (C) : 1 (D) √2: 1 (E) 2: 1

258

GoGMAT Problem

Answers and explanations 1. (E) The maximum possible straight line distance between any two points of the rectangular solid is its volume diagonal. So, đ?‘‘ = √đ?‘Ž + đ?‘? + đ?‘? = √10 + 8 + 6 = √200 = 10√2. So, the answer is E. 2. (D) For the cube with the side đ?‘Ž radius of inscribed sphere equals . So, for our cube radius of inscribed sphere is 3. The volume of the sphere can be found as V = Ď€R = Ď€3 = 36Ď€. The answer is D. 3. (C) As the tank is half full of water, we need to find the volume of the cylinder with height 2 and radius of the base 3. V = Ď€r h = Ď€3 2 = 18Ď€. The answer is C. 4. (B)As the diagram below shows, the height of the cone will be the radius of the hemisphere, so the ratio is 1:1. The answer is B.

259

GoGMAT Problem Explanation

260

Coordinate geometry GoGMAT, Session 12 The Â ďŹ gure  on  the  left  shows  the  (rectangular)  coordinate plane. The horizontal line is called the x-axis and the perpendicular vertical line is called the y-axis. The point at which these two axes intersect, designated O, is called the origin. The axes divide the plane into four quadrants, I, II, III, and IV, as shown. Each point in the plane has an đ?‘Ľ-coordinate and a đ?‘Ś-coordinate.  A  point  is  identiďŹ ed  by  an  ordered  pair (đ?‘Ľ, đ?‘Ś) of numbers in which the đ?‘Ľ-coordinate  is  the Â ďŹ rst  number  and  the  đ?‘Ś-coordinate is the second number. To Â ďŹ nd  the  distance  between  two  points  we  can  use  the  Pythagorean  theorem.

Given coordinates of two points (đ?‘Ľ , đ?‘Ś ) and (đ?‘Ľ , đ?‘Ś ), distance đ?‘‘ between two points is given by: đ?‘‘=

(đ?‘Ľ − đ?‘Ľ ) + (đ?‘Ś − đ?‘Ś )

The coordinates of the midpoint (đ?‘Ľ , đ?‘Ś ) of a line segment with endpoints (đ?‘Ľ , đ?‘Ś ) and (đ?‘Ľ , đ?‘Ś ) are the average of the coordinates of its endpoints đ?‘Ľ =

đ?‘Ľ1 +đ?‘Ľ2 2

and

đ?‘Ś =

đ?‘Ś1 +đ?‘Ś2 2

:

1. (MH) Bob lives in the town A that has point coordinates (1,5). His friend Paul lives in the city B that has point coordinates (−5, −3). Find the straight line distance between two friends. (A) 6 (B) 8 (C) 10 (D) 12 (E) 14

261

Straight line Any straight line on this plane can be described by the equation đ?‘Ś = đ?‘˜đ?‘Ľ + đ?‘?

, where đ?‘? is the đ?‘Ś-intercept (the point at which the line crosses the đ?‘Ś-axis) and đ?‘˜ is the slope of the line. The đ?‘Ľ-intercept is the đ?‘Ľ-coordinate of the point at which the line intersects the đ?‘Ľ-axis. The đ?‘Ľintercept can be found by setting đ?‘Ś = 0 and solving for đ?‘Ľ. Given two points (đ?‘Ľ , đ?‘Ś ) and (đ?‘Ľ , đ?‘Ś ) on a line, the slope đ?‘˜ of the line is: y −y slope = k = x −x Given any two points (đ?‘Ľ , đ?‘Ś ) and (đ?‘Ľ , đ?‘Ś ) with đ?‘Ľ ≠đ?‘Ľ , the equation of the line passing through these points can be found by formula: đ?‘Ľâˆ’đ?‘Ľ đ?‘Śâˆ’đ?‘Ś = đ?‘Ľ −đ?‘Ľ đ?‘Ś −đ?‘Ś

If the slope of a line is negative, the line slants downward from left to right; if the slope is positive, the line slants upward. If the slope is 0, the line is horizontal; the equation of such a line is of the form đ?‘Ś = đ?‘?, since đ?‘˜ = 0.  For  a  vertical  line,  slope  is  not  deďŹ ned,  and  the  equation  is  of  the  form  đ?‘Ľ = đ?‘Ž, where đ?‘Ž is the đ?‘Ľ-intercept. đ?‘Ś

đ?‘Ś

Î&#x;

đ?‘Ś

Î&#x;

đ?‘Ľ

đ?‘Ľ

Î&#x;

đ?‘˜<0

đ?‘˜>0

đ?‘Ś

đ?‘Ľ

đ?‘˜=0

đ?‘Ś

Î&#x;

đ?‘˜=1

đ?‘Ľ

Î&#x;

đ?‘Ľ

đ?‘˜ is not defined

Suppose we have two lines that are defined by equations đ?‘Ś = đ?‘˜ đ?‘Ľ + đ?‘? and đ?‘Ś = đ?‘˜ đ?‘Ľ + đ?‘? . If two lines are parallel then their slopes are equal: đ?‘˜ = đ?‘˜ and đ?‘? ≠đ?‘? . If two lines are coincident they are not parallel: đ?‘˜ = đ?‘˜ and đ?‘? = đ?‘? . If two lines are perpendicular then product of their slopes is -1: đ?‘˜ Ă— đ?‘˜ = −1

262

2. (GC) In the Cartesian coordinate system, Point A has coordinates (6, - 7) and Point B has coordinates (4, 5). If a line is drawn to connect Point A and Point B, does point C lie on line AB? (1) Coordinates of Point C are ( 5, - 1) (2) Point C equidistant from Point A and Point B. Circle on a plane đ?‘Ś In a Cartesian coordinate system, the circle with center (đ?‘Ž, đ?‘?) (đ?‘Ľ, đ?‘Ś) and radius đ?‘&#x; is the set of all points (đ?‘Ľ, đ?‘Ś) such that: (đ?‘Ž, đ?‘?)

(x − a) + (y − b) = r . Î&#x;

đ?‘&#x; đ?‘Ľ

3. (GC) If a circle is defined by an equation (đ?‘Ľ − 3) + (đ?‘Ś − 3) = 6 on the coordinate plane, and there is a square inscribed into the circle cutting it into 5 regions, what is the area of one of the smallest? (A) đ?œ‹ (B) 6đ?œ‹ − √12 (C) 6đ?œ‹ − 2√6 (D) (E)

3

−1 −

GoGMAT Problem

263

Answers and explanations 1. (C) We have to find the distance between two points A and B. Using formulae: AB= 2.

1 − (−5)

+ 5 − (−3)

=

(1 + 5) + (5 + 3) = √36 + 64 = √100 = 10.

(A) The fastest way is to see that we have all the info we need to set up the equation of the line and we can easily check whether C lies on it because Statement 1 gives us that information.  We  don’t  need  to  spend  time  on  calculations.  (1)  is  thus  sufficient. Here is the detailed explanation: đ?‘ đ?‘™đ?‘œđ?‘?đ?‘’ = = (5 + 7)/(4 −  6) = 12/−  2 = −  6. Equation of line:

đ?‘Ś  âˆ’  đ?‘Ś = đ?‘š  (  đ?‘Ľ  âˆ’  đ?‘Ľ ) đ?‘Ś  âˆ’  5 = −  6  (  đ?‘Ľ  âˆ’  4) đ?‘Ś = −  6đ?‘Ľ  +  29 Now ,we know that coordinates of C are ( 5, - 1). for C to lie on AB, (5, - 1) should be one of the solutions for the equation. lets plug in đ?‘Ľ = 5 , we get , đ?‘Ś = −  6  (5) + 29 = −1 So, (1) is sufficient. From  Statement  2,  we  can’t  say  whether  the  point  C  is  on  Line  A.  There  are  many  possibilities. So, (2) is insufficient. The answer is A. 3. (D) To find the answer, we need to know the areas of the square and the circle. From the equation we find that đ?‘&#x; = 6. We  will  get  6Ď€  as  the  area  of  the  circle  and  12  as  the  area  of  the  square (diagonal is 2√6 and by squaring it and dividing by 2, we will get values of the sides: √12).  However,  CD  is  one  of  the  four  outside  segments,  so  the  answer  will  be  (6  Ď€  - 12)/4 or 3/2  Ď€  â€“ 3  or  3(Ď€  /2  â€“ 1). The answer is D. GoGMAT Problem Explanation

264

Home assignment

Lines and angles 1. (MG  E)  If  point  đ?‘‹  is  on  line  segment  đ??´đ??ľ,  all  of  the  following  may  be  true  EXCEPT (A) đ??´đ?‘‹ = đ?‘‹đ??ľ (B) đ??´đ?‘‹ > đ?‘‹đ??ľ (C) đ??´đ?‘‹ < đ?‘‹đ??ľ (D) đ??´đ??ľ > đ?‘‹đ??ľ (E) đ??´đ?‘‹ + đ?‘‹đ??ľ < đ??´đ??ľ

Triangles

1. (OG Â E)

In the triangle above, does đ?‘Ž + đ?‘? = đ?‘? ? (1) đ?‘Ľ + đ?‘Ś = 90 (2) đ?‘Ľ = đ?‘Ś 2. (OG Â H) Â

� � �

The shaded portion of the rectangular lot shown above represents a flower bed. If the area of the bed is 24 square yards and � = � + 2, then � equals (A) (B) (C) (D) (E)

√13 2√13 6 8 10

3. (OG  M)  Starting  from  Town  S,  Fred  rode  his  bicycle  8  miles  due  east,  3  miles  due  south,  2  miles  due  west,  and  11  miles  due  north,  finally,  stopping  at  Town  T.  If  the  entire  region  is  flat,  what  is  the  straight-line  distance  between  Towns  S  and  T? (A) 10 (B) 8√2 (C) √157

265

(D) 14 (E) 24 4. (OG Â M)

In the figure above, V represents an observation point at one end of a pool. From V, an object that is actually located on the bottom of the pool at point R appears to be at point S. If đ?‘‰đ?‘… = 10 feet, what is the distance RS, in feet, between the actual position and the perceived position of the object? (A) 10 − 5√3 (B) 10 − 5√2 (C) 2 (D) (E) 4 5. (MG  M)  Peter  lives  12  miles  west  of  school  and  Bill  lives  north  of  the  school.  Peter  finds  that  the  direct  distance  from  his  house  to  Bill’s  is  6  miles  shorter  than  the  distance  by  way  of  school.  How  many  miles  north  of  the  school  does  Bill  live? (A) 6 (B) 9 (C) 10 (D) 12 (E) None of these 6. (GC  M)  Right  triangle  ABC  has  a  height  BD.  What  is  the  value  of  AB  times  BC? (1) AB is equal to 6 (2) The product of the non - hypotenuse sides is equal to 24 7. (MH  E)  Is  the  triangle  with  angles  đ?‘Ž,  đ?‘?,  đ?‘?  isosceles? (1)180 = (đ?‘Ž + đ?‘?) + 60 (2) đ?‘Ž < 2đ?‘? 8.  (MG  M)  If  a  triangle  of  base  6  has  the  same  area  as  a  circle  of  radius  6,  what  is  the  altitude  of  the  triangle? (A) 6đ?œ‹ (B) 8đ?œ‹ (C) 10đ?œ‹ (D) 12đ?œ‹ (E) 14đ?œ‹

266

9. (MG  E)  The  vertex  angle  of  an  isosceles  triangle  is  p  degrees.  How  many  degrees  are  there  in  one  of  the  base  angles? (A) 180– đ?‘? (B) 90– đ?‘? (C) 180– 2đ?‘? (D) 180– (E)

90–

10. (MG  E)  In  isosceles  triangle  ABC,  what  is  the  value  of  âˆ C? (1) The measure of ∠B is 42 degrees (2) The measure of ∠A is 96 degrees B

A

C

11. In  the  figure  above,  a  triangle  đ??´đ??ľđ??ś  is  a  right.  An  angle  đ??´  is  30  degrees  and  đ??ľđ??ś = 1.  What  is  the  radius  of  the  circle? (A) 1 (B)

√

(C)

√

(D) √2 (E) √3

12.  In  the  figure  to  the  right,  equilateral  triangle  đ??´đ??ľđ??ś  is  inscribed  in  the  circle.  If  the  radius  of  the  circle  is  1,  what  is  the  approximate  the  area  of  the  triangle?  B (A) 1.2 (B) 1.3 (C) 1.4 (D) 1.5 (E) 1.6 A

C

267

Quadrilaterals

1. (MH  M)  A  rectangle  is  equal  in  area  to  a  square  with  sides  of  length  12.  Is  the  diagonal  of  the  rectangle  greater  in  length  than  20? (1) The rectangle has a length of 16. (2) The rectangle has a width of 9. 2. (GC  M)  Into  a  square  with  side  đ??ž  is  inscribed  a  circle  with  radius  đ?‘&#x;.  If  the  ratio  of  area  of  square  to  the  area  of  circle  is  đ?‘ƒ  and  the  ratio  of  perimeter  of  the  square  to  that  of  the  circle  is  đ?‘„.  Which  of  the  following  must  be  true? (A)

>1

(B)

=1

(C)

<

(D)

=

(E)

>

<1

3. (PR  M)  If  rectangle  A  has  width  đ?‘¤  and  length  đ?‘›,  and  đ?‘¤ < đ?‘›  ,  what  is  the  value  of  đ?‘¤? (1) The area of rectangle A is equal to 24. (2) The perimeter of rectangle A is equal to 20. 4. (MG  M)  A  spotlight  on  the  ceiling  is  5  feet  from  one  wall  of  a  room  and  10  feet  from  the  wall  at  right  angles  to  it.  How  many  feet  is  it  from  the  intersection  of  the  two  walls? (A) 15 (B) 5√2 (C) (D) (E)

5√5 10√2 10 √5

5. (MH  E)  What  is  the  area  of  square  floor  X? (1) The perimeter of the floor is a whole-number multiple of 10. (2) The diagonal of the floor measures 5√2 . 6. (GC  M  )To  build  a  rectangular  chicken  pen,  Mike  has  40  meters  of  net.  If  Mike  wants  to  maximize  the  area  of  the  pan,  what  will  be  the  most  favorable  dimensions? (A) 12 Ă— 8 (B) 15 Ă— 8 (C) 10 Ă— 10 (D) 15 Ă— 15 (E) 15 Ă— 5

268

Circles

1. (GC  E)  In  the  figure  below,  if  isosceles  right  triangle  PQR  has  an  area  of  4  and  P  is  the  centre  of  the  circle,  what  is  the  area  of  the  shaded  portion  of  the  figure?

(A) (B) (C) (D) (E)

Ď€ 2Ď€ 2âˆšĎ€ 4Ď€ 8Ď€

2. (GC  M)  In  a  circle  with  centre  đ?‘‹  đ?‘‹đ??ľ  is  the  radius.  There  is  a  chord  đ??´đ??ś  which  intersects  đ?‘‹đ??ľ.  đ??ˇ  is  the  point  of  intersection  between  đ?‘‹đ??ľ  and  đ??´đ??ś.  đ??ľđ??ˇ = 2,  đ??´đ??ś = 12,  đ?‘‹đ??ˇđ??´ = 90  degrees.  What  is  the  circle's  area? (A) 10đ?œ‹ (B) 20đ?œ‹ (C) 50đ?œ‹ (D) 80đ?œ‹ (E) 100đ?œ‹ 3. (OG  M)  What  is  the  number  of  360-degree  rotations  that  a  bicycle  wheel  made  while  rolling  100  meters  in  a  straight  line  without  slipping? (1)  The  diameter  of  the  bicycle  wheel,  including  the  tire,  was  0.5  meter (2)  The  wheel  made  twenty  360-degree  rotations  per  minute 4. (OG  M)

What is the radius of the circle above with center O? (1) The ratio of OP to PQ is 1 to 2 (2) P is the midpoint of chord AB

269

5. (OG Â M)

What is the circumference of the circle above with center O? (1) The perimeter of OXZ is 20 + 10√2 (2)  The  length  of  arc  XYZ  is  5Ď€ 6. (OG  M)  If  the  circle  below  has  center  O  and  circumference  18Ď€,  then  the  perimeter  of  sector  RSTO  is (A) 3Ď€  +  9 (B) 3Ď€  +  18 (C) 6Ď€  +  9 (D) 6Ď€  +  18 (E) 6Ď€  +  24

7. (OG  E)  The  annual  budget  of  a  certain  college  is  to  be  shown  on  a  circle  graph.  If  the  size  of  each  sector  of  the  graph  is  to  be  proportional  to  the  amount  of  the  budget  it  represents,  how  many  degrees  of  the  circle  should  be  used  to  represent  an  item  that  is  15  percent  of  the  budget?  (A) 15 (B) 36 (C) 54 (D) 90 (E) 150 8. (MG  E)  A  square  is  inscribed  in  a  circle  of  area  18đ?œ‹.  Find  a  side  of  the  square. (A) 3 (B) 6 (C) 3 √2 (D) 6 √2 (E) It cannot be determined from the information given. 9.  (MG  M)  In  a  circle  with  center  O,  the  measure  of  arc  RS  =  132  degrees.  How  many  degrees  are  there  in  angle  RSO? (A) 66 (B) 20 (C) 22 (D) 24

270

(E) 48 10. (GC  M)  Assuming  the  Earth's  orbit  around  the  Sun  is  a  circle,  by  how  much  will  the  length  of  the  Earth's  orbit  increase  if  the  radius  of  this  orbit  grows  by  Ď€  /2  meters? (A) 1 meter (B) 2 meters (C) Ď€ meters (D) 2  Ă—  Ď€ meters (E) Ď€ 2 meters 11. (GC  M)  A  circle  is  inscribed  in  a  square.  If  a  diagonal  running  through  the  center  of  the  circle  is  4  cm  long,  what  is  approximate  value  of  the  area  of  the  square  that  is  not  occupied  by  the  circle? (A) 1.7 (B) 2.7 (C) 12 (D) 24 (E) 25 12. (GC  E)  The  radius  of  the  front  wheels  of  the  cart  is  half  that  of  the  rear  wheels.  If  the  circumference  of  the  front  wheels  is  1  meter  and  the  cart  went  1  kilometer,  how  many  revolutions  did  the  rear  wheels  make? (A) 250/  Ď€ (B) 500/  Ď€ (C) 250 (D) 500 (E) 750

13. In  the  figure,  đ??´đ??ľđ??ś  is  an  equilateral  triangle,  and  đ??ˇđ??´đ??ľ  is  a  right  triangle.  What  is  the  area  of  the  circumscribed  circle? (1)  đ??ˇđ??´ = 4 (2)  Angle  ABD  is  30  degrees 14. (MG  E)  If  the  radius  of  a  circle  is  decreased  by  10%,  by  what  percent  is  its  area  decreased? (A) 10 (B) 19 (C) 21 (D) 79 (E) 81

271

15. (MG  M)  A  circle  graph  shows  how  the  MegaTech  corporation  allocates  its  Research  and  Development  budget:  14%  microphotonics;Íž  24%  home  electronics;Íž  15%  food  additives;Íž  29%  genetically  modified  microorganisms;Íž  8%  industrial  lubricants;Íž  and  the  remainder  for  basic  astrophysics.  If  the  arc  of  each  sector  of  the  graph  is  proportional  to  the  percentage  of  the  budget  it  represents,  how  many  degrees  of  the  circle  are  used  to  represent  basic  astrophysics  research? (A) 8 (B) 10 (C) 18 (D) 36 (E) 52 16.  In  the  figure  below,  a  triangle  đ??´đ??ľđ??ś  is  right.  An  angle  đ??´  is  30  degrees  and  đ??ľđ??ś = 1.  What  is  the  radius  of  the  circle? B (A) 1 (B)

√

(C)

√

A

C

(D) √2 (E) √3

Volume Geometry

1. (MH  M)  If  a  cube  has  a  total  surface  area  of  96,  what  is  its  volume? (A) 16 (B) 36 (C) 64 (D) 81 (E) 96 2. (PR  M)  A  certain  cube  floating  in  a  bucket  of  water  has  between  80  and  85  percent  of  its  volume  below  the  surface  of  the  water.  If  between  12  and  16  cubic  centimeters  of  the  cube’s  volume  is  above  the  surface  of  the  water,  then  the  length  of  a  side  of  the  cube  is  approximately (A) 4 (B) 5 (C) 7 (D) 8 (E) 9 3. (PR  E)  If  a  cube  has  a  volume  of  125,  what  is  the  surface  area  of  one  side? (A) 5 (B) 25 (C) 50 (D) 150

272

(E)

625 4. (PR H) A sphere with a radius of 5 is hollowed out at the center. The part removed from the sphere has the same center, and a radius of 3. What fractional part of the original sphere remained? (The formula for the volume of a sphere is πR )

(A) (B) (C) (D) (E)

2/5 16/25 27/125 98/125 3/5 5. (MH M) A hat company ships its hats, individually wrapped, in 8-inch by 10-inch by 12inch boxes. Each hat is valued at $7.50. If the company’s latest order required a truck with at least 288,000 cubic inches of storage space in which to ship the hats in their boxes, what was the minimum value of the order? (A) $960 (B) $1,350 (C) $1,725 (D) $2,050 (E) $2,250 6. (MH M) A truck driver wants to load as many identical cylindrical canisters of olive oil as can fit into the 3-meter x 4-meter x 9-meter storage space of his truck. What is the maximum number of canisters can he load into the truck? (1) Each canister has a volume of 62, 500 cubic centimeters. (2) The height of each canister is four times the radius. 7. (MH M) When Greenville State University decided to move its fine arts collection to a new library, it had to package the collection in 20-inch by 20-inch by 15-inch boxes. If the university pays $0.50 for every box, and if the university needs 3.06 million cubic inches to package the collection, what is the minimum amount the university must spend on boxes? (A) $255 (B) $275 (C) $510 (D) $1,250 (E) $2,550 8. (GC) A cylindrical tank, with radius and height both of 10 feet, is to be redesigned as a cone, capable of holding twice the volume of the cylindrical tank. There are two proposed scenarios for the new cone: in scenario (1) the radius will remain the same as that of the original cylindrical tank, in scenario (2) the height will remain the same as that of the original cylindrical tank. What is the approximate difference in feet between the new height of the cone in scenario (1) and the new radius of the cone in scenario (2)? (A) 13 (B) 25 (C) 30

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(D) 35 (E) 40

Coordinate Geometry 1. In the coordinate plane, a circle has center (2,-3) and passes through the point (5,0). What is the area of the circle? (A) 3đ?œ‹ (B) 3√2đ?œ‹ (C) 3√3 (D) 9đ?œ‹ (E) 18đ?œ‹ 2.  (GC  E)  Line  đ?‘Ś = đ?‘˜đ?‘Ľ + đ?‘?  intersects  X-axis  at  (2,0)  and  Y-axis  at  (0,5).  What  is  ? (A) − (B) − (C) − (D) (E) 3. (GC  M)  The  vertices  of  a  triangle  have  coordinates  (x, 1),  (5,1),  and  (5, y)  where  x < 5  and  y > 1.  What  is  the  area  of  the  triangle? (1) đ?‘Ľ = đ?‘Ś (2) Angle at the vertex (đ?‘Ľ, 1) = angle at the vertex (5, đ?‘Ś) 4. (GC  H)  What  is  the  area  of  a  triangle  with  the  following  vertices  đ??ż(1,3),  đ?‘€(5,1),  and  đ?‘ (3,5)? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 5. (GC  M)  At  what  angle  do  the  lines  đ?‘Ś = đ??žđ?‘Ľ + đ??ż  and  đ?‘Ľ = đ?‘Ś + đ??žđ??ż  intersect? (1) đ??ž = 2 (2) đ??ž = đ??ż 6.  (GC  E)  Which  of  the  following  points  is  not  on  the  line  đ?‘Ś = 5đ?‘Ľ + 3? (A) (1/2, 11/2) (B) (1/3, 14/3) (C) (√8,  3  +  10  Ă—  âˆš2) (D) (√4,  13) (E) (√2,  31/3) 7. (GC  M)  Which  of  the  following  lines  is  parallel  to  line  đ?‘Ľ = 2.66 − 2đ?‘Ś?

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(A) đ?‘Ś = 2đ?‘Ľ + 25 (B) đ?‘Ś = 2.66đ?‘Ľâ€“ 2 (C) đ?‘Ľ + 3đ?‘Ś = 0 (D) đ?‘Ś = − đ?‘Ľâ€“ 32 (E) đ?‘Ś = đ?‘Ľ + 3 8. (GC  M)  Line  đ??ż  passes  through  points  (đ?‘Ž/2, đ?‘?)  and  (đ?‘Ž, −2đ?‘?).  Which  of  the  following  represents  the  slope  of  line  đ??ż? (A) − (B) − (C) − (D) (E) 9. (PR  M)  In  the  rectangle  coordinate  system,  triangle  đ??´đ??ľđ??ś  has  a  vertex  at  point  (0,56).  If  point  đ??ľ  is  at  the  origin,  then  how  many  points  on  line  đ??´đ??ś  have  integer  values  for  both  their  đ?‘Ľ  and  đ?‘Ś  values? (1) The third vertex of triangle đ??´đ??ľđ??ś lies on the đ?‘Ľ-axis, and the triangle has an area of 196 (2) Point đ??´ has a positive đ?‘Ľ coordinate and a đ?‘Ś coordinate of zero 10. (OG H) In the đ?‘Ľđ?‘Ś-plane, if line đ?‘˜ has negative slope and passes through the point (−5, đ?‘&#x;), is the đ?‘Ľ-intercept of the line đ?‘˜ positive? (1) The slope of line đ?‘˜ is -5. (2) đ?‘&#x; > 0

275

# of question

Topic of question (type) Your answer Geometry Line and Angles

1

Correct Answer

E Triangles

1 2 3 4 5 6 7 8 9 10 11 12

A E A A B B E D E B A B Quadrilaterals

1 2 3 4 5 6

D B C C B C (AP) Circles

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

B E A E D B C B D E A D A B D A Volume geometry

1 2 3 4 5

C A B D E

276

6 7 8

C A D Coordinate geometry

1 2 3 4 5 6 7 8 9 10

E C C D A E (AP) D C A E

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Test № 6. Geometry 1. In the figure shown above, a pentagon is composed with three right triangles. What is the 2 perimeter of the pentagon? (A)

8 2 3

(B)

8

6

(C) (D) (E)

8

5

12 15

2

2 2

2. If a triangle has sides a, b, and c, the area of the triangle can yield with the formula: s ( sa )( sb )( sc ), where s is half of the perimeter. If the sides of a triangle are 4, 5, and 7, what

is the area of the triangle? (A)

2 6

(B)

6 3

(C)

4 6

(D)

6 2

(E)

8 3

3. From left to right, points А, В, С, D and E lie on the number line. Is the distance between A and В less than the distance between С and D? (1) The distance between A and С is less than the distance between В and D. (2) The distance between A and D is less than the distance between В and E. 4. If the curve represented by y = x2 – 5x + t intersects with the x-axis at two points and one of the points is (–1, 0), what is the other point? (A) (1, 0) (B) (–2, 0) (C) (5, 0) (D) (6, 0) (E) (3, 0) y

r

x

5. The figure shows the position on a radar screen of a sweeping beam that is rotating at a constant rate in a clockwise direction. In which of the four quadrants will the beam lie 30 seconds from now? (1) In each 30-second period, the beam sweeps through 270°. (2) r = 30.

278

6. Is quadrilateral ABCD a square? (1) AB = ВС = CD = DA. (2) Two diagonals have the same length. 7. If x and у are integer, is x - y even? (1) x = 3y. (2) x – 3y is even. 8. p and q are integers. If p is divisible by 10q and cannot be divisible by 10q + 1, what is the value of q? (1) p is divisible by 25, but is not divisible by 26. (2) p is divisible by 56, but is not divisible by 57. 9. Is mx < m + x? (1) 0 < x < 1. (2) m is a positive integer. 10. A new released book includes hardback and paperback. If the sales price for the hardback is $20, and the sales price for the paperback is $15, what is the average price for all of the books sold during a certain day? (1) The ratio of the number of the hardback sold to the number of paperback sold is 5 to 3. (2) A total of 160 books were sold during the day. 11. Line m and n are perpendicular to each other. If m intersects with y-axis at point (0, –2), at which point does line n intersect with x-axis? (1) Line m intersects with x-axis at (–2, 0). (2) Line n passes through point (0, 10). 12. What is the area of a rectangle? (1) The perimeter of the rectangle is 24. (2) The square of the length of the diagonal is 100. Sun Tree 1 Tree 2 30

x 40

13. As the figure shown above, the taller tree is 30 feet high and has a 40 feet shadow. What is the height of the shorter tree? (1) The shorter tree has a 22 feet shadow. (2) The distance between two trees is 18 feet.

279

14. As the figure above shows, two identical circles are inscribed in a square. If the radius of the circle is r, what is the length of the diagonal of the square? (A)

r(4

2)

(B)

r(2

2)

(C)

2r(1

(D) (E)

4r 2

2)

5r A

D B

E C

15. In the figure above, if DE is parallel to ВС, what is the sum of the degree of angle ADE and angle ACB? (1) AB = BC. (2) Angle A is 70 degree. 16. Do integers p and t have the same number of different prime factors? 5

(1) p 7 t . (2) pt = 35. 17. If n is an integer, what is the value of n? (1) n is positive. (2) n2 = 2n. 18. About 300 employees in Company X usually use a laptop. What percent of the male employees usually use a laptop? (1) 200 female employees usually use a laptop. (2) 60% of the female employees usually use a laptop. 19. Is quadrilateral ABCD a parallelogram? (3) Two of the sides have length of 7. (4) Two of the opposite sides have length of 9. 20. A square and an equilateral triangle have the same perimeter. What is the ratio of the area of the triangle to the area of the square?

280

(A)

9 3 4

(B)

2 3 3

(C)

4 3 9

(D)

2 3 9

(E)

3 2

21. As the figure shown above, four identical circles are tangent to each other and the 4 centers of the circles are the vertex of a square. If the diameter of the circles is 2, what is the area of the shaded region? (A) 16 – 4√2 (B) 16 – 2√2 (C) 8 – 2√2 (D) 4 – √2 (E) 2√2– 4

22. Square R is inscribed in circle C and C is inscribed in square T. Is the circumference of С greater than 10? (3) The side length of R is greater than 2. (4) The side length of T is greater than 4. 23. If one of the sides of a right triangle has length of 10, which of the following could be the length of other two sides of the triangle? I. 6 and 8 II. III. (A) (B) (C) (D) (E)

15 2 40 3

25

and 2

50

and 3 I only II only III only I and III only I, II and III

281

24. If 3x + 5y < 15 and 5x + 3y > 15, which of the following mast be true? II. x>y III. x<y IV. x>3 (A) I only (B) II only (C) III only (D) I and II only (E) I and III only 25. Of the students in a certain class, 40 learn Spanish, 30 learn French, and 20 learn Italian. If each of her students learn at least one of the languages, how many students learn exactly two of the languages? (1) There are 60 students in the class. (2) 10 of the students learn all of the three languages. 26. The side surface of a cylinder is rolled with a rectangular plate. If the height of a cylinder tank is 12 feet and the perimeter of the circular base Is 9 feet, what is the diagonal of the rectangular plate? (A) 13 (B) 15 (C) 18 (D) 20 (E) 21 27. What is the volume of a rectangular solid? (1) Length of one of sides of the solid is 8. (2) One of the faces has area of 30 and perimeter of 22. 28. If the hypotenuse of the right triangle is 10, what is the perimeter of the triangle? (1) The area of the triangle is 25. (2) The triangle is an isosceles triangle. E A

B

D

C

29. In the figure, ADE is an isosceles right triangle above on the square ABCD. If the side length of the square is 10, what is the area of the whole region? (A) 110 (B) 115 (C) 120 (D) 125

282

(E)

133 A

D

B

C

30. In the figure above, ABCD is a rectangle and DA and CB are radii of the circles shown. If AB = 4, what is the perimeter of the shaded region? (A) 2π + 4 (B) 4π + 4 (C) 4π + 8 (D) 8π + 8 (E) 8π + 16 31. Triangle A’s base is 10% greater than the base of triangle B, and A’s height is 10% less than the height of triangle B. The area of triangle A is what percent less or more than the area of triangle B? (A) 9 % less (B) 1 % less (C) Equal to each other (D) 1 % more (E) 9 % more 32. Farmer Brown drives his tractor 40 kilometers. If he travels at r kilometers per hour for one-third of the distance and 2r kilometers per hour for the remainder, how many hours does the entire trip take, in terms of r? (A) (B) (C)

24 r 36 r 80 3r

(D)

20r

(E)

r 36

33. A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere? (A) 10 3 1 (B)

5

(C) 10 2 1 (D) 5 3 1 (E) 5 2 1

283

34. How many integers are between 100 and 150, inclusive, cannot be evenly divided by neither 3 nor 5? (A) 35 (B) 27 (C) 25 (D) 26 (E) 28 Answers: 1. D 2. C 3. A 4. D 5. A 6. C 7. D 8. C 9. C 10. A 11. C 12. C 13. D 14. C 15. B 16. C 17. E 18. E 19. C 20. C 21. D 22. B 23. E 24. A 25. C 26. B 27. C 28. D 29. D 30. A 31. B 32. C 33. D 34. B

284

Lesson  â„–  7.  Combinatorics  &  Probability Combinatorics GoGMAT, Session 13 Combinatorics is a branch of mathematics that helps to give the answer on the question "In how many ways an event can happen?". In this part you'll see word problems again. Two main rules of Combinatorics are Addition and Multiplication rules. Addition rule If event A can happen in (A) ways and event B can happen in (B) ways, then one of events - A or B can happen in (A)+(B) ways: đ??Žđ??œđ??œđ??Žđ??Ťđ??˘đ??§đ??  đ??¨đ??&#x;  đ??€  đ??¨đ??Ť  đ??  = đ??Žđ??œđ??œđ??Žđ??Ťđ??˘đ??§đ??  đ??¨đ??&#x;  đ??€ + đ??Žđ??œđ??œđ??Žđ??Ťđ??˘đ??§đ??  đ??¨đ??&#x;  đ?? = (đ??€) +  (đ?? ).  The word "or" indicates addition. 1. The box contains 5 red balls, 4 blue, 6 green and 3 yellow. In how many ways can Bob take from the box one red or one yellow ball? (A) 3 (B) 5 (C) 8 (D) 10 (E) 18

Multiplication Rule

If you know the number of outcomes in each of the events A, B, C etc, the number of possibilities for both of them occurring together (event A and event B) is multiplication of individual numbers of outcomes: đ??Žđ??œđ??œđ??Žđ??Ťđ??˘đ??§đ??  đ??¨đ??&#x;  đ??€  đ??šđ??§đ???  đ??  = đ??Žđ??œđ??œđ??Žđ??Ťđ??˘đ??§đ??  đ??¨đ??&#x;  đ??€ Ă— đ??Žđ??œđ??œđ??Žđ??Ťđ??˘đ??§đ??  đ??¨đ??&#x;  đ?? = (đ??€) Ă— (đ?? ) The word "and" is the key word that indicates multiplication. In other words, when presented with multiple groups of items from which you are required to make a selection, you will multiply the separate cases together. 2. Tony has 3 apples, 5 pears and 7 plums in his basket. The boy is choosing 2 fruits at random. Find the number of outcomes in which Tony chooses an apple and a plum. (A) 3 (B) 7 (C) 10 (D) 21 (E) 30

Factorial. Arrangements of n different objects.

The  factorial  symbol  â€œ!â€?  is  used  when  you  need  to  multiply  an  integer  by  all  the  positive  integers  that  are less than it. For example: 5! = 5 Ă— 4 Ă— 3 Ă— 2 Ă— 1 n! = n Ă— (n − 1) Ă— (n − 2) Ă— ‌ Ă— 2 Ă— 1

285

Note: 0! = 1! = 1, 2!=2, 3!=6, 4!=24, 5!=120. 0! and 1! are the only odd factorials because all other factorials have 2 as a factor. 3.

What is value of (A) 336 (B) 40 (C) 120 (D) 360 (E) 1680

!

?

!

The number of arrangements of n different objects in a raw is a typical problem that can be solve this way: 1. How many objects we can put at 1-st place? n. 2. How many objects we can put at 2-nd place? n - 1. We can't put the object that already placed at 1st place. ..... n. How many objects we can put at n-th place? 1. Only one object remains. Therefore, the total number of arrangements of n different objects in a raw is đ??? = đ??§ Ă— (đ??§ − đ?&#x;?) Ă— (đ??§ − đ?&#x;?) Ă— ‌ Ă— đ?&#x;? Ă— đ?&#x;? = đ??§! 4. (A) (B) (C) (D) (E)

How many ways can you arrange the letters in the word MICRO? 5 1 120 20 60

Circular arrangements

Let's say we have 6 distinct objects, how many relatively different arrangements do we have if those objects should be placed in a circle.

The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. It means number arrangement in a row n times the number the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have:

286

R=

n! = (n − 1)! n

Or we can say that it does not matter where the first item is placed, since only the order around that item is important. And total possible arrangements of remaining n-1 items are (n-1)! Generally, n different items in a circle can be arranged in (n-1)! ways. 5. Seven family members are seated around their circular dinner table. If only the only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, then how many distinct arrangements around the table are possible? (A) 7 (B) 42 (C) 5040 (D) 524 (E) 720

Permutation with repetition.

If the only concern we have in a question is how some subsets of identical objects can be rearranged, we can use the special formula: (đ??­đ??¨đ??­đ??šđ??Ľ  đ??§đ??Žđ??Śđ??›đ??žđ??Ť  đ??¨đ??&#x;  đ??¨đ??›đ??Łđ??žđ??œđ??­đ??Ź)! (đ??§đ??Žđ??Śđ??›đ??žđ??Ť  đ??˘đ??§  đ??Źđ??Žđ??›đ??Źđ??žđ??­đ??Ź  đ?&#x;?)! Ă— (đ??§đ??Žđ??Śđ??›đ??žđ??Ť  đ??˘đ??§  đ??Źđ??Žđ??›đ??Źđ??žđ??­  đ?&#x;?)! Ă— (đ??§đ??Žđ??Śđ??›đ??žđ??Ť  đ??˘đ??§  đ??Źđ??Žđ??›đ??Źđ??žđ??­  đ?&#x;‘)! ‌ 6. There are three men and four women lining up against a wall from left to right. If the only concern is gender of each individual, how many different ways can we arrange the people against the wall by gender? (A) 8,640 (B) 2,160 (C) 720 (D) 84 (E) 35 7. (A) (B) (C) (D) (E)

In how many ways can you place 3 roses, 4 lilies and 2 daisies in a row in the flower bed? 720 1260 5040 40320 362880

Always together.

Frequently, certain item must always be kept together. To do these questions, you must treat the joined items as they were only one object. 8. In how many ways can you sit 8 people on a bench if 3 of them must sit together? (A) 720 (B) 2,160 (C) 2,400 (D) 4,320 (E) 40,320 9. The choir consists of 5 boys and 6 girls. In how many ways can the singers be arranged in a row, so that all the boys are together? (Assume the only concern is gender). (A) 120

287

(B) 30 (C) 24 (D) 11 (E) 7

Permutation.

A permutation is an ordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose and arrange k objects out of n distinct objects is denoted as: P . Knowing how to find the number of arrangements of n distinct objects we can easily find formula for permutations: 1. The total number of arrangements of n distinct objects is n! 2. Now we have to exclude all arrangements of remaining (n-k) objects ((n-k)!) as the order of remained (n-k) objects doesn't matter. đ??§! đ???đ??§đ??¤ = (đ??§ − đ??¤)! Remember, n is the total number of items, k is the number you want to order and n is always more or equal to k. Colors, names, numbers, letters, positions are the keys that indicate permutation. When item has some identification, like color or letter, an order of it item is important – this is the difference permutation from combination. 10. There are five girls, Amy, Barbara, Candice, Diane, and Elain, playing in the park. If three of these girls will be chosen to be a princess, queen and sorceress, how many different groups of girls can be chosen to play? (A) 120 (B) 90 (C) 60 (D) 40 (E) 10 11. How many four-digit numbers can you form using ten numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) if the numbers can be used only once? (A) 5040 (B) 3556 (C) 4536 (D) 10000 (E) 24

Combination.

In the previous examples, when using the permutations, the order of items to be arranged mattered. If all  you  want  to  do  is  select  items,  and  don’t  care  what  order  they’re  in,  you  can  use  combinations.   A combination is an unordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose k objects out of n distinct objects is denoted as: C đ??‚đ??§đ??¤ =

đ??§! đ??¤! (đ??§ − đ??¤)!

Pay your attention that we always choose k objects out of n, and n is greater or equal to k.

288

12. From a group of 8 secretaries, select 3 persons for promotion. How many distinct selections are there? (A) 24 (B) 56 (C) 336 (D) 8 (E) 3 13. made? (A) (B) (C) (D) (E)

A pizza can have 3 toppings out of a possible 7 toppings. How many different pizzas can be 35 21 3 12 7

At least / at most

These questions will require you to add all the possible cases together but in these problems it will take a lot of time. Know and use whenever you see at least or at most the shortcuts: đ??šđ??­  đ??Ľđ??žđ??šđ??Źđ??­/đ??šđ??­  đ??Śđ??¨đ??Źđ??­ = đ??­đ??¨đ??­đ??šđ??Ľ  đ??œđ??šđ??Źđ??žđ??Ź − đ??Žđ??§đ??°đ??šđ??§đ??­đ??žđ???  đ??œđ??šđ??Źđ??žđ??Ź 14. A committee of 3 people is to be formed from a group of 5 men and 5 women. How many possible committees can be formed if at least one woman is in the committee? (A) 120 (B) 110 (C) 100 (D) 90 (E) 80 15. A three-person committee must be chosen from a group of 7 professors and 10 graduate students. If at least one of the people on the committee must be a professor, how many different groups of people could be chosen for the committee? (A) 70 (B) 560 (C) 630 (D) 1,260 (E) 1,980

Combinations from multiple selection pools.

This type of problems combines multiplication rule with permutations and combinations. 16. A committee of 3 boys and 5 girls is to be formed from a group of 5 boys and 7 girls. How many committees are possible? (A) 420 (B) 210 (C) 105 (D) 35 (E) 15

289

Arrangements allowing the use of place more than once. Sometimes we are interested in arrangements allowing the use of place more than once. Usually it means that objects are independent. Then to calculate number of arrangements we use formula n , where k is the number of objects and n is the number of options for each object. 17. There are three lamps in the hall. Each lamp can be switched on and off independently. In how many ways can the hall be illuminated? (The hall is illuminated when at least one of the lamps in on) (A) 9 (B) 8 (C) 7 (D) 6 (E) 5 18. 7 different objects must be divided among 3 people. In how many ways can this be done if one or two of them can get no objects? (A) 15,1200 (B) 2,187 (C) 3,003 (D) 792 (E) 344

Assurance of occurrence.

There are some problems which ask you to find minimum number of attempts to ensure some outcome. There are no formulas, just count the worst scenario. 19. Of the science books in a certain supply room, 50 are on botany, 65 are on zoology, 90 are on physics. 50 are on geology, and 110 are on chemistry. If science books are removed randomly from the supply room, how many must be removed to ensure that 80 of the books removed are on the same science? (A) 81 (B) 159 (C) 166 (D) 285 (E) 324

290

The order matters (Permutations) Typical problem

Formula

Solution

In how many ways can 5 boys be arranged in a row?

đ?‘›!

5! = 5 Ă— 4 Ă— 3 Ă— 2 Ă— 1 = 120

In how many ways can a first, second and third prize be awarded in a class of 8 students, if no student can get more than 1 prize?

đ?‘›! đ?‘ƒ = (đ?‘› − đ?‘˜)!

đ?‘›  =  8 đ?‘˜  =  3 8! 8! đ?‘ƒ = = = 336 (8 − 3)! 5!

How many arrangements of the letters of the word PARRAMATTA are possible?

n! x! × y! × z! ‌

Total number of letters – 10 Number of  â€œRâ€?  â€“ 2 Number  of  â€œAâ€?  â€“ 4 Number  of  â€œTâ€?  â€“ 2 10! = 37,800 4! Ă— 2! Ă— 2!

In how many ways can 4 people be arranged in a circle?

(đ?‘› − 1)!

In how many ways can four girls and three boys be arranged in a row so that the boys are always together?

Treat the joined objects as one

How many 4-digit integers can be formed using the digits {1, 3, 5}?

đ?‘›

(4  –  1)!  =  3! = 6 We have 5 objects here: 4 girls and a group of 3 boys, treated as one. 5! × 3! = 120 × 6 = 720

đ?‘›  =  3, đ?‘˜  =  4 đ?‘› = 3 = 81

291

The order does NOT matter (Combinations) Typical problem 1. In how many ways can the set of 3 books be selected from 8 different books? 1. A committee of 3 people is to be formed from a group of 5 men and 5 women. How many possible committees can be formed if at least one woman is in the committee? A committee of 3 boys and 5 girls is to be formed from a group of 5 boys and 7 girls. How many committees are possible?

Formula

Solution

n! k! (n − k)!

n = 8, k = 3 8! 8! C = = = 56 3! (8 − 3)! 3! 5!

at least/at most = total cases - unwanted cases

Total number of committees: C Unwanted (male) committees: C C − C = 120 − 10 = 110

C =

The word "and" is the key word that indicates multiplication.

Number of male committees: C Number of female committees: C C × C = 10 × 21 = 210

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GoGMAT Problems

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Answers and explanations 1. We need to find out, in how many ways Bob can take 1 red or one yellow ball from the box. The word "or" indicates addition, so, just add 5 variants for red and 3 - for yellow: 5+3 =8, the answer is C. 2. Here we have the classic multiplication problem: the boy has to choose an apple and a plum. The word "and" is an indicator of multiplication. Thus, let's multiple number of ways to choose an apple (3) by the number of ways to choose a plum (7). 3×7=21, the answer is D. ! 3. To calculate ! we won’t compute 8! and then divide it by 5! That would take too long. If we × × × × × × ×

× × × !

write out the problem, we get = . And 5! cancels, leaving us with × × × × ! 8 × 7 × 6 = 336. The answer is (A). 4. The basic idea is we have 5 objects, and 5 possible positions they can occupy. For the first place, there are five letters to pick from. For the second - we've used up one letter, only 4 variants are left. Third- 3 unused letters remaining. Fourth - 2 letters remaining. We multiply all numbers 5 × 4 × 3 × 2 × 1 = 120. The answer is (C). 5. If all the orderings mattered, the answer would be 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040. But they don’t all matter, so clearly the answer is not C. Which ones don’t matter? Well, imagine you have seven people seated around a table, and they all get up and move one seat to the right. Their relative positions haven’t changed, so according to the rules of the question, this shouldn’t really count as a new ordering. How many times can we shift people around the table before they arrive back at their original seats? We can shift them six times, resulting in seven identical arrangements, counting the initial arrangement. So you have to divide by 7, it will give you 720. The answer is (E). 6. There are two subsets of men and women. We see difference in gender. But in each subsets there are uniform people (we don’t know their names or positions). And we have just to rearrange people, so, we can use the formula. The question tells us that we have three men and four women. So the total number of people ! !× × × we are ‘rearranging’ is seven. Substituting into the formula we get: !× ! = × × × ! = 35. The answer is (E). 7. We have three subsets of flowers. Each subset contains identical flowers. And the only concern is to rearrange flowers, so, we can use the formula. The total number of flowers we need to put in a row is 9. Substituting into the formula we get: ! !× × × × × × × × = = = 1260. The answer is (B). !× !× ! × × × !× × × 8. Let's look at our 8 people in such a way: | | | | | | | |. Treat the three that sit together as one person for the time being: | | ( | | | ) | | |. Now, you have only 6 people (5 and the three that act as one) on 6 places: 6! = 720. You have to remember that the three that sit together can also change places among themselves: 3! = 6. So, the total number of possibilities is 6! × 3! = 4320. The answer is (D). 9. Look at the five boys as at one person. Now there are only 7 people and 6 of them are uniform. To seat the boy (five boys act as one) on seven pleases there are 7 options. And then 6 remaining girls seat on remaining 6 pleases the only way. The total number of ways is 7 × 1 = 7. The answer is (E). 10. This question is permutation since there are roles of girls – their identifications, that means that order is important. There are three girls being chosen to play in the game, so the layout begins with three slots for the girls. –––– –––– –––– . Then we must label the slots that we have laid out. In this case each slot will contain a girl. For the princess we have 5 options. No girls have been chosen yet and all are still available.

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For the queen we now have 4 options. One girl has already been chosen and only four are stillavailable.

For the sorceress we now have 3 options. Two girls have been chosen and that leaves only three girls available.

Once all of the slots have been filled with values, the question is considered to be set up. We now multiply across to find the answer: 5 × 4 × 3 = 60. Also we could find the answer just using the formula for permutation: ! ! !× × × P = ( )! = ! = = 60. The answer is (C). !

11. It seems to be easy, we take 10x9x8x7 since we have 4 positions, and get 5040, but there is a trick to this problem because 5040 will include numbers that start with a 0, and in reality we don't have those. So, we need to subtract the number of the fake 4-digit numbers. Use a Permutation formula: P(9,3) = 9x8x7 = 504. 5040-504 = 4536. Or we can count the number of numbers as we used to do with slots: ___ ___ ___ ___ . For the thousands digit we can use all digits except for zero: _9 ___ ___ ___. For the hundreds digit we can use all remaining 9 digits: one digit has already been chosen and only nine are still available _9_ _9_ ___ ___. For the tens digit there are 8 options. Two digits have been chosen and that leaves only eight digits available _9_ _9_ _8_ ___. For the units digit there are 7 options _9_ _9_ _8_ _7_. To find the answer we will just multiply 9 × 9 × 8 × 7 = 4536. The answer is (C). 12. This question is combination since order is not important. ! ! !× × × × × C = !( )! = ! ! = = × × = 56 ! !

The answer is (B). 13. There are 7 toppings in total, and by selecting 3, we will make different types of pizza. This question is a combination since having a different order of toppings will not make a different pizza. ! ! !× × × × × C = !( )! = ! ! = = = 35 ! ! × × The answer is (A). 14. If at least one woman is on the committee, that means we can have a committee with 3 women, 2 women, or 1 woman. And we can find the combinations for each case separately, then add them all together. But it too long. We can use shortcut instead. The total possible cases would be the number of committees of any 3 people out of 10: C . The unwanted cases would be the number of committees of 3 men without women: C . 10! 5! C − C = − = 120 − 10 = 110 3! 7! 2! 3! The second way is easier because it need two calculations instead of five in the first way. The answer is (B). 15. In the task we see “at least” and it is the key to solution: to find a total without a limit and then subtract the number of situations when there are no professors on the team. ! × × We get C = ! ! = × × = 680 total possible committees. Now, the number of teams with students only is, using the same formula:C =

!

! !

=

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× ×

= 120 Now, 680-120=560 – the number of groups with could be chosen to the com mittee, where at least one professor. The answer is (B). 16. Out of 5 boys, we must choose 3: C = 10. Out of 7 girls, we must choose 5: C = 21. All together: C × C =10× 21 = 210. The answer is (B). 17. Each switch has two possible positions, on and off. Placing a 2 in each of the 3 positions, we have 2 . But among that 8 cases there is one, when all lamps are switched off, and we must subtract this case because we are interested when the hall is illuminated: 2 − 1 = 8 − 1 = 7. The answer is (C). 18. Each of 7 objects can go to 3 different people and by the rule of multiplying we get 3×3×...×3 = 37 ways. The answer is (B). 19. Such problems invite us to provide a foolproof solution that would work in 100% of the cases. Thus, this means we will need to find a solution for the worst case. So after we will have removed all of the botany and geology books as well as 65 on zoology, 50 on botany, and 50 on geology, but we still don't have 80 of the same kind. So, after another 79 on chemistry and 79 on physics still not enough. Now we will have a total of 50+65+50+79+79=323 books removed. Now, however, we need to remove only one book because we will know that we have only two kinds of books left (either chemistry or physics) and any of them will give us a set of 80. Of course in reality it would not be that bad, but we have to take the worst situation to be sure. GoGMAT Problems Explanations × ×

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Probability GoGMAT, Session 13 An event is defined as any outcome that can occur. Probability is defined as the chance that an event will happen. The definition of probability is đ??Šđ??Ťđ??¨đ??›đ??šđ??›đ??˘đ??Ľđ??˘đ??­đ??˛ =

đ??§đ??Žđ??Śđ??›đ??žđ??Ť  đ??¨đ??&#x;  đ??&#x;đ??šđ??Żđ??¨đ??Ťđ??šđ??›đ??Ľđ??ž  đ??žđ??Żđ??žđ??§đ??­đ??Ź đ??§đ??Žđ??Śđ??›đ??žđ??Ť  đ??¨đ??&#x;  đ??­đ??¨đ??­đ??šđ??Ľ  đ??žđ??Żđ??žđ??§đ??­đ??Ź Â

The favorable events are the events of interest. They are the events that the question is addressing. The total events are all possible events that can occur relevant to the question asked. Example. Coin

There are two equally possible outcomes when we toss a coin: a head (H) or tail (T). Therefore, the probability of getting head is 50% or and the probability of getting tail is 50% or Example. Dice

There are 6 equally possible outcomes when we roll a die. The probability of getting any number out of 1-6 is . Example. Â Marbles, Â Balls, Â Cards... 1. We have a jar with 20 green and 80 white marbles. If we randomly choose a marble, what is the probability of getting a green marble? (A) 1/5 (B) 1/10 (C) 1/20 (D) 1/90 (E) 1/100

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Multiplication Rule

Two events are independent if occurrence of one event does not influence occurrence of other events. If two (or more) independent events are occurring, and you know the probability of each, the probability of BOTH (or ALL) of them occurring together (event A and event B and event C etc) is a multiplication of their probabilities. Probability of A and B = p A and B = p(A) ∗ p(B) Probability of A and B and C … and Z = p A and B and C … and Z = = p(A) × p(B) × p(C) × . . . × p(Z) -

The multiplication theorem is used to answer the following questions: What is the probability of two or more events occurring either simultaneously or in succession? For two events A and B: What is the probability of event A and event B occurring? The word "and" is the key word that indicates multiplication of the individual probabilities. 2. If there is a 20% chance of rain, what is the probability that it will rain on the first day but not on the second? (A) 0.04 (B) 0.16 (C) 0.2 (D) 0.4 (E) 0.8 3. There are two sets of integers: {1,3,6,7,8} and {3,5,2}. If Robert chooses randomly one integer from the first set and one integer from the second set, what is the probability of getting two odd integers? (A) (B) (C) (D) (E) 4. Given that there are 5 married couples. If we select only 3 people out of the 10, what is the probability that none of them are married to each other? (A) (B) (C) (D) (E) 5. There are 8 employees including Bob and Rachel. If 2 employees are to be randomly chosen to form a committee, what is the probability that the committee includes both Bob and Rachel? (A) (B) (C)

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(D) (E)

Addition rule

The probability that event “E or Fâ€? occurs is đ?‘ƒ  (đ??¸  đ?‘œđ?‘&#x;  đ??š)  =  đ?‘ƒ  (đ??¸)  +  đ?‘ƒ  (đ??š)  −  đ?‘ƒ  (đ??¸  đ?‘Žđ?‘›đ?‘‘  đ??š) (remember the formula for the union of sets |đ??´ âˆŞ đ??ľ| = |đ??´| + |đ??ľ| − |đ??´ ∊ đ??ľ|). If events E and F are independent, then we can rewrite the addition rule as follows: đ?‘ƒ  (đ??¸  đ?‘œđ?‘&#x;  đ??š)  =  đ?‘ƒ  (đ??¸)  +  đ?‘ƒ  (đ??š)  −  đ?‘ƒ  (đ??¸) Ă— đ?‘ƒ(đ??š) If the event “E and Fâ€?  is impossible (that is, đ??¸ ∊ đ??š has no outcomes), then E and F are said to be mutually exclusive events, and đ?‘ƒ(đ??¸  đ?‘Žđ?‘›đ?‘‘  đ??š)  =  0. Then the general addition rule is reduced to đ?‘ƒ  (đ??¸  đ?‘œđ?‘&#x;  đ??š  )  =  đ?‘ƒ  (đ??¸)  +  đ?‘ƒ  (đ??š  ). This is the special addition rule for the probability of two mutually exclusive events. The addition theorem is used to answer the following questions: What is the probability of one event or another event or both events occurring? What is the probability of event A or event B occurring? The word "or" indicates addition of the individual probabilities. 6. If Jessica rolls a die, what is the probability of getting at least a "3"? (A) (B) (C) (D) (E) 7. What is the probability that a fair die rolled once will land on either 4 or 5? (A) (B) (C) (D) (E) 8. A certain club has 10 members, including Harry. One of the 10 members is to be chosen at random to be the president, one of the remaining 9 members is to be chosen at random to be the secretary, and one of the remaining 8 members is to be chosen at random to be the treasurer. What is the probability that Harry will be either the member chosen to be the secretary or the member chosen to be the treasurer? (A)

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(B) (C) (D) (E)

Probability that a specific event will not occur.

The maximum probability for any event is 1. A probability of 1 means that the event is certain to occur. If you know that the probability of an event (or one of the outcomes) is p, the probability of this event NOT happening (or the probability of it NOT having this given outcome) is (1 − p). When you see the words "at least" in a probability question, chances are it is easier to find the probability of the opposite happening and then subtract from 1. The word "at least" indicates problems to solve which one use the formula đ???(đ??€) =  đ?&#x;?  âˆ’  đ???(đ??§đ??¨đ??­  đ??€). 9. What is the probability that, on three rolls of a single fair die, at least one of the rolls will be a six? (A) (B) (C) (D) (E) (F) 10. There are 3 black balls and 7 white balls in a box. If two balls are chosen randomly, what is the probability that at least one will be black? (A)1/15 (B) 7/15 (C) 8/15 (D) 7/30 (E)23/30 11. A new Machine is bought by a factory to produce 2 parts, A and B. A and B when assembled produce C. During the testing stages of the machine, 1 out of 10 in A is defective and 2 out of 10 in B are defective. What is the probability of C being defective? (A) 0.02 (B) 0.2 (C) 0.28 (D) 0.32 (E) 0.72

Binomial distribution.

Sometimes we'll need to find out the answer for the following question: If the probability that an object has a certain characteristic is p, what is the probability that exactly k of n objects have this characteristic?

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There are đ??ś ways to choose k objects out of n. Then, for all of them we have to guarantee a certain characteristic: đ?‘? Ă— đ?‘? Ă— ‌ = đ?‘? . For the rest (n-k) objects we have to guarantee the absence of the characteristic: (đ?‘? − 1) Ă— (đ?‘? − 1) Ă— ‌ = (đ?‘? − 1) . So, the answer can be calculated using the formula: đ??‚đ??§đ??¤ Ă— đ??Šđ??¤ Ă— (đ?&#x;? − đ??Š)đ??§ đ??¤ 12. The probability that a visitor at the mall buys a pack of candy is 30%. If three visitors come to the mall today, what is the probability that exactly two will buy a pack of candy? (A) 0.343 (B) 0.027 (C) 0.063 (D) 0.189 (E) 0.147 13. In Paris, the probability of rain in April is 40% for every day. Ann and Ben decided to visit Paris on April 4th-7th. To the nearest hundredths, what is the probability that it was raining on exactly 3 days of their visit? (A) 0.04 (B) 0.06 (C) 0.08 (D) 0.15 (E) 0.24 GoGMAT Problems

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Answers and explanations 1. The number of all marbles: N = 20 + 80 =100 The number of green marbles: n = 20 Probability of getting a green marble: p = = = . So, the answer is A. 2. The probability of rain is 0.2; therefore probability of not raining is 1 - 0.2 = 0.8. This yields that the probability of rain on the first day and sunshine on the second day is: P = 0.2 * 0.8 = 0.16. Best answer is (B). 3. There is a total of 5 integers in the first set and 3 of them are odd: {1, 3, 7}. Therefore, the probability of getting odd integer out of first set is . There are 3 integers in the second set and 2 of them are odd: {3, 5}. Therefore, the probability of getting an odd integer out of second set is . Finally, the probability of getting two odd integers is: P = × = . The answer is (A). 4. 1-st person: = 1 - we choose any person out of 10. 2-nd person: - we choose any person out of 8=10-2(one couple from previous choice) 3-rd person: - we choose any person out of 6=10-4(two couples from previous choices). p = 1 × × = . The answer is (D). 5. Probability approach: The probability of choosing Bob or Rachel as a first person in committee is 2/8. The probability of choosing Rachel or Bob as a second person when first person is already chosen is 1/7. The probability that the committee includes both Bob and Rachel is: P = × = = Combinatorial approach: The total number of possible committees is N = C = 28. The number of possible committee that includes both Bob and Rachel is n = 1. Then: P= = = . So, the answer is (B). 6. There are 4 outcomes that satisfy our condition (at least 3): {3, 4, 5, 6}. The probability of each outcome is 1/6. The probability of getting at least a "3" is: P = + + + = . The is answer is (A). 7. This is an “or” problem, because it is asking for the probability that the die will land on either 4 or 5. The probability that the die will land on 4 is 1/6. The probability that the die land on 5 is 1/6. Therefore, the probability that the die will land on either 4 or 5 is + = . The answer is (B). 8. Two probabilities must be calculated here: (1) the probability of Harry’s being chosen for secretary and (2) the probability of Harry’s being chosen for treasurer. (1) If Harry is to be secretary, he first CANNOT have been chosen for president, and then he must be chosen for secretary. The probability that he will be chosen for president is 1/10, so the probability of his NOT being chosen for president is 9/10. Then, the probability of his being chosen for secretary is 1/9. Thus, the probability that Harry will be chosen for secretary is × = . (2) If Harry is to be treasurer, he needs to be NOT chosen for president, then NOT chosen for secretary, and then finally chosen for treasurer. The probability that he will NOT be chosen for president is again 9/10. The probability of his NOT being chosen for secretary is 8/9. The probability of his being chosen for treasurer is 1/8, so the probability that Harry will be chosen for treasurer is × × = . (3) So, finally, the probability of Harry’s being chosen as either secretary or treasurer is

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thus + = . The answer is E. 9. We could list all the possible outcomes of three rolls of a die (1-1-1, 1-1-2, 1-1-3, etc.) and then determine how many of them have at least one six, but this would be very timeconsuming. Instead, it is easier to think of this problem in reverse before solving: What is the probability that not one of the rolls will yield a 6? On each roll, there is a 5/6 probability that the die will not yield a 6. Thus, the probability that on all 3 rolls the die will not yield a 6 is × × = . Now we return to answer the question by subtracting our result from 1. The probability that at least one of the rolls will be a 6 includes every outcome, except when three consecutive nonsixes are rolled. 1− = is the probability that at least one six will be rolled. The answer is (B). 10. We see the words "at least" in a probability question, so it is easier to find the probability of the opposite happening and then subtract from 1. The opposite of at least one ball being black is both balls being white. We can find the probability of drawing two white balls in two ways. Firstly, using probability approach: 7/10 × 6/9 = 7/15. Secondly, with the help of Combinatorics: there are C(2, 10) = 45 total ways to take out 2 balls from the box of 10. We can choose 2 white balls in C(2, 7) = 21 ways. So, the probability of drawing two white balls is 21/45 =7/15. Don't forget to subtract that from 1: 1 − = . So the correct answer is (C). 11. P (A is defective) = 1/10, so P (A is not defective) = 9/10; P(B is defective) = 2/10, so P (B is not defective) = 8/10. C is defective if A is defective, while B is not; or if B is defective and A is not; or if both of them are defective. Counting the probability of each of these events can be time consuming. The only situation that does not suits the question is when A and B are both not defective. Then P(C is not defective) = P (A and B are not defective) =P(A − not) × P(B − not) = × = = 0.72. Thus, P(C is defective) = 1-0.72 = 0.28. The answer is (C). 12. Here, a certain characteristic is buying candies, so, p=0.3. We have 3 visitors in total: n=3, and have to choose 2 out of them: k =2. Substituting numbers in the formula gives us: ! P = C × 0.3 × 0.7 = !× ! × 0.09 × 0.7 = 3 × 0.063 = 0.189. The answer is (D). 13. Now, the option we are interested in is rain, so, p=0.4. We have 4 days in total: n=4, and have to choose 3 out of them: k =3. ! Substituting numbers in the formula gives us: P = C × 0.4 × 0.6 = !× ! × 0.064 × 0.6 = 4 × 0.0384 = 0.1536 ≈ 0.15. The answer is (D).

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GoGMAT Problems Explanations

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Home assignment.

Combinatorics

1. 15 chess players take part in a tournament. Every player plays twice with each of his opponents. How many games are to be played? (A) 190 (B) 200 (C) 210 (D) 220 (E) 225 2. How many even numbers of 3 digits can be formed with the digits 0, 1, 2, 3, 4, 5, and 6? (A) 168 (B) 162 (C) 120 (D) 105 (E) 100 3. There are 4 contestants in the competition for 6 different prizes. A contestant can win only one prize. How many different outcomes are possible at the competition for the four contestants? (A) 360 (B) 720 (C) 4 (D) 6 (E) 1 4. On how many ways can the letters of the word "COMPUTER" be arranged so that all the vowels are together? (A) 3!5! (B) 8! (C) 7! (D) 3!6! (E) 8!-3!6! 5. How many 4 digit numbers begin with a digit that is prime and end with a digit that is prime? (A) 1600 (B) 800 (C) 80 (D) 1440 (E) 16 6. The box contains 5 green candies, 7 red, 15 blue, and 20 yellow. Allan is taking candies from the box at random. What is the least number of candies he has to take to ensure that at least two of them will be yellow? (A) 2 (B) 12 (C) 22

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(D) 29 (E) 47 7. The president of a country and 4 other dignitaries are scheduled to sit in a row on the 5 chairs. If the president must sit in the center chair, how many different seating arrangements are possible for the 5 people? (A) 4 (B) 5 (C) 20 (D) 24 (E) 120 8. Given a selected committee of 8, in how many ways, can the members of the committee divide the responsibilities of a president, vice president, and secretary? (A) 72 (B) 672 (C) 336 (D) 56 (E) 112 9. The president of a company wants to fire 3 persons from the Sales department. How many distinct selections can he make if the Sales department gathers 8 sales managers? (A) 366 (B) 3 (C) 8 (D)112 (E) 56 10. Ten friends meet in the park. If each person shakes hands with his each of friends once, how many handshakes take place? (A) 141 (B) 131 (C) 115 (D) 90 (E) 45 11. How many different words with up to 4 letters can be made with the letters A and B? (A) 2 (B) 4 (C) 16 (D) 30 (E) 32 12. Chef Gundy is making a new “style” of salad, which will contain two kinds of lettuce, one kind of tomato, one kind of pepper and two kinds of squash. If Chef Gundy has 8 kinds of lettuce, 4 kinds of tomatoes, 5 types of peppers, and 4 kinds of squash from which to choose, than how many different “styles” of salad can he make?

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(A) 640 (B) 1120 (C) 2240 (D) 3360 (E) 13440 13. Alan has a flock of sheep from which he will choose 4 to take with him to the livestock show in Houston. If Alan has 15 distinct possible groups of sheep he could take to the show then which of the following is the number of sheep in his flock? (F) 6 (G) 7 (H) 15 (I) 30 (J) 5 14. Four women and six men work in the accounting department. In how many ways can a committee of 3 be formed if it has to include at least one woman? (A) 36 (B) 60 (C) 72 (D) 100 (E) 110 15. There are 3 yellow chips and 2 green chips. When arranged in a row, they form a certain color pattern, for example RBRRB. How many color patterns are possible? (A) 10 (B) 12 (C) 24 (D) 60 (E) 100 16. There are ten lamps in the hall. Each lamp can be switched on and off independently. In how many ways can the hall be illuminated? (The hall is illuminated when at least one of the lamps in on) (A) 10 (B) 100 (C) 128 (D) 1023 (E) 1024 17. (SC) A 6-people committee in a film festival will be selected from 8 directory and 6 film critics. If the committee consists of 4 directors and 2 critics, how many such committees are possible? (A) 2,100 (B) 1,400 (C) 1,050 (D) 840 (E) 700 18. (SC) In how many arrangements can a photographer seat 3 girls and 3 boys in a row of 6 seats if the boys are to have the first, third, and fifth seats? (A) 6

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(B) (C) (D) (E)

9 12 36 720

19. If a customer has to make exactly 1 selection from each of the 5 categories listed below, what is the number of different ice cream sundaes that a customer can create? 12 ice cream flavors; 10 kinds of candy; 8 liquid toppings; 5 kinds of nuts; With or without whip cream. (A) 9600 (B) 4800 (C) 2400 (D) 800 (E) 400 20. How many four-digit numbers can you form using ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) if the digits can be used only once? (A) 9000 (B) 5852 (C) 4940 (D) 4536 (E) 10000 21. A tennis command contains three persons. Such command must be chosen from a group of 7 juniors and 10 graduates. If at least one of the people on the committee must be a junior, how many different tennis commands could be made? (A) 70 (B) 560 (C)630 (D) 1,260 (E) 1,980 22. There are 9 books on a shelf, 7 hard cover and 2 soft cover. How many different combinations exist in which you choose 4 books from the 9 and have at least one of them be a soft cover book? (A) 184 (B) 126 (C) 91 (D) 36 (E) 21 23. How many different signals can be transmitted by hoisting 3 red, 4 yellow and 2 blue flags on a pole, assuming that in transmitting a signal all nine flags are to be used?

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(A)2520 (B) 1260 (C)4!3!2! (D) 9! (E) 24 24. If Bob and Jen are two of 5 participants in a race, how many different ways can the race finish where Jen always finishes in front of Bob? (A) 120 (B) 60 (C) 30 (D) 15 (E) 5 25. Olivier is an abstract painter who is working on a series of paintings. If each of these paintings has three identical blue vertical stripes, two identical red vertical stripes and two identical black vertical stripes spaced evenly across a square canvass, how many distinct paintings could Olivier’s series include? (A) 5040 (B) 720 (C) 210 (D) 96 (E) 6 26. Carly has 3 movies that she can watch during the weekend: 1 Action movies, 1 Comedy, and 1 Drama. However, she needs to watch the Drama 3 times. Assuming Carly has time for 5 movies and intends to watch all of them, in how many ways can she do so? (A) 6 (B) 20 (C) 24 (D) 60 (E) 120 27. There are 11 top managers that need to form a decision group. How many ways are there to form a group of 5 if the President and Vice President are not to serve on the same team? (A) 232 (B) 84 (C) 378 (D) 162 (E)196 28.

There are 5 married couples and a group of three is to be formed out of them; how many

arrangements are there if a husband and wife may not be in the same group? (A) (B) (C) (D) (E)

680 480 80 70 60

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29.

How many numbers between 0 and 1670 have a prime tens digit and a prime units digit?

(A) 268 (B) 272 (C) 202 (D) 112 (E) 262 30.

On a ship, a signal can be made by putting from 1 to 6 different colored flags one above the

other on a pole. How many different signals can be made, if the ship has exactly 6 different colored flags? (A) 6! (B) 6!/5!+6!/4!+6!/3!+6!/2!+6!/1!+ 6!/0! (C) 6+5+4+3+2+1 (D) C (6, 6) (E) P (6, 6) 31. John has 12 clients and he wants to use color coding to identify each client. If either a single color or a pair of two different colors can represent a client code, what is the minimum number of colors needed for the coding? (the order of a pair of colors does not matter) (A) 24 (B) 12 (C) 7 (D) 5 (E) 4 32. Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10? (A) 3 (B) 4 (C) 5 (D)6 (E) 7 33. Seven men and seven women have to sit around a circular table so that no 2 women are together. In how many different ways can this be done? (A) 6!7! (B) 7!7! (C) 6!6! (D) 14! (E) 7!2

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34. Find the number of odd 3 digit integers that can be formed from the digits 0,1,2,3,4,5 if no digit can be repeated. (A) 48 (B) 72 (C) 216 (D) 192 (E) 243 35. Aunt Anny has 5 nephews. She bought 10 chocolates and wants to give them all to the children. In how many ways this can be done if it could happen that one, two, three or four children get no chocolate? (A) 5! (B) 5 (C) 10! (D) 5 (E) 10

Probability

1. (OG) A committee is composed of w women and m men. If 3 women and 2 men are added to the committee, and if one person is selected at random from the enlarged committee, then the probability that a women is selected can be represented by (A) w/m (B) w/(w + m) (C) (w + 3)/(m + 2) (D) (w + 3)/(w+ m + 3) (E) (w + 3)/(w+ m + 5)

2. The probability of Sam passing the exam is 1/4. The probability of Sam passing the exam and Michael passing the driving test is 1/6. What is the probability of Michael passing his driving test? (A) 1/24. (B) 1/2. (C) 1/3. (D) 2/3. (E) 2/5 3.

Xavier, Yvonne, and Zelda each try independently to solve a problem. If their individual 1

1

5

probabilities for success are 4 , 2 , and 8 , respectively, what is the probability that Xavier and Yvonne, but not Zelda, will solve the problem? 11 (A) 8

(B) 7 8

(C) 9

64

313

(D) 5 (E)

64 3 64

4. Sixty percent of the members of a study group are women, and 45 percent of those women are lawyers. If one member of the study group is to be selected at random, what is the probability that the member selected is a woman lawyer? (A) 0.10 (B) 0.15 (C) 0.27 (D) 0.33 (E) 0.45 5.

There is a total of 120 marbles in a box, each of which is red, green, blue, or white. If

1 one marble is drawn from the box at random, the probability that it will be white is 4 and the 1 probability that it will be green is 3 . What is the probability that the marble will be either red or

blue ? (A) 1

6 1 (B) 4 (C) 2 7 (D) 1 3 (E) 5 12

6. Each of the eggs in a bowl is dyed red, or green, or blue. If one egg is to be removed at random. what is the probability that the egg will be green? (1) There are 5 red eggs in the bowl. 1 (2) The probability that the egg will be blue is 3

7. A box contains 100 balls, numbered from 1 to 100. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers on the balls selected from the box will be odd? 1 (A) (B) (C) (D)

4 3 8

1 2 5 8

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(E)

3 4

8. Each of the 350 students either is graduate or postgraduate, what is the probability that one student selected at random is a female postgraduate? (1) The number of postgraduates is 187. (2) The number of the female students is 247. 9. (OG) If x is to be chosen at random from the set {1, 2, 3, 4} and y is to be chosen at random from the set {5, 6, 7}, what is the probability that x × y will be even? (A) 1/6 (B) 1/3 (C) ½ (D) 2/3 (E) 5/6 10. Out of a box that contains 4 black and 6 white mice, three are randomly chosen. What is the probability that all three will be black? (A) 8/125 (B) 1/30 (C) 2/5 (D) 1/720 (E) 3/10 11. (GC) A new Machine is bought by a factory to produce 2 parts, A and B. A and B when assembled produce C. During the testing stages of the machine, 1 out of 10 in A is defective and 2 out of 10 in B are defective. What is the probability of C being defective? (A) 0.02 (B) 0.20 (C) 0.28 (D) 0.32 (E) 0.72

12. There are 8 books lying on a table: 4 are hardback book and the other 4 are paperback books. If 3 books are to be selected at random from the 8 books, what is the probability that at least one of the paperback books will be selected? 1 (A) (B) (C) (D) (E) 13.

2 2 3 11 12 13 14 32 35

There are 20 books in a bookcase. If one book is selected at random, what is the

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probability that the book is either a hardback or a novel? (1) 8 books in the bookcase are novel books and 10 books are hardbacks. (2) 3 books in the bookcase are hardback novels. 14. A jar contains only x black balls and y white balls. One ball is drawn randomly from the jar and is not replaced. A second ball is then drawn randomly from the jar. What is the probability that the first ball drawn is black and the second ball drawn is white? (A) (B)

y

x x

y

x

y

x 1 x y 1

x x

y

(C) xy

x y

y

y 1 x y

y

x

(D) x 1 x

(E)

x x

y y 1

15. A machine is made up of two components, A and B. Each component either works or fails. The failure or nonfailure of one component is independent of the failure or nonfailure of the other component. The machine works if at least one of the components works. If the 2

individual probability that each component works is 3 , what is the probability that the machine works? (A) 1 (B) (C) (D) (E)

9 4 9 1 2 2 3 8 9

16. There are 18 balls in a jar. You take out 3 blue balls without putting them back inside, and now the probability of pulling out a blue ball is 1/5. How many blue balls were there in the beginning? (A) 9 (B) 8 (C) 7 (D) 12 (E) 6 17. In a jar there are 3 red balls and 2 blue balls. What is the probability of drawing at least one red ball when drawing two consecutive balls randomly? (A) 9/10

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(B) 16/20 (C) 2/5 (D) 3/5 (E) ½ 18. If a bottle is to be selected at random from a certain collection of bottles, what is the probability that the bottle will be defective? (1) The ratio of the number of bottles in the collection that are defective to the number that are not defective is 3:500. (2) The collection contains 3,521 bottles. 19. The probability that the typist will make a mistake on a page is 1/100. What is the probability that the typist will type 100 pages without making a single mistake? (A)1 (B) 1/10 (C) 99/100 (D) (99/100)100 (E) (1/100)100 20. Chef Gundy is creating a new dessert that will be made from 3 ingredients. If he has 8 cookies and one flavor of sorbet to choose from, what fraction of the possible arrangements will contain the sorbet? (A) 2/3 (B) 1/3 (C) 1/56 (D) 1/9 (E) 1/84 21. Of the 10 employees in a department, 4 have a master's degree. If an action group with three members is to be selected, at random, what is the probability that the group will include at least one employee who has a master's degree? 2 (A) (B) (C) (D) (E)

5 3 5 1 3 1 6 5 6

22. When a coin is tossed for 6 times, what is the probability that after the first tossing, every outcome will be different from the previous one? 1 (A) (B)

16 1 24

317

(C) (D) (E)

1 32 1 48 1 64

23. The probability that it would be sunny at any day of the next week is 20%. Find the probability that it would be sunny at exactly 6 days of the next week. (A) 0.0000512 (B) 0.000064 (C) 0.000128 (D) 0.0003584 (E) 0.0064 24. If four whole numbers taken at random are multiplied together, what is the probability that the last digit in the product is 1, 3, 7 or 9? (A) 0.4 (B) 0.4 (C)0.5 (D) 0.5 (E) 0.6 25. If n is an integer from 1 to 96, what is the probability for n × (n + 1) × (n + 2) being divisible by 8? (A) 25% (B) 50% (C) 62.5% (D) 72.5% (E) 75% 26. If the probability of rain on any given day in city x is 50% what is the probability it with rain on exactly 3 days in a five day period? (A) 8/125 (B) 2/25 (C) 5/16 (D) 8/25 (E) 3/4

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#of question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Topic of question (type) Your answer COMBINATORICS

Correct Answer C A A D A D D C E E D D A D A D C D A D B C B B C B C C A B D E A A D

319

PROBABILITY 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

E D E C E E C E D B C D C E E E A A D B E C D B C C

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Test № 7. Combinatorics, Probability 1. (1) (2)

If w < x < y < z, is the product of four numbers negative? z > 0. wx < 0.

2. If m and n are positive integers, and 80 = 22m + n, what is the value of n? (1) n < 22. (2) m = 3. 3. (1) (2)

What is the remainder when positive integer n is divided by 3? When n – l is divided by 3, the remainder is 2. When n + l is divided by 3, the remainder is 1.

4.

If m and n are positive integers, what is the value of mn? m n

7 15

(1) (2)

The largest common factor of m and n is 3.

5. (1) (2)

What is the value of m2 – n2? m + n = 4. (m – n)2 = 16.

6.

If x and y are none zero, is x2 y3 0 ? x > 0. y > 0.

(1) (2)

x3 y5

7. There are 33 students in a certain class, 18 are girls. If 23 of the students joined a math club, how many boys did not joined the club? (1) 11 girls joined the club. (2) There are fewer girls than boys in the club. 8. Last year in sales of single-home houses, 60% of already-existing houses was sold while 76% of new-established houses sold. If there are three times as many new-established houses as alreadyexisting houses, what percent of single-home houses has been sold in last year? (A) 60 % (B) 64 % (C) 72 % (D) 78 % (E) 80 % 9. In a certain company, if 75% of the employees usually use a laptop, what percent of the employees usually use a PDA (Personal Digital Assistant)? (1) 60% of the employees who usually use a laptop also use a PDA. (2) 90% of the employees who usually use a PDA also use a laptop. 10. During a certain trading day, the opening price for stock X is $40 and the opening price for stock Y is $25; the closing price for Stock X is $41, and the closing price for stock Y is $28.5. The

321

percentage increase of stock Y is how much greater than the percentage increase of stock X? (A) 115 (B) 11.5 (C) 12 (D) 12.5 (E) 125 11. (1) (2)

If  the  units’  digit  of  integer  x  is  greater  than  1,  what  is  the  units’  digit? The  units’  digit  of  x3 and  the  units’  digit  of  x  are  the  same. The  units’  digit  of  x2 is 6.

12. If point P is inside a circle with center at origin and radius of 2, is point Đš inside the same circle? (1) OP = 1. (2) PK = 3.5.

13. As the figure above shows, four identical circles are inscribed in a square. If the distance between centers of circle A and Đ’ is 4 2 , what is the area of the shaded region? (A) 256 – 72đ?œ‹ (B) 256 – 64đ?œ‹ (C) 256 – 32đ?œ‹ (D) 128 – 32đ?œ‹ (E) 64 – 16đ?œ‹ 14. Each of the 350 students either is graduate or postgraduate, what is the probability that one student selected at random is a female postgraduate? (1) The number of postgraduates is 187. (2) The number of the female students is 247.

(A) (B) (C) (D) (E)

15. An  analyst  will  recommend  a  combination  of  3  industrial  stocks,  2  transportation  stocks,  and  2  utility  stocks.   If  the  analyst  can  choose  from  5  industrial  stocks,  4  transportation  stocks,  and  3  utility  stocks,  how  many  different  combinations  of  7  stocks  are  possible? 12 19 60 180 720 16. A  club  with  a  total  membership  of  30  has  formed  3  committees,  M,  S,  and  R,  which  have  8,  12,  and  5  members,  respectively.  If  no  member  of  committee  M  is  on  either  of Â

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the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees? (A) (B) (C) (D) (E)

(1) (2)

5 7 8 10 12 17. There are two kinds of books on the bookshelf: Spanish and English. If two books are randomly chosen from the bookshelf without replacement, what is the possibility that at least one of the books chosen is English book? The ratio of Spanish books to English book is 3:1. There are less than 20 books on the shelf.

18. When a coin is tossed for 6 times, what is the probability that after the first tossing, every outcome will be different from the previous one? (A) (B) (C) (D) (E)

1 16 1 24 1 32 1 48 1 64

19. There are 4 red marbles and 2 blue marbles in jar A, and there are 3 red marbles and 2 blue marbles in jar B. If one marble is to be selected at random from each jar, what is the probability that exactly one red marble and one blue marble will be selected? (A) (B) (C) (D) (E)

1 5 4 15 1 3 7 15 7 11

20. In how many arrangements can a photographer seat 3 girls and 3 boys in a row of 6 seats if the boys are to have the first, third, and fifth seats? (A) 6 (B) 9 (C) 12

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(D) (E)

36 720

21. A machine is made up of two components, A and B. Each component either works or fails. The failure or nonfailure of one component is independent of the failure or nonfailure of the other component. The machine works if at least one of the components 2 works. If the probability that each component works is 3 , what is the probability that the machine works? 1 (A) 9 4 (B) 9 1 (C) 2 2 (D) 3 8 (E) 9 22. Twelve jurors must be picked from a pool of n potential jurors. If m of the potential jurors are rejected by the defense counsel and the prosecuting attorney, how many different possible juries could be picked from the remaining potential jurors? (1) If one less potential juror had been rejected, it would be possible to create 13 different juries. (2) n = m + 12 23. What is the value of x? (1) (2)

(A) (B) (C) (D) (E)

x4

9

The average (arithmetic mean) of x2, 6x, and 3 is – 2.

24. The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression? 300 120 150 170 270

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Answers: 1. B 2. D 3. D 4. C 5. E 6. A 7. A 8. C 9. C 10. B 11. E 12. C 13. E 14. E 15. D 16. D 17. E 18. C 19. D 20. D 21. E 22. D 23. B 24. C

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Practice test

326

Practice test 1. (Medium) The price of a certain commodity increased at a rate of X% per year between 2000 and 2004. If the price was M dollars in 2001 and N dollars in 2003, what was the price in 2002 in terms of M and N? (A) √(M × N) (B) N × √(N/M) (C) N × √M (D) N × M/√N (E) N × M3/2

2. (Medium) It took the bus three hours to get from town A to town B. What was the average speed of the bus for the entire trip? (1) After one hour the bus finished 1/3 of the distance going at 60 km/h (2) During the second hour the average speed of the bus was 120 km/h, twice its speed during the third hour (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

3. (Easy) Driving 1.5 times slower, Bill was late for school today. What is the usual time it takes Bill to drive to school? (Assume that each day Bill takes the same route). (1) It took Bill 15 more minutes to drive to school today than usually (2) The distance between home and school is 15 miles (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

327

4. (Medium) Is integer N even? (1) N × N = N (2) N = N3 (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

5. (Hard) Two sets are defined as follows: A = {2, 3, 4, 4, 4} B = {0, 1, 2}.

If a number is taken from set A at random and another number is taken from set B at random, what is the probability that the sum of these numbers is a prime integer? (A) 1/15 (B) 2/15 (C) 5/15 (D) 7/15 (E) 9/15

6. (Medium) Is integer R even? (1) R3 + R2 is even (2) 3×R is divisible by 6 (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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(D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

7. (Hard) Among 200 people 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam? (A) 20 (B) 60 (C) 80 (D) 86 (E) 92

8. (Easy) The sequence is defined as follows: A(1) = 1; A(2) = - 1 A(n + 1) = A(n) + 2 × A(n - 1). What is the sum A(1) + A(2) + ... + A(1001) ? (A) - 2 (B) - 1 (C) 0 (D) 1 (E) 2

9. (Easy) Set T consists of odd integers divisible by 5. Is standard deviation of T positive? (1) All members of T are positive (2) T consists of only one member (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient

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(E) Statements (1) and (2) TOGETHER are NOT sufficient

10. (Hard) If quadrilateral ABCD is inscribed into a circumference. What is the value of angle A? (1) AC = CD (2) Measure of angle D is 70 degrees (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

11. (Hard) Is |x - 1| < 1? (1) (x - 1)2 ≤ 1 (2) x2 - 1 > 0 (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

12. (Medium) What is the value of X? (1) X! is odd (2) X is even (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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(D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

13. (Hard) If the farmer sells 75 of his chickens, his stock of feed will last for 20 more days than planned, but if he buys 100 more chickens, he will run out of feed 15 days earlier than planned. If no chickens are sold or bought, the farmer will be exactly on schedule. How many chickens does the farmer have? (A) 60 (B) 120 (C) 240 (D) 275 (E) 300

14. (Medium) The 19th of September 1987 was Saturday. What day was the 21st of September 1990 if 1988 was a leap - year? (A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday

15. (Medium) What is the value of integer J? (1) |J| = J - 1 (2) JJ = 1

(A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

331

16. (Medium) If n = p/q (p and q are nonzero integers) is n an integer? (1) n2 is an integer (2) (2n + 4)/2 is an integer (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

17. (Medium) Two payment schemes are available for customers in the N&K store. The first scheme implies a down payment of 20% of the purchase price and 10 monthly payments of 10% each. The second implies a down payment of 10% and 20 monthly payments of 8% each. If a customer buys a TV for $216, by what percent will he find the first scheme cheaper than the second (approximately)? (A) 14% (B) 27% (C) 30% (D) 34% (E) 35%

18. (Medium) A man cycling along the road noticed that every 12 minutes a bus overtakes him while every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at constant speed, what is the time interval between consecutive buses? (A) 5 minutes (B) 6 minutes (C) 8 minutes (D) 9 minutes (E) 10 minutes

19. (Easy) What is the hundreds digit of 210 + 25 + 1?

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(A) 0 (B) 1 (C) 2 (D) 5 (E) 8

20. (Medium) Richard is 3 years younger than his sister. How old will Richard be in 5 years? (1) Two years ago Richard was twice as young as his sister (2) If Richard's sister were born 2 years earlier, she would now be twice as old as Richard (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

21. (Hard) Mary and Joe are to throw three dice each. The score is the sum of points on all three dice. If Mary scores 10 in her attempt what is the probability that Joe will outscore Mary in his? (A) 24/64 (B) 32/64 (C) 36/64 (D) 40/64 (E) 42/64

22. (Medium) M, N, and O are midpoints of sides AB, BC, and AC of triangle ABC. What is the area of triangle MON? (1) The area of ABC is √3/4 (2) ABC is an equilateral triangle with height √3/2 (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient

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(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

23. (Medium) Is the mean of a non - empty set S bigger than its median? (1) All members of S are consecutive multiples of 3 (2) The sum of all members of S equals 75 (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

24. (Easy) If X is an odd integer, which of the following numbers must be even? (A) X3/3 (B) (X2 - 1)/2 (C) X × p + 1 where p is a prime number (D) (X + 7) × (X - 2)/2 (E) XX - 1

25. (Hard) The diameter of each of the car's wheels is 20 inches. What was the average speed of the car in km/h if after the two-hour journey each wheel made 75,000 revolutions? (1 inch = 0.0252 meter) (A) 32 (B) 41 (C) 59 (D) 74 (E) 88

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26. (Medium) 6 students in the group study different languages as specified: Russian 4 Ukrainian

3

Hebrew

2

Each student studies at least 1 language. It is also known that exactly 3 students learn exactly 2 languages. How many students are studying all languages? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

27. (Hard) What is the last digit of the following number [222] × [315] × [516] × [71]? (A) 6 (B) 5 (C) 2 (D) 1 (E) 0

28. (Hard) A soccer coach riding his bike at 24 km/h reaches his office 5 minutes late. Had he traveled 25% faster, he would have reached the office 4 minutes earlier than the scheduled time. How far is his office from his house? (A) 18 km (B) 24 km (C) 36 km (D) 40 km (E) 72 km

29. (Medium) 20% of employees are women with fair hair. 40% of fair-haired employees are women. What percent of employees have fair hair? (A) 25

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(B) 30 (C) 45 (D) 50 (E) 60

30. (Hard) Among 5 children there are 2 siblings. In how many ways can the children be seated in a row so that the siblings do not sit together? (A) 38 (B) 46 (C) 72 (D) 86 (E) 102

31. (Medium) A, B, and C are points on the plane. Is AB > 15? (1) BC + AC > 14 (2) Area of triangle ABC < 1 (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

32. (Easy) What is (1/16) - 1/4? (A) - 1/2 (B) 2 (C) 2562 (D) 1/2 (E) 4

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33. (Medium) In what proportion should the 30% - solution and the 3% - solution be mixed to obtain 12% - solution? (A) 1:3 (B) 1:2 (C) 2:3 (D) 3:4 (E) 4:5

34. (Medium) A plane takes off from the hill at 750 meters above the sea level and lands some time later in a town located at 50 meters below the sea level. During the first part of its flight the plane gained height at a rate of 50 meters per minute but then it started to descend at a rate of 20 meters per minute. The duration of the first part of the flight was what percent of the total flight time? (1) The duration of the descent is known (2) The total flight time is known

(A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

35. (Medium) If operation $ is defined as $X = X + 2 if X is even, $X = X - 1 if X is odd, What is $(...$($($(15)))...) 99 times? (A) 120 (B) 180 (C) 210 (D) 225 (E) 250

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36. (Easy) What is X? (1) X2 - 1 = X + 1 (2) X + 3 is a prime number (A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient (B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient (E) Statements (1) and (2) TOGETHER are NOT sufficient

37. (Medium) 160 kg of dry wood pulp is needed to produce 50 sheets of paper. How much raw pulp will be needed to produce 200 sheets of paper if raw pulp loses 20 percent of its weight during desiccation? (A) 400 (B) 600 (C) 640 (D) 800 (E) 850

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Answers and explanations for practice test

1. {A} Price in 2003 was M × (1 + X/100)2 = N. From this equation (1 + X/100) = √(N/M). Price in 2002 was M × (1 + X/100) = M × √(N/M) = √(N × M). Let's check the answer: price in 2003 = (price in 2002) × (1 + X/100) = √(N × M) × √(N/M) = N as in the stem. 2. {A} From Statement (1) it follows that 1/3 of the distance amounts to 60 km. Therefore, the entire distance is 180 km and the average speed for the entire trip is 180 km / 3 h = 60 km/h. Statement (2) gives no information about the first hour. The bus could be nearly flying during the first hour (and then its average speed for the entire trip would be high) or it could be crawling (and then its average speed for the entire trip would be low). 3. {A} Statement 1 tells us that on average Bill drives for 15 minutes less. Since he drove 1.5 times slower, the 15 minutes account for the difference. Therefore, it is possible to find the difference. Statement 2 does not tell us much; we don't really need to know the distance and by itself it is insufficient. 4. {E} Statement (1) says that N can be 0 or 1. Not enough Statement (2) says that N can be 0, 1, or - 1. Not enough again. Combine: N can be 0 (even) or 1 (odd). So the answer is E. 5. {D} There is no way how we can obtain a prime bigger than 5. In all, there are three possibilities for the sum to be a prime number (P denotes probability function): P(the sum is 2) = P(2 from A, 0 from B) = 1/5 × 1/3 = 1/15 P(the sum is 3) = P(2 from A, 1 from B) + P(3 from A, 0 from B) = 1/5 × 1/3 + 1/5 × 1/3 = 2/15 P(the sum is 5) = P(3 from A, 2 from B) + P(4 from A, 1 from B) = 1/5 × 1/3 + 3/5 × 1/3 = 4/15. Summing up: P(the sum is prime) = P(the sum is 2) + P(the sum is 3) + P(the sum is 5) = 7/15. 6. {B} Statement (1) is not sufficient, because for R = 2 or 3, R3 + R2 is even; 12 and 36. Statement (2) is sufficient because to be divisible by 6, 3R needs to contain a 2, and thus be even. So the answer is B. 7. {B} The general equation is S + A + R - SA - SR - AR + SAR + N = 100% where N denotes the number of people who like none of the three jams (draw the Venn diagram). To maximize the number of people who like raspberry but not strawberry or apple jam N has to be 0. The new equation can be rewritten as follows: R - AR - SR + SAR = 100% + SA - A - S. The left side of this equation is precisely what the question asks for - the number of people who like raspberry jam but do not like strawberry or apple jam (Venn diagram will help to understand why it is so). Plucking the numbers: R - AR - SR + SAR = 100% + 30% - 44% - 56% = 30%. Finally, 30% of 200 is 60. 8. {D} Let's write out some more terms of this sequence: A(3) = 1; A(4) = - 1; A(5) = 1. It is clear that even terms equal - 1 and odd terms equal 1. Therefore, the sum A(1) + A(2) + ... + A(1001) = 1.

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9. {B} This question tests the basics. Standard deviation of a set is always non - negative and equals 0 only if all members of the set are the same. Statement (1) doesn't help to answer the question. Statement (2) says that there is only one number in T and therefore standard deviation of T is 0 (not positive). 10. {E} Statement (1) gives us nothing useful for solving the problem, only that angle A is bigger than angle D. Statement (2) tells us that angle B = 110 degrees from the equation: angle A + angle C = angle B + angle D = 180 for inscribed quadrilaterals. So neither statement is sufficient. The answer is E. 11. {E} Statement (1) is equivalent to |x - 1| ≤ 1. If x = 0 or x = 2. Statement (1) holds but the principal statement is not true. Statement (2) is not sufficient either. If x = 1.5 the principal statement holds but if x is big, say 5, the principal statement is not true. Although Statement (2) means that x cannot be 0 it does not exclude the possibility of x being 2. Thus, Statement (1) and Statement (2) combined are not sufficient to answer the question. 12. {C} From statement (1) X! is odd, so X can be either 1 or 0 as 0! = 1. Insufficient The statement (2) gives that X is even. Not sufficient at all. Combining two statements, we have to choose even number out of 1 and 0. 0 is even, so the answer is C. 13. {E} Let V denote the volume of feed one chicken consumes per day. Then the total volume of feed in stock will be V × D × C where D is the number of days the feed will last if the number of chickens does not change and C is the current number of chickens. From the stem it follows that V × (D + 20) × (C - 75) = V × D × C; V × (D - 15) × (C + 100) = V × D × C. The first equation simplifies to 20 × C - 75 × D = 1500. The second equation simplifies to ( - 15) × C + 100 × D = 1500. After dividing everything by 5 we get the linear system: 4 × C - 15 × D = 300 ( - 3) × C + 20 × D = 300. After solving it we get C = 300, D = 60. 14. {E} If it were not for the leap - year the 19th of September in 1988 would be Sunday, in 1989 it would be Monday, and in 1990 it would be Tuesday (because there are 365 days in a year and 365 = 52 × 7 + 1). Allowing for the leap - year the 19th of September in 1990 should be Wednesday. Therefore, the 21st of September 1990 should be Friday. 15. {D} From the first statement we have that J can be 1 (it cannot equal to 0 or - 1). Thus (1) is sufficient. From statement (2), we learn that J can be either 1; apparently 00 is a non existent value. 16. {D} Statement (1) states that p2/q2 is an integer and as both are integers p/q must also be an integer. Sufficient. Statement (2) is the same as n + 2 which is an integer. Therefore, n also must be an integer. 17. {C} Please note that the price of $216 is irrelevant to this question and is only cited to trap you into lengthy calculations. For any price X the customer would pay: - By the first scheme: 0.2 × X + 10 × 0.1 × X = 1.2 × X - By the second scheme: 0.1 × X + 20 × 0.08 × X = 1.7 × X.

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The percentage difference = 100% × (1.7 × X - 1.2 × X)/(1.7 × X) = 100% × 0.5/1.7 = approximately 100% × 0.51/1.7 = 30%. 18. {B} Let Vb denote the speed of the bus and Vc the speed of the cyclist. Then the distance between the buses = 4 × (Vb + Vc) = 12 × (Vb - Vc), so Vb = 2 × Vc. Interval between the buses = (distance between the buses)/(speed of the bus) = 4 × (Vb + Vc)/Vb = 4 × (3/2 × Vb)/Vb = 6 minutes. 19. {A} The quickest way is to count: 210 + 25 + 1 = 1024 + 32 + 1 = 1057. The hundreds digit of 1057 is 0. 20. {D} Let’s use R for Richard's age, S for his sister's age. From Statement (1) an equation can be built 2 × (R - 2) = S - 2. As S = R + 3 this equation can be solved for R and thus R + 5 can also be determined. R = 5 and R + 5 = 10. From Statement (2) another equation can be built S + 2 = 2 × R or R + 5 = 2 × R from where R = 5 and R + 5 = 10. 21. {B} This is a look-for-shortcuts problem. To outscore Mary Joe has to score in the range of 11 - 18. The probability to score 3 is the same as the probability to score 18 (1 - 1 - 1 combination against 6 - 6 - 6, if 1 - 1 - 1 is on the tops of the dice the 6 - 6 - 6 is on the bottoms). By the same logic, the probability to score x is the same as the probability to score 21 - x. Therefore, the probability to score in the range 11 - 18 equals the probability to score in the range of 3 - 10. As 3 - 18 covers all possible outcomes the probability to score in the range 11 - 18 is 1/2 or 32/64. 22. {D} Triangle MON is similar to triangle ABC (that is all sides of MON are proportional to the sides of ABC). In fact, MON is one of the four equal triangles that ABC breaks into if lines MN, MO, and NO are drawn. Thus, Statement (1) is sufficient to answer the question - the area of MON = 1/4 × (area of ABC). Information given by Statement (2) is sufficient to find the area of ABC and therefore is also sufficient to find the area of MON. 23. {A} Statement (1) says that members of S are evenly distributed in the set. In this case, the mean = the median = (first member + last member)/2. This holds even if S contains only one member. Statement (2) gives no valuable information. S can be {75} with the answer to the question NO or it could be {1, 2, 72} with the answer to the question YES. 24. {B} (X2 - 1)/2 can be written as (X - 1) × (X + 1)/2 = even × even/2 = even × integer = even. To see that other choices are not necessarily even consider X = 3 (in C consider p = 2). 25. {C} The key thing in this question is to avoid difficult calculations. Note that the answer choices are very much apart so we can round numbers in calculations wherever possible and then choose the answer choice which is closest to the result we obtained. The wheel's radius in meters = 20 inches / 2 × 0.0252 = 0.252 = approximately 0.25 meters. The distance in kilometers = 75000 × 1.5/1000 = 75 × 1.5 = 112.5 = approximately 113 km. The average speed in km/h = 113/2 = 56.5 km/h which is closest to choice C. 26. {A} We have a total of 9 languages taken by 6 students. If 3 of them study 2 languages, we have 3 languages left for 3 people to study. Therefore, since students are studying at least one language, nobody is taking 3. 27. {E} Since we have 2 × 5 in the product the last digit will for sure be 0. 28. {A} This is a good problem. The best choice is backsolving. However, we can create an

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equation as well: 29. {D} Let X denote the percent of fair-haired employees. Then 0.4 × X = women with fair hair = 20% from where X = 50%. 30. {C} Without limitations 5 children can be seated in 5! = 120 ways. What is the number of ways to seat the 5 children so that the siblings DO sit together? The siblings can be regarded as one unit so there are 4! combinations. But within this unit the siblings can sit in two different ways. So the number of ways to seat the 5 children so that the siblings DO sit together is 4! × 2 = 48. Thus, the number of combinations in which the siblings DO NOT sit together is 120 - 48 = 72. 31. {E} Statement (1) allows AB to be of any length. Statement (2) also allows AB to be of any length: if AB is long angle ACB has to be close to 180 degrees to make the area of the triangle smaller than 1. Statements (1) and (2) combined are not sufficient either. AB can be long (in this case angle ACB has to be close to 180) or it can be short (in this case angle ACB has to be close to 0). 32. {B} (1/16) - 1/4 = 161/4 = 2. 33. {B} Let X denote the volume of the 30% - solution and Y the volume of the 3% solution. (0.3 × X + 0.03 × Y)/(X + Y) = 12% or 30 × X + 3 × Y = 12 × X + 12 × Y. From here 18 × X = 9 × Y or X:Y = 1:2. 34. {D} A for time spent in ascent, D for time spent in descent. From the stem it is clear that 20 × D - 50 × A = 750 + 50 = 800. The question asks for A/(A + D) × 100%. Statement (1) says that D is known. From the equation above A can be found and thus A/(A + D) × 100% can be determined. Statement (2) says that A + D is known. This allows expressing A in terms of D and finding D from the equation above. A/(A + D) × 100% can then be determined. 35. {C} $(15) = 14. We have to compute $(...$($($(14)))...) 98 times. Each $ just adds 2 to the previous result. Therefore, $(...$($($(14)))...) 98 times = 14 + 2 × 98 = 210. 36. {E} From Statement (1) X is either 2 or - 1. Be careful not to cancel out the factor (X + 1) for it can equal 0. Statement (2) does not help to determine whether X is - 1 or 2 as both 2 and 5 are primes. 37. {D} Let X denote the required amount of raw pulp. This X kg of raw pulp will turn into 0.8 × X kg of dry pulp after desiccation. Now we can compose a proportion: 0.8 × X/200 = 160/50 from where X = 800.

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Key Glossary Arithmetic

Absolute Value Addend Addition Additive Inverse Approximation Braces Brackets Canceling Cardinal number Carry Circulating decimal Common Denominator Common Factors Common Multiples Complex Fraction Composite Number Consecutive Numbers Cube Cube Root Decimal Fraction Decimal Point Denominator Difference Distributive property Divisibility Even Number Expanded Notation Factor Factorization Finite Fraction Greatest Common Factor T Hundredth Imaginary Numbers Improper Fraction Infinite Integer Interval Invert Irrational Least Common Multiple Lowest Common Denominator Minuend

Модуль Слагаемое Сложение Обратное число Приближение {} [] Сокращение (дробей) Количественное числительное Перенос Периодическая дробь Общий знаменатель Общие множители Общее кратное Сложная дробь Составное число Последовательные числа Куб Кубический корень Десятичная дробь Точка в десятиной дроби Знаменатель Разница Распределительный закон Делимость Четное число Запись числа разложением на разряды Множитель Разложение на множители Конечное число Обыкновенная дробь Наибольший общий делитель Сотые Мнимое число Неправильная дробь Бесконечный Целое число Промежуток Обращение Иррациональное число Наименьшее общее кратное Наименьший общий знаменатель Уменьшаемое (вычитаемое)

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Mixed Number Multiples Multiplicand Multiplicative Inverse Multiplier Natural Negative Number Number Line Numerator Odd Number Operation Operand Order of Operations Ordinal number Parentheses Percentage Periodic Place Positive Number Prime Number Product Proper Fraction Proportion Quotient Ratio Rational Number Real Number Reciprocal Reduce Rounding Scientific Notation Subtrahend Sum Tenth Terminating Decimal Unknown Value Whole Number

Смешанное число Кратное Множимое Взаимно обратные Множитель Натуральное число Отрицательное число Числовая прямая Числитель Нечетное число Операция Операнд Порядок действий Порядковое числительное () Процент Периодическая дробь Разряд Положительное число Простое число Произведение Правильная дробь Пропорция Частное Отношение Рациональное число Вещественное число Обратное (данному) число Упрощение (выражения) Округление Экспоненциальное представление числа Вычитаемое Сумма Десятые Конечное десятичное число Неизвестная величина Значение Целое число

Algebra Algebra Algebraic Fractions Ascending Order Axiom Binomial Binomial Coefficient Binomial Equation

Алгебра Алгебраическая дробь Возрастающий порядок Аксиома Двучлен Коэффициенты двучлена Квадратное уравнение

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Closed Interval Coefficient Constant Cross-multiplication Degree of Polynomial Descending Order Equation Evaluate Exponent Extremes Extract a Root Factorization of Algebraic Equations Fractional Equation Fractional Exponent Half-Open Interval Incomplete Quadratic Equation Leading Coefficient Like Term Linear Equation Literal Equation Monomial Nonlinear Equation Open Interval Polynomial Power Proper Fraction Quadratic Equation Radical Sign Radicand Reduced Equation Solution Set or Solution Square Square Root System of Equations Term Theorem Trinomial Variable

Отрезок Коэффициент Константа Перекрестное умножение Степень многочлена Убывающий порядок Уравнение Вычислять Показатель степени Граничные значения (отрезка) Извлечение корня Разложение (на множители) Дробное уравнение Дробный показатель степени Луч Неполное квадратное уравнение Старший коэффициент Подобные члены Линейное уравнение Буквенное выражение (уравнение) Одночлен Нелинейное уравнение Открытый интервал Многочлен Степень Правильная дробь Квадратное уравнение Радикал Подкоренное выражение Упрощенное уравнение Решение, множество решений Квадрат Квадратный корень Система уравнений Элемент Теорема Трехчлен Переменная

Geometry Acute Angle Acute Triangle Adjacent Angles Alternate Angle Alternate Exterior Angle Alternate Interior Angle

Острый угол Острый треугольник Смежные углы Противолежащий угол Противолежащий внешний угол Противолежащий внутренний угол

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Altitude Altitude of a Triangle Angle Angle of Depression Angle of Elevation Angular Bisector Arс Area Axis of Symmetry Bar Base Bisect Bisector of an Angle Central Angle Center of a Circle Chord Circle Circumcenter Circumcircle Circumference Circumradius Circumscribed Circumscribed circle Complementary Angles Concave Concentric Circles Cone Congruent Congruent Squares Congruent Triangles Consecutive Angles Convex Corresponding Cube Decagon Degree Diagonal of Polygon Diameter Edge Ellipse Equiangular Polygon Equilateral Polygon Equilateral Triangle Exterior Angle Face Height Heptagon Hexagon Hypotenuse

Высота Высота треугольника Угол Угол понижения Угол повышения Биссектриса угла Дуга Область Ось симметрии (Горизонтальная) черта Основа Разделение пополам Биссектриса угла Центральный угол Центр окружности Хорда Окружность Центр окружности Описанная окружность Длина окружности Радиус окружности Описанный Описанная окружность Дополняющие углы до 90° Вогнутый Концентричные окружности Конус Конгруэнтный Равные квадраты Равные треугольники Последовательные углы Выпуклая фигура Соответствующий Куб Десятиугольник Градус Диагональ многоугольника Диаметр Ребро Эллипс Равноугольный многоугольник Равносторонний многоугольник Равносторонний треугольник Внешний угол Грань или плоская поверхность Высота Семиугольник Шестиугольник Гипотенуза

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Incongruent Inscribed Angle Interior Angles Intersecting Lines Isosceles Right Triangle Isosceles Triangle Legs Line Segment Median Midpoint Minute Nonagon Oblique Oblong Obtuse Angle Obtuse Triangle Octagon Oval Parallel Lines Parallelogram Parallelepiped Pentagon Perimeter Perpendicular Lines Π Plane Plane Figure Plane Geometry Point Polygon Prism Pythagorean Theorem Quadrilateral Radius Ray Rectangle Regular Polygon Rhombus Right Angle Right Circular Cylinder Right Triangle Scalene Triangle Segment Similar Solid Geometry Square Straight Angle Straight Line Supplementary Angles

Несравнимый Вписанный угол Внутренние углы Пересекающиеся линии Равнобедренный прямоугольный треугольник Равнобедренный треугольник Катеты Отрезок Медиана Средняя точка Минута Девятиугольник Неортогональный Вытянутый Тупой угол Тупой треугольник Восьмиугольник Овал Параллельные линии Параллелограмм Параллелепипед Пятиугольник Периметр Перпендикулярные прямые Число Пи (~3.14) Плоскость Плоская фигура Геометрия плоскости Точка Многоугольник Призма Теорема Пифагора Четырехугольник Радиус Луч Прямоугольник Правильный многоугольник Ромб Прямой угол Прямой круговой цилиндр Прямой треугольник Разносторонний треугольник Сегмент Подобный Пространственная геометрия (стереометрия) Квадрат Развернутый угол Прямая Смежные углы

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Surface Area Tangent to A Circle Trapezoid Triangle Vertex Vertical Angles Vertices Volume

Площадь поверхности Касательная Трапеция Треугольник Вершина Вертикальные углы Вершины Объем

Coordinate Geometry Abscissa Analytic Geometry Cartesian Coordinates Coordinate Axes Coordinate Plane Cross Point Hyperbola Intercept Open Ray Ordered Pair Ordinate Origin Parabola Quadrants Rectangular Coordinates Slope Transversal X-Axis X-Coordinate Y-Axis Y-Coordinate

Абсцисса (х-координата) Аналитическая геометрия Прямоугольные координаты Оси координат Координатная плоскость Точка пересечения Гипербола Точка пересечения с осью Открытый луч Упорядоченная пара Ордината (у-координата) Начало координат Парабола Квадрант (четверть) Прямоугольные (декартовые) координаты Наклон прямой Пересекающая линия Ось абсцисс Абсцисса Ось ординат Ордината

Word Problems Balance Biannual Compound Interest Dividend Percentage Change Quarterly Rate Semiannual Simple Interest Velocity

Итог Двухгодичный Сложный процент Дивидент Процентное изменение Квартально Скорость Полугодовой Простой процент Скорость

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Statistics Element Empty Set Equal Sets Equivalent Sets Mean (Arithmetic) Median Mode Null Set Range Roster Set Subset Universal Set

Элемент Пустое множество Равные множества Одинаковые множества Средняя величина Срединный Мода Пустое множество Разброс Перечень элементов множества Множество Подмножество Универсальное Множество

Combinatorics & Probability Combinations Dependent Events Independent Events Intersection of Sets Permutations Probability Union of Sets Venn Diagram

Сочетание Зависимые события Независимые события Пересечение множеств Перестановка Вероятность Объединение множеств Диаграмма Венна

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Appendix. Arithmetic to Memorize Operation Addition Subtraction Multiplication Division

Symbol + ÷

Result Sum Difference Product Quotient

Example 16 is the sum of 12 and 4 8 is the difference of 12 and 4 48 is the product of 12 and 4 3 is the quotient of 12 and 4

16=12+4 8=12-4 48=12 4 3=12:4

English words Is, was, will be, had, has, will have, is equal to, is the same as Plus, more than, sum, increased by, added to, exceeds, received, got, older than, farther than, greater than Minus, fewer, less than, difference, decreased by, subtracted from, younger than, gave, lost Times, of, product, multiplied by Divided by, per, for, quotient

Mathematical meaning Equals

Symbol =

Addition

+

Subtraction

-

Division

:,

More than, greater than At least Fewer than, less than At most What, how many, etc

Inequality Inequality Inequality Inequality Unknown quantity

> ≥ < ≤ X (or some other variable)

Multiplication

Multiplication table 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

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Fractions to decimals 1 = 0.2 5

3 = 0.6 5

1 = 0.25 4

3 = 0.75 4

1 = 0.125 8

3 = 0.375 8

2 = 0.4 5

4 = 0.8 5

5 = 0.625 8

3 = 0.6 5

1 = 0. (3) 3

2 = 0. (6) 3

Powers of integers 21 = 2

31 = 3

41 = 4

51 = 5

22 = 4

32 = 9

42 = 16

52 = 25

23 = 8

33 = 27

43 = 64

53 = 125

24 = 16

34 = 81

44 = 256

54 = 625

25 = 32

35 = 243

45 = 1024

55 = 3125

26 = 64 27 = 128 28 = 256 29 = 512 210 = 1024

Squares of integers 1 =1

11 = 121

2 =4

12 = 144

3 =9

13 = 169

4 = 16

14 = 196

5 = 25

15 = 225

6 = 36

16 = 256

7 = 49

17 = 289

8 = 64

18 = 324

9 = 81

19 = 361

10 = 100

20 = 400

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