Fresh 2

Question 1

“There is no integer n such that x is divisible by n and

1 < n < x.”

What does this statement mean?

Answer 1

x is prime.

If there is no integer smaller than x and greater than 1 that will deliver an integer when x is devided by it, then x has only two factors: 1 and itself. x = prime

Question 2

Solve using Test Cases strategy

If p, q and r are integers, is pq + r even?

(1) p + r is even
(2) q + r is odd

Answer 2

E

Question 3

You score 80 in your mid-term exam and 100 on your final exam. Only these two exams make up your final grade of 92. How heavily did your teacher weight the final exam?

Answer 3

60%

Let the weighting of the 100 grade be x.

80(1 - x) + 100x = 92

20x = 12

x = 3/5 = 60%

Question 4

Write the expression of 3 less than X.

Answer 4

X - 3

Question 5

x > y

Is 1/x > 1/y?

Answer 5

No.

When numbers are reciprocals, the inequality sign changes. For example x = 4 and y = 3

4 > 3 and 1/4 < 1/3

Question 6

Greg, Marcia, Peter, Jan, Bobby and Cindy go to a movie and sit next to each other in 6 adjacent seats in the front row of the theater. If Marcia and Jan will not sit next to each other, in how many different arrangements can the six people seat?

Answer 6

Arrangements with constraints

First, we calculate the total possibilities of arrangements:

6! = 720

Now we take the constraints. It will be easier to calculate the number of cases in which Jan and Marcia would sit together and then subtract. Here woulc be usefull to use the glue method (like Marcia and Jan are glued as one person). So, the posibilities would equal 5! = 120

Now, 120 * 2 = 240 (since they can sit as J - M or M - J)

720 - 240 = 480 possible combinations

Question 7

If n is a positive integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten

Answer 7

E.

Test cases:

Can it be 0? Yes, 10^3=1,000.
Can it be 1? Yes, 1^3=1.
Can it be 2? Yes, 8^3=512.
Can it be 3? Yes, 7^3=343.
Can it be 4? Yes, 4^3=64.
Can it be 5? Yes, 5^3=125.
Can it be 6? Yes, 6^3=…6.
Can it be 7? Yes, 3^3=27.
Can it be 8? Yes, 2^3=8.
Can it be 9? Yes, 9^3=…9.

Question 8

Three grades of milk are 1 percent, 2 percent and 3 percent fat by volume. If x gallons of the 1 percent grade, y gallons of the 2 percent grade, and z gallons of the 3 percent grade are mixed to give x+y+z gallons of a 1.5 percent grade, what is x in terms of y and z?

Answer 8

We know that 1%x + 2%y + 3%z = 1.5%(x + y + z)

Thus,

x + 2y + 3z = 1.5x + 1.5y + 1.5z

0.5x = 0.5y + 1.5z (multiply by 2)

x = y + 3z

Question 9

If x and k are integers and (12^x)[4^(2x+1)] = (2^k)(3²), what is the value of k?

Answer 9

k = 14

Question 10

10^x = (4^y)(5^z)

If x, y, and z are integers, is x even?

Answer 10

Yes.

10^x = (4^y)(5^z)

(2^x)(5^x) = (2^2y)(5^z)

Thus, x = 2y and z = x.

2y is even (any number multiplied by 2 is even), x must be even too.

Question 11

If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers?

(1) The range of the n integers is 14
(2) The greatest of the n integers is 17

Answer 11

D.

Question 12

What is the difference of the meaning between:

1. “Point x lies on the circle.”
2. “Point x lies in the circle.”

Answer 12

1. On the circle means that x lies on the circles border, on its circumference.
2. In the cirle means that point x lies inside the circle, not on its circumference.

Question 13

Renee has a bag of 6 candies, 4 of which are sweet and 2 of which are sour. Jack picks two candies simultaneously and at random. What is the chance that exactly 1 of the candies he has picked is sour?

Answer 13

Probability trees

1. First candy will be wither sweet or sour. 2/6 chance it will be sour and 4/6 chance it will be sweet.
2. 1. If the first candy we pick is sour, than the second candy has 1/5 chance to be sour and 4/5 chance to be sweer. Thus, the probability to get only one sour candy when the first candy is sour = 2/6 * 4/5 = 4/15
3. 2. If the first candy we pick is sweet, than the second candy has 2/5 chance to be sour and 3/5 to be sweet. Thus, the probability to get only one sour candy when the first candy is sweet = 4/6 * 2/5 = 4/15

It results that the probability of getting only one sour candy = 4/15 + 4/15 = 8/15

Question 14

What is the GCF and LCM of 18 and 32?

Answer 14

First we have to factor out each term.

18 = 3 * 3 * 2

32 = 2 * 2 * 2 * 2 * 2

GCF = product of the smallest number of factors, thus = 2

LCM = the product of the greatest number of factors, thus = 2⁵ * 3² = 32 * 9 = 288

GCF = 2

LCM = 288

Check Number Properties p. 84

Question 15

Alicia lives in a town whose streets are on a grid system, with all streets running east-west or north-south without breaks. Her school, located on a corner, lies three blocks south and three blocks east of her home, also located on a corner. If Alicia s equally likely to choose any possible path from home to school, and if she only walks south or east, how many possible routes can she take to school?

Answer 15

Disguised combinatorics.

Alicia goes 6 blocks and she has to make 6 choices whether to go south or east, but she has only 3 choices south and 3 choices east, she does not go further.

Thus,

6! / 3!3! = 20

Question 16

If n and k are positive integers, is n divisible by 6?

(1) n = k(k + 1)(k - 1)
(2) k – 1 is a multiple of 3

Answer 16

A.

(1) We know that n is the product of 3 consecutive integers. It is sufficient. Try test cases.
(2) It tells us nothing about n

Question 17

At a certain pizzeria, 1/8 of the pizzas sold in one week were mushroom and 1/3 of the remaining pizzas sold were pepperoni. If n of the pizzas sold were pepperoni, how many were mushroom (in terms of n) ?

Answer 17

Remaining pizzas = 1 - 1/8 = 7/8

n = (1/3)(7/8)total

n = (7/24)total

total = (24/7)n

nr of mushroom pizzas = (1/8)(24/7)n = (3/7)n

Question 18

Danny scored on the first exam 75 and on the second 92. If first exam counts to 40% of final score and second exam counts to 60% of the final score, what is Dannys final score?

Answer 18

75 * 0.4 + 92 * 0.6 = 85

Question 19

The length of a rectangular floor is twice its width. The floor is partially covered by a rectangular carpet whose length is the same as the lenfth of the floor and whose width is two feet less than the width of the floor. If the area of the carpet is 160 sq. feet, what is the length, in feet, of the floor?

A. 8
B. 16
C. 20
D. 24
E. 32

Answer 19

C.

160 = (w - 2)(2w)

w = 10

10 * 2 = 20

Question 20

Expressions

Salesperson A’s compensation for any week is $360 plus 6 percent of the portion of A’s total sales above $1,000 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 salepeople earn the same compensation?

Answer 20

15.000$

360 + 0.06(x - 1000) = 0.08x

0.02x = 300

x = 15000

Question 21

A total of 30 percent of the geese included in a certain migration study were male. If
some of the geese migrated during the study and 20 percent of the migrating geese were
male, what was the ratio of the migration rate for the male geese to the migration rate for
the female geese?
[Migration rate for geese of a certain sex = (number of geese of that sex migrating) /
(total number of geese of that sex)]

A. 1/4
B. 7/12
C. 2/3
D. 7/8
E. 8/7

Answer 21

B.

Let there be g geese, of which 0.3g are male and 0.7g are female
Lets say m geese migrated, 0.2m are male and 0.8m are female

Migration Rate Male = 0.2m / 0.3g = (2/3)*(m/g)
Migration Rate Female = 0.8m / 0.7g = (8/7)*(m/g)

Ratio of migration rates = (2/3) / (8/7) = 7/12

Question 22

In a mayoral election, Candidate X received 1/3 more votes than candidate Y, and Candidate Y received 1/4 fewer votes than Z. If Z received 24,000 votes how many votes did candidate X received?

Answer 22

X = 4/3Y

Y = 3/4Z

X = 4/3 * 3/4Z

X = Z

Thus, X = 24.000

Question 23

For every integer k from 1 to 10, inclusive the “k”th term of a certain sequence is given by (−1)^(k+1) * (1/2^k) if T is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/4

Answer 23

D.

Try to input the first terms to see the tendency. It is very slowly growing from 1/4. Trying 3 - 4 terms is enough to understand that the sum wont get above 1/2 but will be bigger than 1/4.

Question 24

What is the probability that the sum of two number cubes will yield a 10 or lower?

Answer 24

Two number cubes = 36 total posibilities.

It is easier to find the posibilities that the cubes will yield a 11 or higher and to subtract the result from 1.

To have a 11 we will need: 6 + 5 or 5 + 6 2 cases.

To have a 12 we will need: 6 + 6 1 case.

Thus, 1 - 3/36 = 33/36 = 11/12

Question 25

A box contains 3 types of toys: soldiers, dolls and cars. Equal number of each. If a child gets 3 toys out of the box, each time replacing them, what is the chance that at least one toy will be a soldier?

Answer 25

1 - P(x) trick It will be much easier to calculate the chance of not getting a soldier than to calculate the chance of getting at least one soldier. The chance of not getting a soldier = 2/3 \* 2/3 \* 2/3 = 8/27 Now we subtract 8/24 from one and we get **19/27.**

Question 26

When I see: "is ; are ; was ; were ; will be ; gives ; yields ; sold for ; has ; costs ; adds up to ; the same as or as much as" I know it is \_\_\_\_\_.

Answer 26

equals

Question 27

In the figure shown, what is the value of x? (1) The length of line segment QR is equal to the length of line segment RS (2) The length of line segment ST is equal to the length of line segment TU

Answer 27

x = 45 ## Footnote **C.**

Question 28

If x and y are integers greater than 1, is x a multiple of y? (1) 3y² + 7y = x (2) x² - x is a multiple of y

Answer 28

**A.** Use smart numbers and test cases. Check gmatclub forum for more info.

Question 29

In the Speed \* Time = Distance grids, what can be added and what can be not to find the totals?

Answer 29

We can add Distance and Time columns We can never add Speed

Question 30

A driver completed the first 20 miles of a 40 miles trip at an average speed of 50 miles per hour. At what average speed must the driver complete the remaining 20 miles to achieve an average speed of 60 miles per hour for the entire 40-miles trip? ( Assume that the driver did not make any stops during the 40-miles trip)

Answer 30

Solve using the S T D (Speed Time Distance) grid. ## Footnote **75 miles/hr**

Question 31

When I see: "per ; out of ; ratio of ; quotient of ; percent (divide by 100) ; divided by or each" I know it is \_\_\_\_\_.

Answer 31

division

Question 32

Answer 32

**B.**

Question 33

How many total factors does 252 has?

Answer 33

18

Question 34

A magician has five animals in his magoc hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?

Answer 34

**2/5** First we have to find out total number of possible pairs: 5! / (2!3!) = 10 Possible rabits pair: 2! / 2! = 1 Possible doves pairs = 3! / (2!1!) = 3 Thus, we have 4/10 = **2/5** probability to have a matched pair.

Question 35

What is the process of solving any problem?

Answer 35

1. Read carefully. 2. Define what is asked and write it in a circle. 3. Simplify algebraicly or translate english to math 4. Solve 5. Double check what is asked for and the solving (if you have time left)

Question 36

When a number is multiplied by its reciprocal the result always is \_\_\_\_\_\_\_. Explain.

Answer 36

The result is always **1.** That is the main rule for finding reciprocals. For example reciprocal of 5 is 1/5, 5 \* 1/5 = 1.

Question 37

x is a multiple of 5 y is not a multiple of 5 Is the sum of *x* and *y* a multiple of 5?

Answer 37

**No.** When a multiple of a number is added to a non multiple of the same number, the result will never be the multiple of that number. Same with subtraction.

Question 38

What is the reciprocal of: (square root of 3) / 3

Answer 38

**The reciprocal is square root of 3.** Because: square root of 3 \* (square root of 3) / 3 = 3/3 = 1

Question 40

If k³ is divisible by 240, what is the least possible value of integer k? a. 12 b. 30 c. 60 d. 90 e. 120

Answer 40

**b** 240 = 2⁴ \* 5 \* 3 The answer choice must have at least one of each prime factors of 240.

Question 41

In how many different ways can the letters in word "LEVEL" be arranged?

Answer 41

**30** There are 2 letter that are repeated. L and E. The number of arrangements without restrictions = 5!. Letter that is repeated creates a restriction of 2!. Thus, 5! / (2!2!) = 5 \* 2 \* 3 = 30

Question 42

Coordinate plane Slope = 2/3 1. How many points does *y* move? 2. How many points does *x* move?

Answer 42

1. y moves 2 points 2. x moves 3 points

Question 43

If the LCM of *a* and 12 is 36, what are the possible values of *a*?

Answer 43

9, 18 or 36

Question 44

A rectangular photograph is surrounded by a border that is 1 inch wide on each side. The total area of the photograph and the border is M square inches. If the border had been 2 inches wide on each side, the total area would have been (M + 52) square inches. What is the perimeter of the photograph, in inches? A. 34 B. 36 C. 38 D. 40 E. 42

Answer 44

**D.**

Question 45

A BJJ club consists of 9 members. It will send 4 members to a competition. How many different groups of members could the club send?

Answer 45

126 **Combinatorics** 9! / (4!5!) = (9 \* 8 \* 7 \* 6) / (4 \* 3 \* 2) = 3 \* 2 \* 7 \* 3 = 126

Question 46

**Odd/Even** Is the sum of two primes odd or even, if neither of them is 2?

Answer 46

**Its even.** Any prime number, except 2 is odd and *odd + odd = even*.

Question 47

A miniature gumball machine contains 7 blue, 5 green, and 4 red gumballs, which are identical except for their colors. If the machine dispenses three gumballs at random, what is the probability that it dispenses one gumball of each color?

Answer 47

**Combinatorics + Domino effect** Domino effect = one action affects the probability of another. We need a result of 1 blue, 1 green and 1 red gumball. These can drop in random order, so we the total number of combinations of these 3 gumbals = 3! = 6 (1) When each gumball drops, it affects the probability of the next gumball drop. Thus, the probability = prob of blue \* prob of red \* prob of green = 7/16 \* 4/15 \* 5/14 = 1/24 Now, 1/24 \* 6 = **1/4**

Question 48

The integers r, s and t all have the same remainder when divided by 5. What is the value of t? (1) r + s = t (2) 20 \<= t \<= 24

Answer 48

**C.** (1) By testing cases we can find out that the remainder of every term when divided by 5 will be 0. Cannot find out the value of t - Insufficient. (2) Insufficient. (1) + (2) 20 \<= t \<= 24 there is only one term when divided by 5 gets the remainder 0 and precisely 20.

Question 49

If n denotes a number to the left of 0 on the number line such that the square of n is less than 1/100, then the reciprocal of n must be A. less than -10 B. between -1 and -1/10 C. between -1/10 and 0 D. between 0 and 1/10 E. greater than 10

Answer 49

**A.** n is negative n² \< 1/100 - 1/10 \< n Now we turn them to reciprocals bi dividing both sides by 1: 1/(-1/10) \> 1/n -10 \> 1/n When you turn the numbers to reciprocals, the inequality sign changes.

Question 50

**Expressions** A survey of employers found that during 1993 employment costs rose 3.5 percent, where employment costs consist of salary costs and fringe-benefit costs. If salary costs rose 3 percent and fringe-benefit costs rose 5.5 percent during 1993, then fringe-benefit costs represented what percent of employment costs at the beginning of 1993 ?

Answer 50

**20% (1/5)** E - employment costs S - salary F - fringe - benefit costs E = F + S 35 (F + S) = 30S + 55F S = 4F E = 4F + F = 5F Thus, F/E = F/5F = 1/5

Question 51

If n is positive is (square root of n) \> 100 ? (1) (square root of n - 1) \> 99 (2) (square root of n + 1) \> 101

Answer 51

**B.** n \> 100 \* 100 = 10.000 (1) n - 1 \> 99 \* 99 n - 1 \> 9801 Not sufficient. There can be many values of n less or greater than 10.000 (2) n + 1 \> 101 \* 101 n + 1 \> 10201 Thus, n is greater than 10.000. Sufficient

Question 52

When I see: "increased by ; more than ; combined ; together ; total of ; sum ; added to ; and or plus" I know it is ______ .

Answer 52

addition.

Question 53

If x+y=8z, then how would you represent the average (arithmetic mean) of x, y, and z, in terms of z?

Answer 53

3z

Question 54

(x + y) / xy = 1 y = ? Simplify.

Answer 54

(x + y) / xy = 1 x + y = xy x = xy - y x = y(x - 1) **y = x / (x - 1)**

Question 55

The ratio of the length of a side of S to a side of T is 4:5. Is the perimeter of square S greater than the perimeter of equilateral triangle T?

Answer 55

Basically the question is asking wheter 4S \> 3T, which can be expressed as S/T \> 3/4. 4/5 is greater than 3/4, thuse **Yes, the perimeter of S is greater.**

Question 56

On a certain sight-seeing tour, the ratio of the number of women to the number of children was 5 to 2. The ratio of the number of children to the number of men was 5 to 11. The number of women on the sight-seeing tour was less than 30. What was the number of men on the sight-seeing tour? Explain.

Answer 56

**22** Number of children = c Number of women = w Number of men = m w/c = 5/2 For a factor x, w = 5x and c = 2x c/m = 5/11 For a factor y, c = 5y, m = 11y 5y = 2x w = 5x \< 30 x \< 6 x can be 5, 4, 3, 2 or 1, and it must keep the expression 5y = 2x as integers (because there cannot be half kids, half men or women). Only x = 5 fits 5 \* 2 = 2 \* 5, thus y = 2 m = 11y = 22

Question 57

In the sequence S of numbers, each term after the first two terms is the sum of the two immediately preceding terms. What is the 5th term of S? (1) The 6th term of S minus the 4th termequals 5. (2) The 6th term of S plus the 7th term equals 21.

Answer 57

**A.** We know that S_n = S_n-1 + S_n-2 (1) S₆ - S₄ = 5 S₅ + S₄ - S₄ = 5 S₅ = 5 Sufficient. (2) S₆ + S₇ = 21 S₅ + S₄ + S₅ + S₄ + S₅ = 21 Since we dont know S₄ we cannot solve. Not sufficient.

Question 58

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 xy will be even?

Answer 58

We have 4 nr in the first set and 3 in the second, thus the total number of possible pairs = 4 \* 3 = 12 To find the number of even products we will find the number of odd results and will subtract it from 1. **1 - P(x)** For a product to be odd, both multiplies must be odd. It will happen in 4 cases: 1 \* 7; 1 \* 5; 3 \* 5; 3 \* 7 Thus, the probability of a odd result = 4/12 = 1/3 1 - 1/3 = **2/3**

Question 59

Tickets to a play cost $10 for children and $25 for adults. If 90 tickets were sold, were more adult tickets sold than children's tickets? 1) The average revenue per ticket was $18 2) The revenue from ticket sales exceeded $1600

Answer 59

**D.** 1) We have 2 equations, thus can solve for variables. Sufficient. 2) If all were adults, total revenue = 2250$ If all were kids, total revenue = 900$ The average = 1575$ We have total revenue \> 1600$, thus we can conclude that the play was attended by more adults than kids.

Question 60

**Word problems** A certain manufacturer produces items for which the production costs consist of annual fixed costs totaling $130,000 and variables costs averaging $8 per item. If the manufacturer’s selling price per item is $15, how many items the manufacturer produce and sell to earn an annual profit of $150,000?

Answer 60

profit = revenue - cost per item - total cost 150. 000 = 15x - 8x - 130.000 280. 000 = 7x **x = 40.000**

Question 61

**Odd/Even** If c and d are odd integers, is c - 3d even?

Answer 61

**Yes.** odd - (odd \* odd) = even

Question 62

If the sum of the last 3 integers in a set of 7 consecutive integers is 258, what is the sum of the first 4 integers?

Answer 62

**330** 258/3 = 86 (6th integer) 85 + 86 + 87 = 256 81 + 82 + 83 + 84 = 330

Question 63

Is the product of 5 consecutive integers divisible by 5? Explain.

Answer 63

**Yes.** The product of 5 consecutive integers is always divisible by 5! or by any of its factors 5 \* 4 \* 3 \* 2 \* 1. Try cases: 3 \* 4 \* 5 \* 6 \* 7. Divisible. 11 \* 12 \* 13 \* 14 \* 15. Divisible.

Question 64

Is the sum of 11 consecutive integers divisible by 11? Explain.

Answer 64

**Yes.** For any set with an odd number of terms, the sum of terms will always be divisible by the number of terms.

Question 65

Is the sum of 26 consecutive integers divisible by 26? Explain.

Answer 65

**No.** For any set with an even number of terms, the sum of its terms will never be divisible by the number of its terms.

Question 66

If x is an even integer, is x(x + 1)(x + 2) divisible by 4? Explain.

Answer 66

**Yes.** We have a sum of consecutive integers. Knowing that x is even we can assert that x + 2 is even too. We have at least 2 factors of, thus the product will be divisible by 4.

Question 67

One week, a certain truck rental lot had a total of 20 trucks, all of which were on the lot Monday morning. If 50% of the trucks that were rented out during the week were returned to the lot on or before Saturday morning of that week, and if there were at least 12 trucks on the lot that Saturday morning, what is the greatest number of different trucks that could have been rented out during the week?

Answer 67

**16** Let x be the number of trucks that were rented. 20 - x trucks were rented 0.5x trucks returned On Saturday morning were: (20 - x) + 0.5x = 12 trucks 20 - 0.5x = 12 - 0.5x = -8 x = 16

Question 68

Answer 68

3k² - 3

Question 69

Write as an algebraic expression: "Ricky is now twice as old as he was 8 years ago."

Answer 69

r = 2(r - 8)

Question 70

When n is divided by 6, the remainder is 1. What are the possible remainders when the positive integer n is divided by 12?

Answer 70

**1, 7** Possible values for n = 1, 7, 13, 19 ....

Question 71

x + y = 2x Express x in terms of y.

Answer 71

x = y

Question 72

Does the following numbers have terminating decimals? Explain why. 13/2¹² 26/5⁵

Answer 72

**Yes, both have terminating decimals because:** The rule states that any number divided by any form of the terms 2ⁿ5^m or by their product yield a number with terminating decimals.

Question 73

Is *xy* even if x/y is even?

Answer 73

**Yes** If x/y is even than x/y = 2n x = 2ny xy = 2ny² xy has a 2 as a factor, thus xy = even

Question 74

If n = 4p and p is a prime number greater than 2. How many different positive even divisors does n include?

Answer 74

**Four.** 2, 4, 2n, 4n

Question 75

Answer 75

B. Find out the needed slope. Look for it in the answer options.

Question 76

If m^-1 = -(1/3), than m^-2 = ?

Answer 76

m = 3 ## Footnote **m^-2 = 1/9**

Question 77

Answer 77

Question 78

Is x positive if x - 1 is a perfect square?

Answer 78

**Yes.** x - 1 = n² for some number n x = n² + 1, Thus whether n is negative or positive x will definetly be positive

Question 79

The sequence of four numbers a₁, a₂ , a₃ and a₄ is such that each number after the first is a₁−1 greater than preceding number . What is the value of a1? (1) a₂=15 (2) a₄=29

Answer 79

**D.** a₂ = a₁ + a₁ - 1 = 2a₁ -1 a₃ = 2a₁ - 1 + a₁ - 1 = 3a₁ -2 a₄ = 4a₁ - 3