GRE Math Practice Test 21

Test 21

GRE MATH

Complete GRE MATH Preparation Materialhttp://studymaterialcollection.blogspot.com/2015/12/complete-gre-math-preparation-material.html

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GRE Math Tests

406

Questions: 24 Time: 45 minutes

[Multiple-choice Question – Select One Answer Choice Only] 1. If 42.42 = k(14 + m/50), where k and m are positive integers and m < 50, then what is the value of k + m ?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

[Multiple-choice Question – Select One or More Answer Choices] 2. If c and d are two integers and @ is defined in a and b as @(a, b) = (a + 1)(b + 2), and @(c, d) equals

the product of 3 and 5, then what could be the value c + d ?

(A) –11 (B) 0 (C) 5 (D) 6 (E) 11

Quantitative Comparison Question] 3. Column A X is a 3-digit number and Y is a 4-

digit number. All the digits of X are greater than 4, and all the digits of Y are less than 5.

Column B

The sum of the digits of X The sum of the digits of Y

Test 21 — Questions

407

[Multiple-choice Question – Select One Answer Choice Only] 4. How many positive integers less than 500 can be formed using the numbers 1, 2, 3, and 5 for the

digits?

(A) 48 (B) 52 (C) 66 (D) 68 (E) 84

Quantitative Comparison Question] 5. Column A x > 0 Column B 1/2x 2x

[Multiple-choice Question – Select One or More Answer Choices] 6. The following are the measures of the sides of five different triangles. Which of them represents a

right triangle?

(A) 3, 4, 5 (B) 1, 5, 4 (C) 3, 4, 5 (D) 3, 7, 4 (E) 4, 8, 10

GRE Math Tests

408

[Multiple-choice Question – Select One Answer Choice Only] 7. In the figure, ABCD is a rectangle and E is a point on the side AB. If AB = 10 and AD = 5, what is the

area of the shaded region in the figure?

(A) 25 (B) 30 (C) 35 (D) 40 (E) 45

[Multiple-choice Question – Select One Answer Choice Only] 8. In the figure, lines l and k are parallel. If a is an acute angle, then which one of the following must be

true?

(A) b > 10 (B) b > 15 (C) b < 20 (D) b < 30 (E) b > 45

A B

D

E

C

b°

(b + 30)°

a°

l

k

O


409

[Multiple-choice Question – Select One Answer Choice Only] 9. In the figure, ABC is inscribed in the circle. The triangle does not contain the center of the circle O.

Which one of the following could be the value of x in degrees?

(A) 35 (B) 70 (C) 85 (D) 90 (E) 105

[Multiple-choice Question – Select One Answer Choice Only] 10. In the figure, which one of the following is the measure of angle ?

(A) (B) (C) (D) (E) It cannot be determined from the information given

x y

(x, y)

.O

x°

A

B C


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GRE Math Tests

410

Quantitative Comparison Question] 11. Column A 1 < p < 3 Column B p2 2p

Quantitative Comparison Question] 12. Column A Kate ate 1/3 of a cake, Fritz ate

1/5 of the remaining cake, and what was left was eaten by Emily.

Column B

Fraction of the cake eaten by Emily

5/7

[Multiple-choice Question – Select One or More Answer Choices] 13. If x and y constrained by the following two different patterns:

3x + y = x + 3y (for all x + y > 0), and x + 3y = 2x + y (for all x + y < 0)

then what is the value of 2x – y when x = 2 ?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4


411

[Numeric-Entry]

14. The average of x + 2, x + 3, and 2x + 5 is x + x3+[][]?

[Multiple-choice Question – Select One Answer Choice Only] 15. Kelvin takes 3 minutes to inspect a car, and John takes 4 minutes to inspect a car. If they both start

inspecting different cars at 8:30 AM, what would be the ratio of the number of cars inspected by Kelvin and John by 8:54 AM of the same day?

(A) 1 : 3 (B) 1 : 4 (C) 3 : 4 (D) 4 : 3 (E) 7 : 4

Quantitative Comparison Question] 16. Column A Column B x(x2)4 (x3)3

GRE Math Tests

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[Multiple-choice Question – Select One Answer Choice Only]

17. If |3x| 2, what is the value of

9x 2 43x + 2

9x 2 43x 2

?

(A) –9 (B) –4 (C) 0 (D) 4 (E) 9

[Multiple-choice Question – Select One Answer Choice Only] 18. A cyclist travels at 12 miles per hour. How many minutes will it take him to travel 24 miles?

(A) 1 (B) 2 (C) 30 (D) 60 (E) 120

[Multiple-choice Question – Select One Answer Choice Only] 19. The combined salaries of three brothers is $90,000. Mr. Big earns twice what Mr. Small earns, and

Mr. Middle earns 1 1/2 times what Mr. Small earns. What is the smallest salary of the three brothers?

(A) 20,000 (B) 22,000 (C) 25,000 (D) 30,000 (E) 40,000


413

[Multiple-choice Question – Select One or More Answer Choices] 20. Mark and Steve sell items to each other. The seller offers a round figured number of items of the

value of the items given by the buyer. For example, if Mark buys toffees from Steve by offering 3.8 toffees worth of goods, then Steve gives him the round figure 4 toffees. In this mode, once they could buy 2 Indian Mangoes for 3 Fuji Applies but could not buy 3 Fuji Applies for 2 Indian Mangoes. Which of the following could be the ratio of the prices of Indian Mangos and Fuji Apples?

(A) 2 : 3 (B) 3 : 2 (C) 3 : 4 (D) 4 : 3 (E) 3 : 7 (F) 7 : 3

[Multiple-choice Question – Select One Answer Choice Only] 21. Set S is the set of all numbers from 1 through 100, inclusive. What is the probability that a number

randomly selected from the set is divisible by 3?

(A) 1/9 (B) 33/100 (C) 34/100 (D) 1/3 (E) 66/100

Quantitative Comparison Question] 22. Column A The number of distinct

elements in set A is 8, and the number of distinct elements in set B is 3.

Column B

The number of distinct elements common to set A and set B

The number of distinct elements in set A that are not in set B

GRE Math Tests

414

[Multiple-choice Question – Select One Answer Choice Only] 23. The sum of the first n even, positive integers is 2 + 4 + 6 + + 2n is n(n + 1). What is the sum of

the first 20 even, positive integers?

(A) 120 (B) 188 (C) 362 (D) 406 (E) 420

[Multiple-choice Question – Select One Answer Choice Only] 24. A ship is sinking and 120 more tons of water would suffice to sink it. Water seeps in at a constant rate

of 2 tons a minute while pumps remove it at a rate of 1.75 tons a minute. How much time in minutes has the ship to reach the shore before is sinks?

(A) 480 (B) 560 (C) 620 (D) 680 (E) 720



Test 21 — Solutions

415

Answers and Solutions Test 21:

Question Answer 1. E 2. A, C 3. D 4. D 5. D 6. C, D 7. A 8. D 9. E 10. E 11. D 12. B 13. C, D 14. 10/3 15. D 16. C 17. B 18. E 19. A 20. B, D 21. B 22. B 23. E 24. A

If you got 18/24 correct on this test, you are likely to get 750+ on the actual GRE by the time you complete all the tests in the book. 1. We are given that k is a positive integer and m is a positive integer less than 50. We are also given that 42.42 = k(14 + m/50). Suppose k = 1. Then k(14 + m/50) = 14 + m/50 = 42.42. Solving for m yields m = 50(42.42 – 14) = 50 28.42, which is not less than 50. Hence, k 1. Now, suppose k = 2. Then k(14 + m/50) = 2(14 + m/50) = 42.42, or (14 + m/50) = 21.21. Solving for m yields m = 50(21.21 – 14) = 50 7.21, which is not less than 50. Hence, k 2. Now, suppose k = 3. Then k(14 + m/50) = 3(14 + m/50) = 42.42, or (14 + m/50) = 14.14. Solving for m yields m = 50(14.14 – 14) = 50 0.14 = 7, which is less than 50. Hence, k = 3 and m = 7 and k + m = 3 + 7 = 10. The answer is (E).

GRE Math Tests

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2. According to the definition,

@(c, d) = (c + 1)(d + 2)

We are given that

@(c, d) = the product of 3 and 5, which is 15 (c + 1)(d + 2) = 15

Since c and d are integers, so are c + 1 and d + 2. Now, the product of two integers is 15 only when they are either 3 and 5 or when they are –3 and –5.

c + 1 = 3 & d + 2 = 5 c = 2 & d = 3 c + d = 5 Select choice (C).

c + 1 = –3 & d + 2 = –5 c = –4 & d = –7 c + d = –11 Select choice (A).

c + 1 = 5 & d + 2 = 3 c = 4 & d = 1 c + d = 5

c + 1 = –5 & d + 2 = –3 c = –6 & d = –5 c + d = –11

The answer is (A) and (C). 3. Column A: Since all the digits of the 3-digit number X are greater than 4, each digit must be greater than or equal to 5. Hence, the sum of the three digits must be greater than or equal to 3 5 = 15. Also, since the maximum value of each digit is 9, the maximum possible value of the sum of its digits is 3 9 = 27. Hence, we have the inequality 15 Column A 27. Column B: Since all the digits of the 4-digit number Y are less than 5, each digit must be less than or equal to 4. Hence, the sum of the four digits must be less than or equal to 4 4 = 16. Also, the minimum value of the sum of the digits of a 4-digit number is 1 (for example, for 1000, the sum of the digits is 1). Hence, we have the inequality 1 Column B 16. Since the inequality for Column A, 15 Column A 27, and the inequality for Column B, 1 Column B 16, have a common range (numbers between 15 and 16, inclusive, satisfy both inequalities), an inequality between the two columns cannot be derived. Hence, we cannot know which column is greater. The answer is (D). Method II: Let's construct the smallest and largest numbers possible for X and Y. Since all the digits of X are greater than 4, each digit must be greater than or equal to 5. So, X will be as small as possible when it consists of only 5's and will be as large as possible when it consists of only 9's:

Smallest X Largest X 5 5 5 9 9 9

Sum: 5 + 5 + 5 = 15 Sum: 9 + 9 + 9 = 27

Since all the digits of Y are less than 5, each digit must be less than or equal to 4. So, Y will be as small as possible when it consists of a leading 1 following by three zeros and will be as large as possible when it consists of only 4's:

Smallest Y Largest Y 1 0 0 0 4 4 4 4 Sum: 1 + 0 + 0 + 0 = 1 Sum: 4 + 4 + 4 + 4 = 16

If we chose the smallest sum for X (15) and the largest sum for Y (16), then Column B is greater. And if we chose the largest sum for X (27) and the smallest sum for Y (1), then Column A is greater. This is a double case and therefore the answer is (D).


417

4. A number less than 500 will be 1) a single-digit number, or 2) a double-digit number, or a 3) triple-digit number with left-most digit less than 5. Let the compartments shown below represent the single, double and three digit numbers.

The compartment for the single-digit number can be filled in 4 ways (with any one of the numbers 1, 2, 3, and 5). Each of the two compartments for the double-digit number can be filled in 4 ways (with any one of the 4 numbers 1, 2, 3, and 5) each. Hence, the two-digit number can be made in 4 4 = 16 ways. Regarding the three-digit number, the left most compartment can be filled in 3 ways (with any one of the numbers 1, 2, and 3). Each of the remaining two compartments can be filled in 4 ways (with any one of the numbers 1, 2, 3, and 5) each. Hence, total number of ways of forming the three-digit number equals

3 4 4 = 48 Hence, the total number of ways of forming the number is 4 + 16 + 48 = 68. The answer is (D). 5. Intuitively, one expects 2x to be larger than the fraction 1/2x. But that would be too easy to be the answer to a hard problem. Now, clearly, 1/2x cannot always be greater than 2x, nor can it always be equal to 2x. Hence, the answer is (D). Let’s also solve this problem by substitution. If x = 1, then 1/2x = 1/2 and 2x = 2. In this case, Column B is

greater. But if x = 1/2, then

12x

=1

2 12( )

=11

= 1 and

2x = 2 12

= 1 . In this case, the columns are equal.

This is a double case and the answer is (D).

1 or 2 or 3 or 5

------- (Total 4 ways)

+ +

1 or 2 or 3 or 5


1 or 2 or 3 or 5


1 or 2 or 3 or

5 not allowed -------

(Total 3 ways)

1 or 2 or 3 or 5


= 4 + (4 4) + (3 4 4) = 4 + 16 + 48 = 68 Compartment

for single-digit number

Compartments for double-digit number.

Compartments for triple-digit number.

1 or 2 or 3 or 5


GRE Math Tests

418

6. A right triangle must satisfy The Pythagorean Theorem: the square of the longest side of the triangle is equal to the sum of the squares of the other two sides. Hence, we look for the answer-choice that satisfies this theorem:

Choice (A):

5( )2

3( )2

+ 4( )2

. Reject. Choice (B): 52 12 + 42. Reject. Choice (C): 52 = 32 + 42. Correct.

Choice (D):

7( )2

= 3( )2

+ 4( )2. Correct.

Choice (E): 102 82 + 42. Reject. The answer is (C) and (D). 7. The shaded region contains two triangles AED and EBC. Hence, the area of the shaded region equals the sum of the areas of these two triangles. Now, the formula for the area of a triangle is 1/2 base height. Hence, the area of AED = 1/2 AE AD, and the area of EBC = 1/2 EB BC = 1/2 EB AD (since BC = AD). Hence, the area of the shaded region equals 1/2 AE AD + 1/2 EB AD = 1/2 AD(AE + EB) = 1/2 AD AB [since AB = AE + EB] = 1/2 5 10 = 25. The answer is (A). 8. Draw a line parallel to both of the lines l and k and passing through O.

We are given that a is an acute angle. Hence, a < 90. Since angles p and b + 30 are alternate interior angles, they are equal. Hence, p = b + 30. Similarly, angles q and b are alternate interior angles. Hence, q = b. Since angle a is the sum of its sub-angles p and q, a = p + q = (b + 30) + b = 2b + 30. Solving this equation for b yields b = (a – 30)/2 = a/2 – 15. Now, dividing both sides of the inequality a < 90 by 2 yields a/2 < 45. Also, subtracting 15 from both sides of the inequality yields a/2 – 15 < 30. Since a/2 – 15 = b, we have b < 30. The answer is (D). 9. A chord makes an acute angle on the circle to the side containing the center of the circle and makes an obtuse angle to the other side. In the figure, BC is a chord and does not have a center to the side of point A. Hence, BC makes an obtuse angle at point A on the circle. Hence, A, which equals x°, is obtuse and therefore is greater than 90°. Since 105 is the only obtuse angle offered, the answer is (E).

b°

(b + 30)°

q°

l

k

O p° m


419

10. There are two possible drawings: Case I Case II

In Case I, 11. If p = 2, then both columns equal 4. If p 2, then the columns are unequal. This is a double case, and therefore the answer is (D). 12. We are given that Kate ate 1/3 of the cake. So, the uneaten part of the cake is 1 – 1/3 = 2/3. Hence, regardless of how much Fritz ate, Emily could not have eaten more than 2/3 of the cake. Hence, Column A is less than 2/3; and since 2/3 < 5/7, Column B is larger. The answer is (B). 13. Pattern I: x + y > 0. Subtracting x and y from both sides of the given equation 3x + y = x + 3y yields

3x + y – x – y = x + 2y – x – y 2x = 2y x = y

Now, 2x – y = 2x – x = x = 2 (we are given that x = 2). Hence, select (C). Pattern II: x + y < 0. Subtracting y and x from both sides of the given equation x + 3y = 2x + y yields

x + 3y – x – y = 2x + y – x – y 2y = x y = x/2

Now, 2x – y = 2x – x/2 = 3x/2 = 3(2)/2 = 3 (we are given that x = 2). Hence, select (D). The answer is (C) and (D). 14. The sum of x + 2, x + 3, and 2x + 5 is

(x + 2) + (x + 3) + (2x + 5) = 4x + 10 Now,

Average = Sum/3 = (4x + 10)/3 = 4/3 x + 10/3 = x + x/3 + 10/3 Hence, grid in the fraction 10/3.

(x, y)(x, y)



GRE Math Tests

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15. Kelvin takes 3 minutes to inspect a car, and John takes 4 minutes to inspect a car. Hence, after t minutes, Kelvin inspects t/3 cars and John inspects t/4 cars. Hence, the ratio of the number of cars inspected by them is t/3 : t/4 = 1/3 : 1/4 = 4 : 3. The answer is (D). Method II There are 24 minutes between 8:30 AM and 8:54 AM. Since Kelvin takes 3 minutes to inspect a car, he can inspect 8 (= 24/3) cars in 24 minutes. Since John takes 4 minutes to inspect a car, he can inspect 6 (= 24/4) cars in 24 minutes. Forming the ratio of Kelvin to John yields 8/6 = 4/3. The answer is (D).

16. Column A: x(x2)4 = x x2 4 = x1 x8 = x1 + 8 = x9. Column B: (x3)3 = x3 3 = x9. The answer is (C).

17.

9x 2 43x + 2

9x 2 43x 2

= 9x2 4( ) 13x + 2

13x 2

by factoring out the common term 9x2 – 4

= 9x 2 4( ) 3x 2( ) 3x + 2( )3x + 2( ) 3x 2( )

= 9x 2 4( ) 3x 2 3x 23x( )2 22

= 9x2 4( ) 49x2 4

Since |3x| 2, (3x)2 4, and therefore 9x2 – 4 0.

Hence, we can safely cancel 9x2 – 4 from numerator and denominator.

= –4 The answer is (B). 18. Since the answer is in minutes, let's convert the cyclist's speed (12 miles per hour) into miles per minute. Since there are 60 minutes in an hour, his speed is 12/60 = 1/5 miles per minute. Remember that Distance = Rate Time. Hence,

24 = 15

t

Solving for t yields t = 5 24 = 120. The answer is (E). [If you forgot to convert hours to minutes, you may have mistakenly answered (B).]


421

19. Let s be the salary of Mr. Small. Since Mr. Big earns twice what Mr. Small earns, the salary of Mr. Big is 2s; and since Mr. Middle earns 1 1/2 times what Mr. Small earns, the salary of Mr. Middle equals (1 1/2)s = 3s/2. Since s < 3s/2 and s < 2s, Mr. Small earns the smallest salary. Summing the salaries to 90,000 (given) yields

2s + 3s/2 + s = 90,000

9s/2 = 90,000

s = 90,000 2/9 = 20,000 The answer is (A). 20. Let i and a be the prices of the Indian Mangoes and Fuji Apples, respectively. There are two possibilities: Possibility I: The exchange method dictates that if Fuji Apples were bought for Indian Mangoes, then

2.5a < 2m < 3.5a Dividing the inequality by 2 yields

2.5a/2 < 2m/2 < 3.5a/2 1.25a < m < 1.75a

So, the value of a mango could be in the range 1.25 times to 1.75 times an apple. Possibility II: The exchange method dictates that if Indian Mangoes were bought for Fuji Apples, then

1.5m < 3a < 2.5m Splitting the inequality yields 1.5m < 3a. Dividing by 1.5 yields m < 2a. Splitting the inequality yields 3a < 2.5m. Dividing by 2.5 yields 1.2a < m. Here, the range of the price of mango is between 1.2a and 2a. The wider range of the price of mango is from 1.2a to 2a. That is, m/a must be between 1.2 and 2. Choice (A): 2 : 3; m/a = 2/3 = 0.666. Reject. Choice (B): 3 : 2; m/a = 3/2 = 1.5. Accept. Choice (C): 3 : 4; m/a = 3/4 = 0.7. Reject. Choice (D): 4 : 3; m/a = 4/3 = 1.333. Accept. Choice (E): 3 : 7; m/a = 3/7 < 1.2. Reject. Choice (F): 7 : 3; m/a = 7/3 = 2.33 > 2. Reject. The answer is (B) and (D).

GRE Math Tests

422

21. The count of the numbers 1 through 100, inclusive, is 100. Now, let 3n represent a number divisible by 3, where n is an integer. Since we have the numbers from 1 through 100, we have 1 3n 100. Dividing the inequality by 3 yields 1/3 n 100/3. The possible values of n are the integer values between 1/3 ( 0.33) and 100/3 ( 33.33). The possible numbers are 1 through 33, inclusive. The count of these numbers is 33. Hence, the probability of randomly selecting a number divisible by 3 is 33/100. The answer is (B). 22. We are given that set A contains 8 elements and set B contains just 3 elements, so the greatest possible number of elements common to set A and set B is 3. There are 5 more elements in set A than in set B, so there are at least 5 elements in set A that cannot be in set B. Hence, Column B is always larger than Column A. The answer is (B). 23. We are given a formula for the sum of the first n even, positive integers. Plugging n = 20 into this formula yields

n(n + 1) = 20(20 + 1) = 20(21) = 420 The answer is (E). 24. We have that water enters the ship at 2 tons per minute and the pumps remove the water at 1.75 tons per minute. Hence, the effective rate at which water is entering the ship is 2 – 1.75 = 0.25 tons per minute. Since it takes an additional 120 tons of water to sink the ship, the time left is (120 tons)/(0.25 tons per minute) = 120/0.25 = 480 minutes. The answer is (A).

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