Hard Math Practice: Set 1

This document contains 49 hard math problems. Be warned: these problems are extremely challenging and should probably not be approached until you have mastered the material in your GMAT manual and in the GMAT Official Guide. If you would like to do these problems sooner, it is our suggestion that you ask your GMAT teacher first.

1. Tammie has 10 cards numbered 1 through 10. If she deals two to Tarrell without replacing any of them what is the probability that Tarrell will get both a 2 and a 3?

(A) 51

(B) 451

(C) 501

(D) 901

(E) 4514

2. If x and y are greater then zero, then what is the value of x2y ?

(1) 161

81

41

21

y

(2) x has exactly two distinct positive factors, one of which is even.

3. If 0y , then is x = 0?

(1) xy = x

(2)x

x

2

4. If n is an integer greater than 5.3, then n! must be divisible by which of the following numbers?

(A) 7(B) 11(C) 12(D) 13(E) 14

5. A die with x sides has consecutive integers on its sides. If the probability of NOT

getting a 4 on either of two tosses is 4936

, how many sides does the die have?

(A) 4(B) 5(C) 7(D) 8(E) 13

6. If for all positive integers x and y, y > x, then is !!

xy

even?

(1) y = 13(2) y – x = 2

7. The fraction 77!11

is equivalent to which of the following?

(A)237 532

(B)249 532

(C)248 532

(D)2248 7532

(E) 11532 248

8. At the same time that Rick opened an account with $150 at bank A, Mary Jane opened an account at bank B with $150. Mary Jane’s account has 10% simple annual interest and Rick’s gives 12% annual interest compounded quarterly. If neither Rick nor Mary Jane make any additional deposits or withdrawals, then what percent more does Rick have in his account after a year?

(A)

150 1.03 4 150 1.10 150 1.10

(B) 150(.12)(2)-150(.10)(2)

(C)

150 .12 4 150 .10 150 .10

(D)

150 1.03 4 150 1.10

(E)

150 1.12 4 150 1.10 150 1.10

9. If x and n are positive integers, is n! + x, divisible by x?

(1) n > x(2) n is not a prime number

10. Is a prime?

(1) x! = a(2) x > 2

11. One hour after Chris leaves her home, Charlie leaves his home to intercept her. If Charlie travels twice as fast as Chris, how long will it take Charlie to intercept Chris?

(1) Chris lives 90 miles due south of Charlie

(2) Charlie’s rate is 30 mph

12. Which of the following expressions contain reciprocal expressions?

I.21

x and xx

II.2x and x

III.3x and

31

x

(A) I only(B) II only(C) I and II only (D) II & III only(E) I & III

13. Within rat colony A, 410 new rats are born every other day and

510 die each day. If rat

colony A has 610 rats (after all deaths and births) at the end of a certain day, then how

long until colony A will have less than 510 members?

(A) exactly 5 days(B) exactly 9 days(C) between 8 and 9 days(D) exactly 10 days(E) more than 10 days

14. 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?

(A) 640(B) 1120(C) 2240(D) 3360(E) 13440

15. 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?

(A) 30(B) 15(C) 7(D) 6(E) 5

16. In a drawer of shirts 8 are blue, 6 are green and 4 are magenta. If Mason draws 2 shirts at random, what is the probability at least one of the shirts he draws will be blue?

(A) 15325

(B) 15328

(C) 175

(D) 94

(E) 1712

17. Which of the following are roots of the equation

0

1245 2

x

xxx

?

(A) -2, 0, 5, -12(B) 0, -5, 2, 12(C) -2, 0, 2, 5, - 12(D) -5, -2, 0, 2(E) 0, 4, 5

18. A certain consulting firm employs 8 men and 4 women. In March, 3 employees are selected at random to represent the company at a convention, what is the probability that the representatives will NOT all be men?

(A) 5514

(B) 83

(C) 5541

(D) 32

(E) 5554

19. Kurt, a painter, has 9 jars of paint 4 of which are yellow, 2 are red and the remaining jars are brown. Kurt will combine 3 jars of paint into a new container to make a new color which he will name according to the following conditions:

Brun Y if the paint contains 2 jars of brown paint and no yellow.Brun X if the paint contains 3 jars of brown paint.Jaune X if the paint contains at least 2 jars of yellow.Jaune Y if the paint contains exactly 1 jar of yellow.

What is the probability that the new color will be Jaune?

(A) 425

(B) 4237

(C) 211

(D) 94

(E) 95

20. 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

3

34

rV )

(A) 52

(B) 2516

(C) 12527

(D) 12598

(E) 53

21. Jean drew a gumball at random from a jar of pink and blue gumballs. Since the gumball she selected was blue and she wanted a pink one, she replaced it and drew another. The second gumball also happened to be blue and she replaced it as well. If

the probability of her drawing the two blue gumballs was 499

, what is the probability that the next one she draws will be pink?

(A) 491

(B) 74

(C) 73

(D) 4916

(E) 4940

22. If two candies are drawn at random from a jar with only red and blue candies and not replaced there is a 70% chance of selecting at least one red. If there is a total of 5 candies in the jar, how many of them are red?

(A) 5(B) 4(C) 3(D) 2(E) 1

23. 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

24. If a and b are distinct integers and x = a + b and y = a – b, then which of the following expresses xy + y, in terms of a and b?

(A) abb 22 2

(B) baba 22

(C) ba

(D) ba

(E) baba 22

25.

15141312 2

8

2

4

2

2

2

1

(A)102

1

(B)122

1

(C)152

15

(D)102

2

(E)162

23

26. All votes cast in a recent presidential election were for either the incumbent or the challenger. The challenger received 5.4 million votes and the incumbent received 5 million. If after a recount of the votes and the addition of previously uncounted absentee ballots, the incumbent had 5.2 million votes while the challenger had 5.4 million, then the percentage of the total number of votes that were for the challenger

(A) decreased by approximately 10 %(B) decreased approximately 1%(C) neither increased nor decreased(D) increased approximately 1%(E) increased approximately 2%

27. 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

28. How many 4 digit numbers begin with a digit that is prime and end with a digit that is prime?

(A) 16(B) 80(C) 800(D) 1440(E) 1600

29.

2

224

3

2

1

3

1

2

148

(A) 2716

(B) 2761

(C) 361

(D) 129(E) 183

30.

2

22

5

2550

(A) 23

(B) 23

(C) 5021

(D) 5021

(E) 2521

31. If x is a positive integer, is x a multiple of 5?

(1) 5 is a factor of 2x

(2) 3x + 5 is a multiple of 5

32. What is the greatest common divisor of positive integers a and b?

(1) a and b share exactly one common factor(2) a and b are both prime numbers

33. Does positive integer a equal 5?

(1) a is a not factor of 6006(2) 5 is the largest divisor of a

34. Does s = t ?

(1) ts

(2) s is both a factor and multiple of t

35. In a recent election, candidate A received x and candidate B received y of the votes of the 7.34 million votes counted before the absentee ballots were tallied. What was the percent change in the number of votes A received after the absentee ballots were tallied?

(1) x – y = 391(2) the difference between the number of votes received by each of the three candidates

in the election before the absentee ballots were tallied and the number received after the absentee ballots were tallied was 5000.

36. What is the value of xyz?

(1) y! = 6 and x! > 720(2) z is the least even integer greater than –1

37. Is yx positive?

(1) x is positive(2) y is negative

38. If a is not equal to zero, is 3a a number greater than 1?

(1) 20 a

(2) ab = a

39. When a die that has one of six consecutive integers on each of its sides is rolled twice, what is the probability of getting the number 1 on both rolls?

(1) the probability of NOT getting an eight is 1(2) the probability of NOT getting a seven is 25/36

40. In the rectangle coordinate system, triangle ABC has a vertex at point (0, 56). If point B is at the origin, then how many points on line AC have integer values for both their x and y values?

(1) The third vertex of triangle ABC lies on the x-axis, and the triangle has an area of 196(2) Point A has a positive x coordinate and a y coordinate of zero

41. 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(ß) 42(C) 294(D) 720(E) 5040

42. The product of all prime numbers less than 29 is approximately equal to which of the following?

(A)4102

(ß)

2106

(C)

2108

(D)

2109

(E)

21010

43. An ice cube is floating in a glass of water with between

1

6 and

1

7 of its mass above water and the rest submerged below the water's surface. The ratio of the part of the mass above water to the part of the mass below water is between

(A) 51

and

1

6

(B)

1

6 and

1

7

(C) 65

and 75

(D) 6 and 7

(E)

6

7 and

7

6

44. Is xx 239

= 1?

(1) the product of x and positive integer y is not x(2) x is a integer

45. Of 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(D) 21(E) -21

46. A quarterly interest rate of 5% over a 12 month period is equal to an annual interest rate of approximately

(A) 60% (B) 33% (C) 22%(D) 20%(E) 15%

47.

111

10545

(A) 451

(B) 401

(C)

2

9(D) 5(E) 45

48. Which of the following is NOT equal to

.009 3

.0003 3 ?(A)

2.7104

(B)

33 23 53

(C)

.00027108

(D)

.033 108

(E)

1

331

10 3

49.

3a 1 31 a3 a

(A)

3 a3a

(B)

3 a3a

(C)

3a 2 1

(D)

3a

3 a

(E)

a

1 a

Answer Key:

1. B2. C3. B4. C5. C6. B7. C8. A9. A10. C11. E12. A13. D14. D15. D16. E17. D18. C19. B20. D21. B22. D23. C24. B25. A26. B27. A28. E29. B30. C31. D32. A33. B34. B35. E36. B37. A38. E39. B40. A41. D42. C43. A44. C45. D46. C47. E48. D49. A