Hard Math #3 Problems

p 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 ClassSize 8 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.

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

2.

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?

1 5 1 45 1 50 1 90 14 45

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

(2)

1 1 1 1 ? ? ? 2 4 8 16 x has exactly two distinct positive factors, one of which is even.

3.

If y ? 0 , then is x = 0?

(1)

xy = x x ? x 2

(1)

(2)

y?

4.

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

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

7 11 12 13 14

5.

A die with x sides has consecutive integers on its sides. If the probability of NOT 36 getting a 4 on either of two tosses is , how many sides does the die have? 49

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

4 5 7 8 13

6.

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

(1) (2)

y = 13 y–x=2

7.

The fraction

(A)

27 33 52 29 3 4 52 28 3 4 52 2 8 3 4 5 27 2 2 8 3 4 5 211

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

y! even? x!

11! is equivalent to which of the following? 77

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?

150?1.03? ? 150?1.10? 150 ?1.10? 150(.12)(2)-150(.10)(2) 150?.12 ??4? ? 150 ?.10? 150 ?.10 ? 4

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

150 ?1.03? ? 150 ?1.10 ?

(E)

150?1.12? ? 150 ?1.10? 150?1.10?

9.

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

(1) (2)

n>x n is not a prime number

10.

Is a prime?

(1) (2)

x! = a x>2

4

4

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? 1

I. II.

x 2 and

x x

x ? 2 and x 1

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

x 3 and x 3 I only II only I and II only II & III only I & III

13. Within rat colony A, 10 4 new rats are born every other day and 10 5 die each day. If rat colony A has 10 6 rats (after all deaths and births) at the end of a certain day, then how long until colony A will have less than 10 5 members? (A) (B) (C) (D) (E)

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

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

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

25 153 28 153 5 17 4 9 12 17

17.

Which of the following are roots of the equation

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

-2, 0, 5, -12 0, -5, 2, 12 -2, 0, 2, 5, - 12 -5, -2, 0, 2 0, 4, 5

?

?

x?x ? 5? x 2 ? 4 ? 0? x ? 12

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

14 55 3 8 41 55 2 3 54 55

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

5 42 37 42 1 21 4 9 5 9

20.

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

21.

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

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 4 sphere remained? (The formula for the volume of a sphere is V ? ? r 3 ) 3

2 5 16 25 27 125 98 125 3 5

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 9 the probability of her drawing the two blue gumballs was , what is the probability 49 that the next one she draws will be pink?

1 49 4 7 3 7 16 49 40 49

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

5 4 3 2 1

23.

If –2 < a < 11 and 3 < b < 12, then which of the following is NOT true?

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

1 < a + b < 23 -14 < a – b < 8 -7 < b – a < 14 1 < b + a < 23 -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)

2b 2 ? 2ab a 2 ? b2 ? a ? b a?b ?a ? b a 2 ? b2 ? a ? b

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

25.

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

1 2

12

?

2 13

2

?

4 14

2

?

8 2 15

?

1 2 10 1 2 12 15 2 15 2 2 10 23 2 16

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

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

4 5 7 8 9

28. How many 4 digit numbers begin with a digit that is prime and end with a digit that is prime? (A) (B) (C) (D) (E)

29.

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

30.

16 80 800 1440 1600

? 1 1 1 ? 48 ?? ? ? ?? 4 2 ?2 3 22 ? ? 32 16 27 61 27 61 3 129 183

?

50 5 ? 2 ? 2 ? 2 5

(A)

3 2

(B)

?

(C) (D)

3 2 21 ? 50 21 50

2

?

21 25

(E)

?

31.

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

(1)

5 is a factor of

(2)

x 2 3x + 5 is a multiple of 5

32.

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

(1) (2)

a and b share exactly one common factor a and b are both prime numbers

33.

Does positive integer a equal 5?

(1) (2)

a is a not factor of 6006 5 is the largest divisor of a

34.

Does s = t ?

(1) (2)

s ?t 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) (2)

x – y = 391 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) (2)

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

37.

Is x ? y positive?

(1) (2)

x is positive y is negative

38.

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

(1) (2)

0?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) (2)

the probability of NOT getting an eight is 1 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) 196 (2)

The third vertex of triangle ABC lies on the x-axis, and the triangle has an area of

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) (ß) (C) (D) (E)

7 42 294 720 5040

42.

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

(A) (ß) (C) (D) (E)

2 ? 10 4 6 2? 10 8 2? 10 9 2? 10 10 2? 10

Point A has a positive x coordinate and a y coordinate of zero

43.

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

1 1 and of its mass above 6 7 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 An ice cube is floating in a glass of water with between

1 1 and 6 5 1 1 and 6 7 5 5 and 6 7 6 and 7 6 7 and 7 6

? ?

3?2x

44.

Is 9 x

(1) (2)

the product of x and positive integer y is not x 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) (B) (C) (D) (E)

150 108 42 21 -21

46.

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

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

60% 33% 22% 20% 15%

= 1?

47.

? 45 ?1 ? 5 ? 1 ? ? ? ? ? 10 ? ?

?1

(D) (E)

1 45 1 40 2 9 5 45

48.

Which of the following is NOT equal to

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

2.7 ? 10 3 3 3 3 ?2 ?5 8 .00027 ? 10 3 8 .03 ? 10 1 1 ?3 ? ?3 3 10

(A) (B) (C)

(E)

49.

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

4

3a

?1

?1

?3 a ? 3? a

3? a 3a 3? a 3a ?2 3a ? 1 3a 3? a a 1? a

?.009 ?3 ? ?.0003?3

Answer Key: 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. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

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

p Changes to Hard Math Practice: Set 1

5.

A die with x sides has consecutive integers on its sides. If the probability of NOT getting a 4 36 on either of two tosses is , how many sides does the die have? 49

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

4 5 7 8 13

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?

150?1.03? ? 150?1.10? 150 ?1.10? 150(.12)(2)-150(.10)(2) 150?.12 ??4? ? 150 ?.10? 150 ?.10 ? 4

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

150 ?1.03? ? 150 ?1.10 ?

(E)

150?1.12? ? 150 ?1.10? 150?1.10?

4

4

Question changed so that the simple interest only must be calculated over one year instead of 2, which might then raise the question as to whether it is simple interest or compound interest 11.

Chris’ sex now no longer changes halfway thru the problem.

19. Some spelling mistakes (FYI: the names are the French words for the colors) fixed and a definition cleared up: 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?

23.

If –2 < a < 11 and 3 < b < 12, then which of the following is NOT true?

(C)

-7 < b – a < 14

33.

QUESTION REWORDED to make a only positive: Does positive integer a equal 5?

(1) (2)

a is a not factor of 6006 5 is the largest divisor of a

39.

QUESTION REWORDED: … what is the probability of getting the number 1 on both rolls?

p Hard Math Practice: Set 2 Ready for even more hard math practice? This document contains 49 additional hard math problems. Be warned: these problems are extremely challenging and should probably not be approached until you have mastered the material in your ClassSize 8 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 2 ? z ? , then what is the value of z? 3 3

1.

If

(1) (2)

When positive integer x is divided by 2 the result is z When positive even integer y is divided by 12 the result is z

2.

If a ? 3 and

(A) (B)

?5 5

(C)

3

(D) (E)

126 25

3.

Nine students are split into three equal teams to develop reports on one of three problems: shortage of skilled labor, violence in schools, and low standardized test scores. How many different teams of students are possible?

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

5040 1680 1512 504 168

4.

Eight Alaskan Huskies are split into pairs to pull one of four sleds in a race. How many different assignments of Huskies to sleds are possible?

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

32 64 420 1680 2520

a 4 ? 3a 3 ? a ? 3 ? ? 126 , then what is the value of a? a?3

126

5.

The length of an edge of cube A is 5% greater than the length of an edge of cube B. If the volume of cube B is 27 cubic centimeters, then what is nearest to the volume of cube A?

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

23.1 25.65 27.125 28.35 31.25

6.

How many distinct positive prime factors of 9150 are greater than 0 but less than 1000?

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

7 6 5 4 3

7.

If n is an integer greater than 0, what is the remainder when 9 12n? 3 ? 10 is divided by 10

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

0 1 2 7 9

8.

If a ? ( 2 3 )( 4 3 )( 5 9 ) and b ? ( 4 6 )( 5 6 )( 6 9 ) , then which of the following values is less than 3

ab ?

(A)

( 2 7 )( 5 5 )( 6 3 )

(B)

( 2)( 4 3 )( 5 5 )( 6 3 )

(C)

( 210 )( 3 3 )( 5 5 )

(D)

( 212 )( 5 5 )( 6 )

(E)

( 2 6 )( 5 3 )( 6 7 )

9.

How many positive divisors does ab have?

(1) (2)

a is a prime number and b is a positive integer a3 ? b

10.

What is the sum of positive integers x and y?

(1)

x 2 ? 2 xy ? y 2 ? 16

(2)

x 2 ? y2 ? 8

11.

What is the ratio of x to y?

(1) (2)

.8 x ? .5 y ? 40 .625 y ? 50 ? x

12.

?a? If a, b and c are positive integers, is ? ? ? 1 ? ?b?

(1) (2)

b–a=9 c>1

13.

Is integer n greater than 3?

(1)

? 1 ? ? ? ? .0001 ?100 ?

(2)

? 1 ? ? ? ?100 ?

14.

What’s the remainder when integer n is divided by 10?

(1) (2)

when n is divided by 110 the remainder is 75 when n is divided by 100 the remainder is 25

15.

If Set Q contains only positive integers, is average of the numbers in Set Q is equal to the median of the set?

(1) (2)

The numbers in Set Q are consecutive and odd Set Q contains an odd number of numbers

16.

Is the tens digit of two-digit positive integer p divisible by 3?

(1) (2)

p – 5 is a multiple of 3 p – 11 is a multiple of 3

17.

If x is greater than 0 but less than 10 and k ? x 9 , what is the value of integer k?

(1)

x 2 has a units digit of 1 1 x?2 ? 50

c

n

(2)

n ?1

? .001

18.

Is m a multiple of 6?

(1) (2)

more than 2 of the first 5 positive integer multiples of m are multiples of 3 fewer than 2 of the first 5 positive integer multiples of m are multiples of 12

19.

Does x + y = xy?

(1) (2)

x is neither a positive integer nor a negative integer y is neither a positive integer nor a negative integer

20.

Is x odd?

(1) (2)

a – b = x and b – c = x, where a, b, c & e are distinct prime numbers x is the product of 2 consecutive prime numbers

21.

A jar contains only nickels, dimes and quarters. If a coin is drawn from the jar at random, what is the probability that it will be either a nickel or a quarter?

(1)

The probability the coin will be a nickel or a dime is

(2)

The probability the coin will be a dime is

22.

If a is a positive integer, is a + b a even integer?

(1) (2)

x axb ? 1 x ?1

23.

Working at their individual rates, Marcus and Latrell can build a certain brick house in 7.5 and 5 hours, respectively. When they work together they are paid $35 per hour. If they share their pay in proportion to the amount of work each does, then what is Marcus’ hourly pay for building the house?

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

$3 $6 $7 $14 $21

1 2

1 5

24.

(A) (B) (C)

If a, b and x are integers greater than zero, then which of the following must be greater than a ? a?b

a? x a? b? x a? x a? b? x 2a 2 a ? 2b ? x 2

(D) (E)

? a ? ? ? ?a ? b? a?1 a? b?1

25.

If a and b are integers greater than -1, then what is the value of a b ?

(1) (2)

a ? 1 is undefined b=2

26.

Is m 3 an integer?

(1)

m 2 is an integer

(2)

27.

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

m is an integer

Three students, Mark, Peter and Wanda are all working on the same math problem. If their 1 2 3 individual probabilities of success are , and , respectively, then what is the 4 5 8 probability that at least one of the students will get the problem correct?

39 40 77 80 23 32 9 32 3 80

28.

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

Scott, Jean and Warren are all building wooden models for an architectural presentation at 1 1 noon tomorrow. If their individual probabilities of finishing on time are x, and , 3 7 respectively, then what is the probability that Warren will finish on time but Jean and Scott will not?

21 x ? 38 21 12x 21 2 ? 2x 21 2x ? 2 21 x 21

29.

This year at the Massachusetts Academy the boys are all 2, 3, 5 or 7 years of age. If the product of the ages of the boys in a given class is 10,500, then how many 5 year olds are in that class?

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

0 3 5 125 2100

30.

How many integers between 0 and 1570 have a prime tens digit and a prime units digit?

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

295 252 236 96 76

31.

James and Logan are taking batting practice. If their individual probabilities of hitting a homerun are x and y, respectively, then what is the probability that James will not hit a homerun but Logan will?

(A)

x ? xy

(B) (C)

x ? y2 y ? xy

(D) (E)

? y? ? ? ?x ? 1 ?y x

32.

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

On this years Westchester basketball team, the players are all either 5, 7, or 11 years of age. If the product of the ages of the players on the team is 18,865, then what is the probability that a randomly selected team member will NOT be 7?

3 7 2 5 16 37 3 5 49 55

33.

At a local zoo, 1 porcupine and 8 ferrets consume 223 lbs of feed per week. At the same zoo, 1 porcupine and 8 beavers consume 343 lbs of feed per week. If a beaver consumes 1.6 times as much feed as a ferret, then how much feed does 8 porcupines consume in a week?

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

23 120 184 200 320

34.

Every 28,847 minutes Earvin meets his son, John, at the batting cage. If he met his son at 5:32pm today and it is still that time then when is the next time he will meet his son?

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

10 days from now at 9:32pm 10 days from now at 6:19pm 20 days from now at 6:17am 20 days from now at 6:17pm 20 days from now at 6:19pm

35.

On the first of the year, James invested x dollars at Proudstar bank in an account that yields 2% in interest every quarter year. At the end of the year, in which he made no additional deposit or withdrawals, he had y dollars in the account. If James had invested the same amount in an account which pays interest on a yearly basis, what must the interest rate be for James to have y dollars at the end of the year?

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

2.04% 6.12% 8% 8.25% 10%

36.

Before leaving home for the town of Madison, Pete checks a map which shows that Madison is 5 inches from his current location, Gardensquare. Pete arrives in Madison 2.5 hours later and drove at an average speed of 60 miles per hour. What scale, in inches per mile, is the map drawn at?

(A)

(D) (E)

1 3 1 30 1 10 2 30

37.

The number 70 6 is how many times the number 35 6 ?

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

2 6 9 35 64

38.

In March, Kurt ran an average of 1.5 miles an hour. If by June he had increased his pace by 10 seconds per mile, then which of the following expresses the number of hours it would take Kurt to complete one mile in June

(B) (C)

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

3590 60 2 2410 60 2 2390 60 2 3585 60 60 2 3590

39.

A teacher is assigning 6 students to one of three tasks. She will assign students in teams of at least one student and all students will be assigned to teams. If each task will have exactly one team assigned to it, then which of the following are possible combinations of teams to tasks?

I. II. III.

90 60 45

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

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

40.

On March 15th, the population of the city of Madrigoon was .15 billion people. On May 1st , an earthquake struck Madrigoon and destroyed .01% of the 30 million homes. If an equal number of people lived in each home and 50% of the people whose homes were destroyed moved to another city, then how many people moved to another city?

(A)

(E)

7.5 ? 10 4 1.5 ? 10 4 7.5 ? 10 3 .15 ? 10 4 .15 ? 10 3

41.

If x is an integer greater than zero but less than integer n, is x is a factor of n?

(1) (2)

n is divisible by all integers less than 10 x is not a multiple of a prime number

42.

In a group of 8 semifinalists, all but 2 will advance to the final round. If in the final round only the top 3 will be awarded medals then how many groups of medal winners are possible?

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

20 56 120 560 720

(B) (C) (D)

43.

Dr. McCoy designed a space shuttle that can theoretically travel at a maximum velocity of 8 times the speed of light. If the speed of light is 300 million meters per second, then which of the following is the theoretical maximum speed, in meters per second, of Dr. McCoy’s shuttle?

(A)

(E)

2 .4 ? 10 3 2 .4 ? 10 8 2 .4 ? 10 9 3 ? 10 6 3 ? 10 9

44.

If 6 is a factor of a and 21 is a factor of b, is ab a multiple of 70?

(1) (2)

a is a multiple of 4 b is a multiple of 15

45.

If the speed of sound in air is 3.316 ? 10 2 meters per second and the speed of sound in water is 1,500 meters per second, then approximately how many times faster does sound travel in water than in air?

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

2 3 5 10 20

46.

Of the 150 students at Hunter High, 45 are in the glee club and 72 are in the key club. If the number who are in neither group is twice the number who are in both groups, how many are in both groups?

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

22 33 44 55 66

47.

If set N contains only consecutive positive integers, what is the sum of the numbers in set N?

(1) (2)

Nineteen times the sum of the first number in the set and the last number in the set is 1729 There are 38 numbers in the set.

48.

If P is a set of consecutive integers, is there an even number of integers in set P?

(1) (2)

The sum of the integers in set P is 0 The product of the integers in set P is 0

(B) (C) (D)

49.

For the line whose equation is the slope of that line would be

(A)

1 m

(B)

?

(C) (D) (E)

m ?m m?2

1 m

y? 2?b 2 ? m ? , m is not zero. If the line is rotated 90°, then x x

Answer Key 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. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

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

p Changes to Hard Math Practice: Set 1

5.

A die with x sides has consecutive integers on its sides. If the probability of NOT getting a 4 36 on either of two tosses is , how many sides does the die have? 49

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

4 5 7 8 13

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?

150?1.03? ? 150?1.10? 150 ?1.10? 150(.12)(2)-150(.10)(2) 150?.12 ??4? ? 150 ?.10? 150 ?.10 ? 4

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

150 ?1.03? ? 150 ?1.10 ?

(E)

150?1.12? ? 150 ?1.10? 150?1.10?

4

4

Question changed so that the simple interest only must be calculated over one year instead of 2, which might then raise the question as to whether it is simple interest or compound interest 11.

Chris’ sex now no longer changes halfway thru the problem.

19. Some spelling mistakes (FYI: the names are the French words for the colors) fixed and a definition cleared up: 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?