0<a< <9

Quantity B
Quantity A
9–a –a
35.
For all values of the integer p such that 1.9 < |p| < 5.3,
the function f(p) = p2.

Quantity A Quantity B
36. f(p) for the greatest value of p f(p) for the least value of p

37. If and are reciprocals and < 0, which of the following must be true?

(A) ab < 0

(B) < –1

(C) <1

(D)

(E)

38. If mn < 0 and , which of the following must be true?

(A) km + ln < (mn)2

(B) kn + lm < 1
(C) kn + lm > (mn)2
(D) k + l > mn
(E) km > –ln
39. If the reciprocal of the negative integer x is greater than the sum of y and z, then which
of the following must be true?
(A) x > y + z
(B) y and z are positive.
(C) 1 > x(y + z)
(D) 1 < xy + xz

(E) >z–y
39. In a certain sequence, the term an is defined by the formula an = 2 × an – 1 for each
integer n ≥ 2. If a1 = 1, what is the positive difference between the sum of the first 10
terms of the sequence and the sum of the 11th and 12th terms of the same sequence?
(A) 1
(B) 1,024
(C) 1,025
(D) 2,048
(E) 2,049
The operation @ is defined by the equation a@b = (a – 1)(b – 2).
x@5 = 3@x

Quantity A Quantity B
40. x 1

41. The wait time in minutes, w, for a table at a certain restaurant can be estimated by the
formula w = d2 + kn, where d is the number of diners in the party, k is a constant, and n
is the number of parties ahead in line at the beginning of the wait. If a party of 4 has an
estimated wait time of 40 minutes when 6 other parties are ahead of it, how many
minutes would the estimated wait time be for a party of 6 if there are 3 parties ahead of
it?
(A) 28
(B) 33
(C) 39
(D) 42
(E) 48
A certain sequence is defined by the formula an = an – 1 – 7.
a7 = 7

Quantity A Quantity B
42. a1 –35
 f(x) = m where m is the number of distinct prime factors of x.

Quantity A Quantity B
49. f(30) f(64)

50. The sequence a1, a2, a3, …, an is defined by an = 9 + an – 1 for each integer n ≥ 2. If a1
= 11, what is the value of a35?

51. In sequence Q, the first number is 3, and each subsequent number in the sequence is
determined by doubling the previous number and then adding 2. In the first 10 terms of
the sequence, how many times does the digit 8 appear in the units digit?

52. For which of the following functions f(x) is f(a + b) = f(a) + f(b)?
(A) f(x) = x2
(B) f(x) = 5x
(C) f(x) = 2x + 1
(D) f(x) =
(E) f(x) = x – 2
Sam invests a principal of $10,000, which earns interest over a period of years.

Quantity A Quantity B
The final value of the investment after 2 The final value of the investment after 4
years at 8% interest, compounded years at 4% interest, compounded
53. annually annually

54. The number of years it would take for the value of an investment to double, at 26%
interest compounded annually, is approximately which of the following?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
31. What is the value of (3 – )2 + (3 + )2 ?

(A)

(B)

(C) 36

(D)

(E) 162
32. If , what is the value of m?

(A) –2
(B) –1
(C) 0
(D) 1
(E) 2

rs =

Quantity A Quantity B

33.
Quantity A Quantity B

34.
35. Which of the following fractions has the greatest value?

(A)

(B)

(C)

(D)

(E)
 Quantity A Quantity B
36. m n
42. Jane has a 40-ounce mixture of apple juice and seltzer that is 30% apple juice. If she
pours 10 more ounces of apple juice into the mixture, what percent of the mixture will
be seltzer?
(A) 33%
(B) 44%
(C) 50%
(D) 56%
(E) 67%
Half of the shirts in a closet are white and 30% of the remaining shirts are gray.

Quantity A
Quantity B
The percent of the shirts in the closet that
20%
43. are not white or gray.

The length and width of a painted rectangle were each increased by 10%.

Quantity A
Quantity B
The percent increase in the area of the
10%
44. painted rectangle

45. If 35% of x equals 140, what is 20% of x?

(A) 9.8
(B) 39.2
(C) 80
(D) 320
(E) 400
46. A population of a colony of bacteria increases by 20 percent every 3 minutes. If at
9:00am the colony had a population of 144,000, what was the population of the colony
at 8:54am?
(A) 100,000
(B) 112,000
(C) 120,000
(D) 121,000
(E) 136,000
 The price of an item is greater than $90 and less than $150.

Quantity A Quantity B
The price of the item after a 10%-off The price of the item after a $10-off
47. discount and then a $20-off discount discount and then a 20%-off discount

48. The number that is 20 percent less than 300 is what percent greater than 180?
(A) 25

(B) 33

(C) 50

(D) 66

(E) 75
49. A tank that was 40% full of oil was emptied into a 20-gallon bucket. If the oil fills
35% of the bucket’s volume, then what is the total capacity of the tank, in gallons?
(A) 8.75
(B) 15
(C) 16
(D) 17.5
(E) 19
50. If 150 were increased by 60% and then decreased by y percent, the result would be
192. What is the value of y?
(A) 20
(B) 28
(C) 32
(D) 72
(E) 80
51. If x is 150% greater than 200, x is what percent greater than 50% of 500?
(A) 0
(B) 20
(C) 50
(D) 100
(E) 200
24. Jason deposits money at a bank on a Tuesday and returns to the bank 100 days later to
withdraw the money. On what day of the week did Jason withdraw the money from the
bank?
(A) Monday
(B) Tuesday
(C) Wednesday
(D) Thursday
(E) Friday
25. x and h are both positive integers. When x is divided by 7, the quotient is h with a
remainder of 3. Which of the following could be the value of x?
(A) 7
(B) 21
(C) 50
(D) 52
(E) 57
26. When x is divided by 10, the quotient is y with a remainder of 4. If x and y are both
positive integers, what is the remainder when x is divided by 5?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
27. What is the remainder when 1317 + 1713 is divided by 10?

28. If n is an integer and n3 is divisible by 24, what is the largest number that must be a
factor of n?
(A) 1
(B) 2
(C) 6
(D) 8
(E) 12
(C)

(D)

(E)
 a, b, and c are consecutive integers such that a < b < c < 4.

Quantity A Quantity B
29. The range of a, b, and c The average of a, b, and c

is a prime number, xy is even, and x > 4y > 0.

Quantity A Quantity B
30. y 1

x is even, is a prime number, and x + y = 11.

Quantity A Quantity B
31. x y

The product of positive integers f, g, and h is even and the product of

integers f and g is odd.

Quantity A Quantity B
32. The remainder when f is divided by 2 The remainder when h is divided by 2

33. If x is odd, all EXCEPT which one of the following must be odd?
(A) x2 + 4x + 6
(B) x3 + 5x + 3
(C) x4 + 6x + 7
(D) x5 + 7x + 1
(E) x6 + 8x + 4
x2 > 25 and x + y < 0

Quantity A Quantity B
34. x y

The positive integer a is divisible by 2 and 0 < ab < 1.

Quantity B
Quantity A
b
35.
21. Last year, a magazine charged a $50 subscription fee. This year, the price will be
increased by $10. If the magazine could lose 4 subscribers this year and still collect the
same revenue as it did last year, how many subscribers did the magazine have last
year?
(A) 20
(B) 21
(C) 22
(D) 23
(E) 24
22. A rectangular public park has an area of 3,600 square feet. It is surrounded on three
sides by a chain link fence. If the entire length of the fence measures 180 feet, how
many feet long could the unfenced side of the rectangular park be?
Indicate all such lengths.

30
40
60
90
120

The corner store sells yams and plantains by the pound. A pound of plantains
cost $0.30 less than twice the cost of a pound of yams.

Quantity A
Quantity B
The cost of two pounds of yams and two
The cost of three pounds of plantains
23. pounds of plantains

24. The perimeter of a rectangular patio is 268 feet and its length is 168% of its width.
What is the area of the patio, in square feet?
(A) 4,000
(B) 4,200
(C) 4,320
(D) 4,600
(E) 4,760
(D) 28
(E) 35
 A team of 8 chefs produce 3,200 tarts in 5 days. All chefs produce tarts at
the same constant rate.

Quantity A Quantity B
The number of chefs needed to produce The number of days that 4 chefs need to
25. 3,600 tarts in 3 days produce 4,800 tarts

26. Working together at their respective constant rates, robot A and robot B polish 88

pounds of gemstones in 6 minutes. If robot A’s rate of polishing is that of robot B,

how many minutes would it take robot A alone to polish 165 pounds of gemstones?
(A) 15.75
(B) 18
(C) 18.75
(D) 27.5
(E) 30
27. Car A started driving north from point X traveling at a constant rate of 40 miles per
hour. One hour later, car B started driving north from point X at a constant rate of 30
miles per hour. Neither car changed direction of travel. If each car started with 8
gallons of fuel, which is consumed at a rate of 30 miles per gallon, how many miles
apart were the two cars when car A ran out of fuel?
(A) 30
(B) 60
(C) 90
(D) 120
(E) 150
28. One robot, working independently at a constant rate, can assemble a doghouse in 12
minutes. What is the maximum number of complete doghouses that can be assembled
by 10 such identical robots, each working on separate doghouses at the same rate for 2

hours?

(A) 20
(B) 25
(C) 120
(D) 125
(E) 150
29. Working continuously 24 hours a day, a factory bottles Soda Q at a rate of 500 liters
per second and Soda V at a rate of 300 liters per second. If twice as many bottles of
Soda V as of Soda Q are filled at the factory each day, what is the ratio of the volume
of a bottle of Soda Q to a bottle of Soda V?

(A)

(B)

(C)

(D)

(E)
(B) a + b + 3c

(C) 720 – a – b – 3c

(D) 720 – a – b – 2c

(E) 540 – a–b– c

17. a, b, and c are three consecutive odd integers such that a < b < c. If a is halved to
become m, b is doubled to become n, c is tripled to become p, and k = mnp, which of
the following is equal to k in terms of a?
(A) 3a3 + 18a2 + 24a
(B) 3a3 + 9a2 + 6a

(C) a + 16

(D) 6a2 + 36a + 24

(E) a3 + 6a2 + 4a
18. If m pencils cost the same as n pens, and each pencil costs 20 cents, what is the cost, in
dollars, of 10 pens, if each pen costs the same amount? (100 cents = 1 dollar)

(A)

(B)

(C)

(D)

(E) 200mn
19. Randi sells forklifts at a dealership where she makes a base salary of $2,000 per
month, plus a commission equal to 5% of the selling price of the first 10 forklifts she
sells that month, and 10% of the value of the selling price of any forklifts after that. If
all forklifts have the same sale price, s, which of the choices below represents Randi’s
monthly pay, P, as a function of number of forklifts sold, f, in months in which she
sells more than 10 forklifts? (Assume Randi’s pay is made up entirely of base salary
and commission, and no deductions are taken from this pay.)
(A) P = 2,000 + 0.05sf + 0.10sf
(B) P = 2,000 + 0.05sf + 0.10s(f – 10)
(C) P = 2,000 + 0.05s + 0.10s(f – 10)
(D) P = 2,000 + 0.5s + 0.10sf – 10
(E) P = 2,000 + 0.5s + 0.10s(f – 10)
32. If Beth has more money than Ari, and each person has an integer number of dollars,
which of the following could be the combined value of Beth and Ari’s money?

Indicate all such values.

$12
$54
$72
$200

33. If salesperson A sold 35% more motorcycles than salesperson B, which of the
following could be the total number of motorcycles sold by both salespeople?
Indicate all such total numbers of motorcycles.

47
70
135
235

34. A zoo has twice as many zebras as lions and four times as many monkeys as zebras.
Which of the following could be the total number of zebras, lions, and monkeys at the
zoo?
Indicate all such totals.

14
22
28
55
121

35. In nation Z, 10 terble coins equal 1 galok. In nation Y, 6 barbar coins equal 1 murb. If
a galok is worth 40% more than a murb, what is the ratio of the value of 1 terble coin to
the value of 1 barbar coin?

(A)

(B)

(C)
(D)

(E)
30.

Weekly Revenue Per Product Category at Office Supply Store X

Product Category Weekly Revenue

Pens $164
Pencils $111
Legal pads $199
Erasers $38
Average (arithmetic mean) of categories above $128

According to the chart above, the average (arithmetic mean) revenue per week per
product category is $128. However, there is an error in the chart; the revenue for Pens
is actually $176, not $164. What is the new, correct average revenue per week per
product category?
(A) $130
(B) $131
(C) $132
(D) $164
(E) $176
Set M consists of 20 evenly spaced integers, 10 numbers of which are positive and 10
of which are negative.

Quantity A
Quantity B
The average (arithmetic mean) of all the
0
31. numbers in set M

The average (arithmetic mean) of 3x, x, and y is equal to 2x.

Quantity A Quantity B
32. 2x y
27. A number of scientists’ salaries were reported; physicists’ salaries clustered around a
mean of $100,000 and biologists’ clustered around a mean of $70,000. Which of the
following statements could be true, according to the graph above?
Indicate all such statements.

Some biologists earn more than some physicists.

Both biologists’ and physicists’ salaries are normally distributed.
The range of salaries is greater than $150,000.

28. The graph on the left above represents the number of family members per family in
Town X, while the graph on the right above represents the number of family members
per family in Town Y. The median family size for Town X is equal to the median
family size for Town Y. The horizontal and vertical dimensions of the boxes above are
identical and correspond to the same measurements. Which of the following statements
must be true?
Indicate all such statements.

The range of family sizes measured as the number of family members is larger in
Town X than in Town Y.
Families in Town Y are more likely to have sizes within 1 family member of the
mean than are families in Town X.
The data for Town X has a larger standard deviation than the data for Town Y.
 The probability of rain in Greg’s town on Tuesday is 0.3. The probability
that Greg’s teacher will give him a pop quiz on Tuesday is 0.2. The events
occur independently of each other.

Quantity A
Quantity B
The probability that either or both events
The probability that neither event occurs
36. occur

37. A certain city has a chance of rain occurring on any given day. In any given 3-day

period, what is the probability that the city experiences rain?

(A)

(B)

(C)

(D)

(E) 1
38. Five students, Adnan, Beth, Chao, Dan, and Edmund are to be arranged in a line. How
many such arrangements are possible if Beth is not allowed to stand next to Dan?
(A) 24
(B) 48
(C) 72
(D) 96
(E) 120
39. A polygon has 12 edges. How many different diagonals does it have? (A diagonal is a
line drawn from one vertex to any other vertex inside the given shape. This line cannot
touch or cross any of the edges of the shape. For example, a triangle has zero diagonals
and a rectangle has two.)
(A) 54
(B) 66
(C) 108
(D) 132
(E) 144
 Quantity A Quantity B
The number of possible 4-person teams The number of possible 2-person teams
40. that can be selected from 6 people that can be selected from 6 people

Quantity A Quantity B
The number of ways 1st, 2nd, and 3rd The number of ways 1st, 2nd, 3rd, 4th,
place prizes could be awarded to 3 out of and 5th place prizes could be awarded to
41. 6 contestants 5 contestants

An inventory of coins contains 100 different coins.

Quantity A Quantity B
The number of possible collections of 56 The number of possible collections of 44
coins that can be selected (the order of coins that can be selected (the order of
42. the coins does not matter) the coins does not matter)

An office supply store carries an inventory of 1,345 different products, all of which it
categorizes as “business use,” “personal use,” or both. There are 740 products
categorized as “business use” only and 520 products categorized as both “business
use” and “personal use.”

Quantity A
Quantity B
The number of products characterized as
600
43. “personal use”
Problem Set M

46. What percent of owner-occupied housing units are households with fewer than four
people?
(A) 11.1%
(B) 14.5%
(C) 25.6%
(D) 74.4%
(E) 88.9%
47. Among the owner-occupied housing units represented in the chart above,
approximately how many households are 5-person households?
(A) 1 million
(B) 2 million
(C) 3 million
(D) 4 million
(E) 5 million
 Quantity A Quantity B
16. x+y+z 270

17. A 2-meter by 2-meter sheet of paper is to be cut into 2-centimeter by 10-centimeter

rectangles. What is the maximum number of such rectangles that can be cut from the
sheet of paper? (1 meter = 100 centimeters)

A parallelogram has two sides with length 10 and two sides with length 5.

Quantity A Quantity B
18. The area of the parallelogram 30

19. What is the area of a regular hexagon of side length 4?

(A) 4

(B) 6

(C) 12

(D) 24

(E) 36
22. If a solid right circular cylinder with height 9 and radius 2 is cut as shown into three
new cylinders, each of equal and uniform height, how much new surface area is
created?
(A) 4π
(B) 12π
(C) 16π
(D) 24π
(E) 36π

x > 60°

Quantity A Quantity B
The ratio of the length of arc ABC to the
23. circumference of the circle
35. Triangle ABC has an area of 9. If AC is three times as long as CB, what is the length of
AB?
(A) 6

(B)

(C)

(D)
(E) 15

Quantity A Quantity B
36. CB 7

In the figure above, side lengths AB, BD, and DC are all equal.
 Quantity A Quantity B
37. x y
38. In the figure above, what is the value of x?
(A) 2.5

(B)

(C) 5

(D)

(E)

39. Which of the following statements, considered independently, provide sufficient

information to calculate the area of triangle ABC?

Angle ACB equals 90°

AB = 17
ABC is a right triangle
 In the coordinate plane, points (a, b) and (c, d) are equidistant from the origin.
|a| > |c|

Quantity A Quantity B
21. |b| |d|

In the coordinate plane, lines j and k are parallel and the product of their
slopes is positive.
The x-intercept of line j is greater than the x-intercept of line k.

Quantity A Quantity B
22. The y-intercept of line j The y-intercept of line k

Quantity A Quantity B
23. The area of parallelogram KLMN The area of quadrilateral JKLM

24. Which of the following could be the equation of a line parallel to the line 3x + 2y = 8?

(A) y = x+7

(B) y = – x+7

(C) y = x+7

(D) y = – x+7
 In the figure above, a right triangle is inscribed in a circle with an area of 16π cm2.

Quantity A
Quantity B
The hypotenuse of the triangle, in
8
16. centimeters

17. A rectangular box has a length of 6 centimeters, a width of 8 centimeters, and a height
of 10 centimeters. What is the length of the diagonal of the box, in centimeters?
(A) 10
(B) 12
(C)
(D)
(E) 24

18. Julian takes a 10-inch by 10-inch square piece of paper and cuts it in half along the
diagonal. He then takes one of the halves and cuts it in half again from the corner to the
midpoint of the opposite side. All cuts are represented in the figure with dotted lines.
What is the perimeter of one of the smallest triangles, in inches?
(A) 10
(B)
 The sequence of numbers a1, a2, a3, …, an, … is defined by for

each integer n ≥ 1.

Quantity A Quantity B
The sum of the first 32 terms of this The sum of the first 31 terms of this
31. sequence sequence

32. Each of 100 balls has an integer value from 1 to 8, inclusive, painted on the surface.
The number nx of balls representing integer x is defined by the formula nx = 18 – (x –
4)2. What is the interquartile range of the 100 integers?
(A) 1.5
(B) 2.0
(C) 2.5
(D) 3.0
(E) 3.5
The operator ! is defined such that a!b = ab × b–a.

Quantity A Quantity B

33.
34. What is the ratio of the sum of the odd positive integers between 1 and 100, inclusive,
and the sum of the even positive integers between 100 and 150, inclusive?
(A) 2 to 3
(B) 5 to 7
(C) 10 to 13
(D) 53 to 60
(E) 202 to 251
35. For integer n ≥ 3, a sequence is defined as an = (an – 1)2 – (an – 2)2 and an > 0 for all
positive integers n. The first term a1 is 2, and the fourth term is equal to the first term
multiplied by the sum of the second and third terms. What is the third term, a3?
(A) 0
(B) 3
(C) 5
(D) 10
(E) 16
36. In a certain sequence, each term beyond the second term is equal to the average of the
previous two terms. If a1 and a3 are positive integers, which of the following is not a
possible value of a5?

(A) –

(B) 0

(C)

(D)

(E)

37. The operator @ is defined by the following expression: a@b = where

ab ≠ 0. What is the sum of the solutions to the equation x@2 = ?

(A) –1
(B) –0.75
(C) –0.25
(D) 0.25
(E) 0.75
x is a non-negative number and the square root of (10 – 3x) is greater than x.

Quantity A Quantity B
38. |x| 2