For the positive integers a, b, c, and d, a is half of b, which is one-third of c.
The value
of d is three times the value of c.
Quantity A Quantity B
21.
22. If x2 – y2 = 0 and xy ≠ 0, which of the following must be true?
Indicate all such statements.
x=y
|x| = |y|
=1
3x + 6y = 27
x + 2y + z = 11
Quantity A Quantity B
23. z+5 x + 2y – 2
24. If (x – y) = and (x + y) = , what is the value of x2 – y2?
(A) 3
(B) 6
(C) 9
(D) 36
(E) It cannot be determined from the information given.
a≠b
Quantity A
Quantity B
1
25.
|x|y > x|y|
Quantity A Quantity B
26. (x + y)2 (x – y)2
27. Which of the following could be the graph of all values of x that satisfy the inequality
4 – 11x ≥ ?
(A)
(B)
(C)
(D)
(E)
–1 < a < 0 < |a| < b < 1
Quantity A Quantity B
28.
x > |y| > z
Quantity A Quantity B
29. x+y |y| + z
30. The integers k, l, and m are consecutive even integers between 23 and 33. Which of the
following could be the average (arithmetic mean) of k, l, and m?
(A) 24
(B) 25
(C) 25.5
(D) 28
27. In a certain sequence, the term an is defined by the formula an = an – 1 + 10 for each
integer n ≥ 2. What is the positive difference between a10 and a15?
(A) 5
(B) 10
(C) 25
(D) 50
(E) 100
28. For a physical fitness test, scores are determined by the expression 2ps – 45m, where p
and s are the numbers of push-ups and sit-ups an athlete can do in one minute for each
activity and m is the number of minutes the athlete takes to run a mile. During the test,
Abraham did 21 push-ups and 30 sit-ups and ran the mile in 10 minutes. Javed got the
same score, but did 4 more push-ups and ran the mile in 12 minutes. How many sit-ups
did Javed do in one minute?
(A) 16
(B) 19
(C) 25
(D) 27
(E) 35
29. If a#b = a2 – a, where b ≥ 0, what is the value of (–4)#4?
(A) –36
(B) –28
(C) 12
(D) 28
(E) 36
30. The expression x$y is defined as , where y ≠ 0. What is the value of 9$(6$2)?
(A)
(B)
(C)
(D) 18
(E) 108
31. Amy deposited $1,000 into an account that earns 8% annual interest compounded
every 6 months. Bob deposited $1,000 into an account that earns 8% annual interest
compounded quarterly. If neither Amy nor Bob makes any additional deposits or
withdrawals, in 6 months how much more money will Bob have in his account than
will Amy have in hers?
(A) $40
(B) $8
(C) $4
(D) $0.40
(E) $0.04
32. The half-life of an isotope is the amount of time required for 50% of a sample of the
isotope to undergo radioactive decay. The half-life of the carbon-14 isotope is 5,730
years. How many years must pass until a sample that starts out with 16,000 carbon-14
isotopes decays into a sample with only 500 carbon-14 isotopes?
(A) 180 years
(B) 1,146 years
(C) 5,730 years
(D) 28,650 years
(E) 183,360 years
33. f(x) = and g(x) = 3x – 2. If f(g(x)) = 1, what is the value of x?
(A)
(B)
(C)
(D) 1
(E)
34. A certain investment doubled in value every 9 years. If Saidah had $25,125 in the
investment when she was 27 years old, what was the value of the investment when she
retired at 63 years old?
(A) $50,250
Quantity A Quantity B
19. x 4
20. What is the value of ?
Give your answer as a fraction.
21. In a decimal number, a bar over one or more consecutive digits means that the pattern
of digits under the bar repeats without end. What fraction is equal to 7.583?
Give your answer as a fraction.
Quantity A Quantity B
22.
23. What is the value of ?
(A)
(B)
(C)
(D)
(E)
26. After a 15% discount, the price of a computer was $612. What was the price of the
computer before the discount?
(A) $108.00
(B) $520.20
(C) $703.80
(D) $720.00
(E) $744.00
At the end of April, the price of fuel was 40% greater than the price at the beginning of
the month. At the end of May, the price of fuel was 30% greater than the price at the
end of April.
Quantity A Quantity B
27. The price increase in April The price increase in May
28. Aloysius spends 50% of his income on rent, utilities, and insurance, and 20% on food.
If he spends 30% of the remainder on video games and has no other expenditures, what
percent of his income is left after all of the expenditures?
(A) 30%
(B) 21%
(C) 20%
(D) 9%
(E) 0%
29. In 1970, company X had 2,000 employees, 15% of whom were women, and 10% of
these women were executives. In 2012, the company had 12,000 employees, 45% of
whom were women. If 40% of those women were executives, what was the percent
increase in the number of women executives from 1970 to 2012?
30. 75% of all the boys and 48% of all the girls at Smith High School take civics. If there
are 20% fewer boys than there are girls in the school, what percent of all the students
take civics?
Airline A and airline B both previously charged $400 for a certain flight. Airline A then
reduced its price by 25%. Airline B responded by reducing its price by 55% but adding
$150 in fees. Then, airline A increased its reduced price by 10%.
Quantity A Quantity B
31. The final price of the flight on airline A The final price of the flight on airline B
p is 75% of q and p equals 2r.
Quantity A Quantity B
32. 0.375q r
0 < x < 100
Quantity A Quantity B
33. x% of 0.5% of 40,000 0.05% of 2,000% of 40x
Profit Per Student (in Dollars) at Dan’s Dojo, 2000–2004
2000 60
2001 80
2002 80
2003 100
2004 162
34. At Dan’s Dojo, the percent increase from 2004 to 2005 (not shown) was the same as
the percent increase from 2000 to 2001. What was the profit per student for 2005?
35. If x is 0.5% of y, then y is what percent of x?
(A) 199%
(B) 200%
(C) 2,000%
(D) 19,900%
(E) 20,000%
is divisible by 6.
16. If 5k + 1 = 2,000, what is the value of 5k + 1?
(A) 399
(B) 401
(C) 1,996
(D) 2,000
(E) 2,001
17. If 311 = 9x, what is the value of x?
18. If , what is the value of ?
Give your answer as a fraction.
19. Which of the following is equal to ?
(A)
(B)
(C) 2
(D) 5
(E) 10
20. If , , and 38(3) = 3c, what is the value of a + b + c?
19. If x is a positive integer, which one of the following could be the remainder when 73x
is divided by 10?
Indicate all such remainders.
0
1
2
3
4
5
6
7
8
9
20. If x, y, and z are integers, y + z = 13, and xz = 9, which of the following must be true?
(A) x is even
(B) x = 3
(C) y is odd
(D) y > 3
(E) z < x
Quantity A
Quantity B
The least prime number greater than
The greatest prime number less than 16
21. 13
22. The average (arithmetic mean) of 11 integers is 35. What is the sum of all the integers?
23. What is the sum of all the integers from 1 to 80, inclusive?
(A) 3,200
(B) 3,210
(C) 3,230
(D) 3,240
(E) 3,450
(D) $36
(E) $40
14. During a sale, the local outlet of the Chasm sold three times as many jeans as chinos.
If they made twice as much profit for a pair of chinos as for a pair of jeans, and sold no
other items, what percent of their profits during the sale came from chinos?
(A) 16 %
(B) 20%
(C) 40%
(D) 60%
(E) 83 %
15. Marisol is twice as old as Vikram. Eight years ago, Marisol was 6 years younger than
three times Vikram’s age at that time. How old will Marisol be in 5 years?
16. The length of a rectangle is two more than twice its width, and the area of the
rectangle is 40. What is the rectangle’s perimeter?
17. Marcy bought one pair of jeans at 70% off and one blouse at 40% off. If she paid $12
more for the blouse than for the jeans, and she spent a total of $84, what was the
original price of the jeans?
(A) 76
(B) 96
(C) 100
(D) 120
(E) 124
18. Wall-to-wall carpeting is installed in a certain hallway. The carpeting costs $4.25 per
square foot. If the perimeter of the hallway (in feet) is equal to 44% of the area of the
hallway (in square feet) and the hallway is 50 feet long, how much did the carpeting
cost?
(A) $182.50
(B) $212.50
(C) $505.25
(D) $1,062.50
Each working at a constant rate, Rachel assembles a brochure every 10
minutes and Terry assembles a brochure every 8 minutes.
Quantity A
The number of minutes it will take Quantity B
Rachel and Terry, working together, to 40
17. assemble 9 brochures
18. With 4 identical servers working at a constant rate, a new Internet search provider
processes 9,600 search requests per hour. If the search provider adds 2 more identical
servers, and server work rate never varies, the search provider can process 216,000
search requests in how many hours?
(A) 15
(B) 16
(C) 18
(D) 20
(E) 24
19. If Sabrina can assemble a tank in 8 hours, and Janis can assemble a tank in 13 hours,
then Sabrina and Janis working together at their constant respective rates can assemble
a tank in approximately how many hours?
(A) 21
(B) 18
(C) 7
(D) 5
(E) 2
20. Phil collects virtual gold in an online computer game and then sells the virtual gold for
real dollars. After playing 10 hours a day for 6 days, he collected 540,000 gold pieces.
If he immediately sold this virtual gold at a rate of $1 per 1,000 gold pieces, what were
his average earnings per hour, in real dollars?
(A) $5
(B) $6
(C) $7
(D) $8
(E) $9
11. Cheese that costs c cents per ounce costs how many dollars per pound? (16 ounces = 1
pound and 100 cents = 1 dollar)
(A)
(B)
(C)
(D)
(E) 1,600c
12. A bag of snack mix contains 3 ounces of pretzels, 1 ounce of chocolate chips, 2 ounces
of mixed nuts, and x ounces of dried fruit by weight. What percent of the mix is dried
fruit, by weight?
(A)
(B)
(C)
(D)
(E)
13. At her current job, Mary gets a 1.5% raise twice per year. Which of the following
choices represents Mary’s current income y years after starting the job at a starting
salary of s?
(A) s(1.5)2y
(B) s(0.015)2y
(C) s(1.015)2y
(D) s(1.5)
At a certain college, the ratio of students to professors is 8 : 1 and the ratio of students to
administrators is 5 : 2. No person is in more than one category (for instance, there are no
administrators who are also students).
Quantity A Quantity B
The fractional ratio of professors to
24. administrators
25. In a certain country, 8 rubels are worth 1 schilling, and 5 schillings are worth 1 lemuw.
In this country, 6 lemuws are equivalent in value to how many rubels?
(A)
(B) 30
(C) 40
(D) 48
(E) 240
26. Team A and team B are raising money for a charity event. The ratio of money
collected by team A to money collected by team B is 5 : 6. The ratio of the number of
students on team A to the number of students on team B is 2 : 3. What is the ratio of
money collected per student on team A to money collected per student on team B?
(A) 4 : 5
(B) 5 : 4
(C) 5 : 6
(D) 5 : 9
(E) 9 : 5
27. Jarod uses of an ounce of vinegar in every 2 cups of sushi rice that he prepares. To
prepare 7 cups of sushi rice in the same proportion, how many ounces of vinegar does
Jarod need?
(A)
(B)
(C)
19. The average (arithmetic mean) of 13 numbers is 70. If the average of 10 of these
numbers is 90, what is the average of the other 3 numbers?
(A) –130
(B)
(C) 30
(D) 90
(E) 290
20. Town A has 6,000 citizens and an average (arithmetic mean) of 2 radios per citizen.
Town B has 10,000 citizens and an average of 4 radios per citizen. What is the average
number of radios per citizen in both towns combined?
Give your answer as a fraction.
The average (arithmetic mean) weight of 4 people is 85 kilograms. Two of the people
weigh 75 and 90 kilograms, respectively.
Quantity A
Quantity B
The average (arithmetic mean) weight of
85 kilograms
21. the other two people, in kilograms
22. Fiber X cereal is 55% fiber. Fiber Max cereal is 70% fiber. Sheldon combines an
amount of the two cereals in a single bowl of mixed cereal that is 65% fiber. If the
bowl contains a total of 12 ounces of cereal, how much of the cereal, in ounces, is Fiber
X?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 9
In a normally distributed set of data, the mean is 12 and the standard deviation is less
than 3.
Quantity A
Quantity B
The number of data points in the set
60% of the total number of data points
21. between 9 and 15
Quantity A Quantity B
The standard deviation of the dataset 10, The standard deviation of the dataset 10,
22. 20, 30 20, 20, 20, 20, 20, 30
The graph represents the normally distributed scores on a test. The shaded area
represents approximately 68% of the scores.
Quantity A Quantity B
23. The mean score on the test 550
24.
A and B are graphical representations of normally distributed random variables X and
Y, respectively, with relative positions, shapes, and sizes as shown. Which of the
following must be true?
Indicate all such statements.
24. A baby has x total toys. If 9 of the toys are stuffed animals, 7 of the toys were given to
the baby by her grandmother, 5 of the toys are stuffed animals given to the baby by her
grandmother, and 6 of the toys are neither stuffed animals nor given to the baby by her
grandmother, what is the value of x?
25. How many integers between 2,000 and 3,999 have a ones digit that is a prime number?
26. A group of 12 people who have never met are in a classroom. How many handshakes
are exchanged if each person shakes hands exactly once with each of the other people
in the room?
(A) 12
(B) 22
(C) 66
(D) 132
(E) 244
27. A class consists of 12 girls and 20 boys. One quarter of the girls in the class have blue
eyes. If a child is selected at random from the class, what is the probability that the
child is a girl who does not have blue eyes?
(A)
(B)
(C)
(D)
(E)
28. A certain coin with heads on one side and tails on the other has a probability of
landing on heads. If the coin is flipped three times, what is the probability of flipping 2
tails and 1 head, in any order?
(A)
(B)
(C)
(D)
(E)
29. A number cube has six faces numbered 1 through 6. If the cube is rolled twice, what is
the probability that at least one of the rolls will result in a number greater than 4?
(A)
(B)
(C)
(D)
(E)
30. 100 tiles are labeled with the integers from 1 to 100 inclusive; no numbers are
repeated. If Alma chooses one tile at random, replaces it in the group, and chooses
another tile at random, what is the probability that the product of the two integer values
on the tiles is odd?
(A)
Problem Set H
28. On average, the machine operators that produce the fewest defective parts per 1,000
have how many hours of experience?
(A) 40
(B) 4,000
(C) 8,000
(D) 12,000
(E) 16,000
29. On average, the machine operators with approximately how many hours of experience
have the same defective part rate as those with 12,000 hours of experience?
(A) 2,000
(B) 2,700
(C) 4,400
(D) 8,400
(E) 12,800
30. On average, approximately how many hours of experience do machine operators who
produce the most defective parts per 1,000 have?
(A) 40
(B) 4,000
(C) 8,000
(D) 12,000
(E) 16,000
31. Of the individual machine operators who recorded a defective part rate of 4.2%,
approximately how many hours of experience did the least experienced operator have?
(A) 2,300
(B) 5,000
(C) 7,700
(D) 9,800
(E) 15,100
Problem Set I
32. What is the ratio of male athletes to female athletes on the track and field roster?
(A)
(B)
(C)
(D)
(E)
33. All athletes are on only one varsity sports roster EXCEPT those who are on both the
Track and Field team and the Cross Country team. If there are 76 male athletes in total
on the varsity sports rosters, how many male athletes are on both the Track and Field
team and the Cross Country team?
(A) 11
(B) 17
(C) 37
(D) 54
(E) 76
34. On which varsity sports rosters do male athletes outnumber female athletes?
Indicate all such rosters.
Volleyball
Track and Field
Tennis
Golf
Cross Country
Basketball
35. What is the ratio of female tennis players to male basketball players on the varsity
sports rosters?
(A)
(B)
(C)
(D)
(E)
7. What is the area of a square with a diagonal measuring 6 ?
Quantity A Quantity B
The area of a parallelogram with a base of The area of a trapezoid with two parallel
8. length 4 and height of 3.5 sides of lengths 5 and 9 and a height of 2
Quantity A Quantity B
9. x y
The perimeter of square W is 50% of the perimeter of square D.
Quantity A Quantity B
The ratio of the area of square W to the
10. area of square D
11. A 10-inch by 15-inch rectangular picture is displayed in a 16-inch by 24-inch
rectangular frame. What is the area, in inches, of the part of the frame not covered by
the picture?
(A) 150
(B) 234
(C) 244
(D) 264
(E) 384
A rectangular box has edges of length 2, 3, and 4.
Quantity A Quantity B
12. Twice the volume of the box The surface area of the box
11. A sector of a circle has a central angle of 120°. If the circle has a diameter of 12, what
is the area of the sector?
(A) 4π
(B) 8π
(C) 12π
(D) 18π
(E) 36π
Within a circle with radius 12, a sector has an area of 24π.
Quantity A
Quantity B
The measure of the central angle of the
90
12. sector, in degrees
In the circle with center O, the area of the shaded sector is of the area of the full
circle.
Quantity A Quantity B
13. 2x 75
25. What is the value of x in the figure above?
(A) 5
(B) 10
(C) 18
(D) 30
(E) 54
26. An isosceles right triangle has an area of 50. What is the length of the hypotenuse?
(A) 5
(B)
(C)
(D) 10
(E)
27. In the figure above, what is the length of side AB?
(A) 5
(B)
(C)
(D) 8
(E)
28. In the right triangle above, what is the length of AC?
(A) 9
(B) 10
(C) 12
(D) 13
(E) 15
Quantity A Quantity B
29. x+y 110
11. Which of the following could be the equation of the parabola in the coordinate plane
above?
(A) y = x2 + 3
(B) y = (x – 3)2 + 3
(C) y = (x + 3)2 – 3
(D) y = (x – 3)2 – 3
(E) y = (x + 3)2 + 3
12. Which of the following could be the equation of the parabola in the coordinate plane
above?
(A) y = –x – 1
(B) y = x2 + 1
(C) y = –x2 – 1
(D) y = –x2 + 1
(E) y = –(x – 1)2
Lines l and m are parallel.
Quantity A Quantity B
10. x + 2y 180
Lines l1 and l2 are parallel.
a > 90
Quantity A Quantity B
11. a+f+g b+e+h
Body Mass Index (BMI) is calculated by the formula , where w is
weight in pounds and h is height in inches.
Quantity A
Quantity B
The number of pounds gained by a 74-
The number of pounds lost by a 65-inch-
inch-tall person whose BMI increased by
tall person whose BMI decreased by 1.2.
18. 1.0.
19. How many positive five-digit integers contain the digit grouping “57” (in that order)
at least once? For instance 30,457 and 20,574 are two such integers to include, but
30,475 and 20,754 do not meet the restrictions.
(A) 279
(B) 3,000
(C) 3,500
(D) 3,700
(E) 4,000
Quantity A Quantity B
The average of all the multiples of 5 The average of all the multiples of 10
20. between 199 and 706 between 199 and 706
AC || FD
Quantity A Quantity B
21. a + d – c – 90 90 – e – b – f
22. A man travels to his home from his current location on the rectangular grid shown
above. If he may choose to travel north or east at any corner, but may never travel south
or west, how many different paths can the man take to get home?
(A) 12
(B) 24
(C) 32
(D) 35
(E) 64
23. A bag contains 3 white, 4 black, and 2 red marbles. Two marbles are drawn from the
bag. If replacement is not allowed, what is the probability that the second marble drawn
will be red?
(A)
(B)
(C)
(D)
(E)
x<0
Quantity A Quantity B
24. ((25x)–2)3 ((5–3)2)–x
