The Redesigned SAT®
Mathematics Sample Sets
Information for users of assistive technology
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individuals who use screen readers, text readers, or other assistive technology. You
may wish to consult the manual or help system of your application software to
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Headings
Some questions include passages or other material that you may find it useful to
return to or skip past. To assist in this kind of navigation, the test documents use
headings as follows.
Heading level 3: section titles
Heading level 4: directions for a group of questions or references to material on
which one or more questions will be based (for example, “Question 3 is based on
the following text:”)
Heading level 5: question numbers, which directly precede the associated questions
Heading level 6: indications of skippable content (For example, you may prefer to
skip some sections of this script, such as those that provide figure descriptions or
possible answers in context for questions that involve revision. This content is
identified at the beginning by the phrase “Begin skippable content” and at the end
by the phrase “End skippable content.” These phrases are formatted as level-6
headings.)
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Pronunciation
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pronounce the text where doing so would not inappropriately change test content.
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Tables
Some questions may include tables. Use the table-navigation features of your
application software.
Figures
This document may include figures, which appear on screen. Following each figure
on screen is text describing that figure. Readers using visual presentations of the
figures may choose to skip parts of the text describing the figure that begin with
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description.”
Your application software may speak unhelpful information when you arrive at the
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speaking the unwanted information.
Mathematical Equations and Expressions
Some of these documents include mathematical equations and expressions. Some
of the mathematical equations and expressions are presented as graphics. In cases
where a mathematical equation or expression is presented as a graphic, a verbal
presentation is also given and the verbal presentation comes directly after the
graphic presentation. The verbal presentation is in green font to assist readers in
telling the two presentation modes apart. Readers using audio alone can safely
ignore the graphical presentations, and readers using visual presentations may
ignore the verbal presentations.
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Mathematics—Calculator
The directions and question numbers below are representative of what students will
encounter on test day. Some math sections allow the use of a calculator, while
others do not, as indicated in the directions.
Turn to Section 4 of your answer sheet to answer the questions in this section.
For questions 1 through 30, solve each problem, choose the best answer from the
choices provided, and fill in the corresponding circle on your answer sheet. For
questions 31 through 38, solve the problem and enter your answer in the grid on
the answer sheet. Please refer to the directions before question 31 on how to enter
your answers in the grid. You may use any available space in your test booklet for
scratch work.
1. The use of a calculator is permitted.
2. All variables and expressions used represent real numbers unless otherwise
indicated.
3. Figures provided in this test are drawn to scale unless otherwise indicated.
4. All figures lie in a plane unless otherwise indicated.
5. Unless otherwise indicated, the domain of a given function f is the set of all real
numbers x for which f  x  f of x is a real number.
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Begin skippable figure descriptions.
The figure presents information for your reference in solving some of
the problems.
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Reference figure 1 is a circle with radius r. Two equations are presented below
reference figure 1.
A equals pi times the square of r.
C equals 2 pi r.
Reference figure 2 is a rectangle with length ℓ and width w. An equation is
presented below reference figure 2.
A equals ℓ w.
Reference figure 3 is a triangle with base b and height h. An equation is presented
below reference figure 3.
A equals one-half b h.
Reference figure 4 is a right triangle. The two sides that form the right angle are
labeled a and b, and the side opposite the right angle is labeled c. An equation is
presented below reference figure 4.
c squared equals a squared plus b squared.
Special Right Triangles
Reference figure 5 is a right triangle with a 30-degree angle and a 60-degree angle.
The side opposite the 30-degree angle is labeled x. The side opposite the 60-degree
angle is labeled x times the square root of 3. The side opposite the right angle is
labeled 2 x.
Reference figure 6 is a right triangle with two 45-degree angles. Two sides are
each labeled s. The side opposite the right angle is labeled s times the square root
of 2.
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Reference figure 7 is a rectangular solid whose base has length ℓ and width w and
whose height is h. An equation is presented below reference figure 7.
V equals ℓ w h.
Reference figure 8 is a right circular cylinder whose base has radius r and whose
height is h. An equation is presented below reference figure 8.
V equals pi times the square of r times h.
Reference figure 9 is a sphere with radius r. An equation is presented below
reference figure 9.
V equals four-thirds pi times the cube of r.
Reference figure 10 is a cone whose base has radius r and whose height is h.
An equation is presented below reference figure 10.
V equals one-third times pi times the square of r times h.
Reference figure 11 is an asymmetrical pyramid whose base has length ℓ and
width w and whose height is h. An equation is presented below reference
figure 11.
V equals one-third ℓ w h.
Additional Reference Information
The number of degrees of arc in a circle is 360.
The number of radians of arc in a circle is 2 pi.
The sum of the measures in degrees of the angles of a triangle is 180.
End skippable figure descriptions.
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For student-produced response questions, students will also see the following
directions:
For questions 31 through 38, solve the problem and enter your answer in the grid,
as described below, on the answer sheet.
1. Although not required, it is suggested that you write your answer in the boxes at
the top of the columns to help you fill in the circles accurately. You will receive
credit only if the circles are filled in correctly.
2. Mark no more than one circle in any column.
3. No question has a negative answer.
4. Some problems may have more than one correct answer. In such cases, grid only
one answer.
1
three and one-half must be recorded as 3.5
2
1
three point five or 7 / 2. seven slash two. (If 3 three and one-half is entered
2
31
into the grid as
, three, one, slash, two, it will be interpreted as ,
2
1
thirty-one halves, not 3 . three and one-half).
2
5. Mixed numbers such as 3
6. Decimal answers: If you obtain a decimal answer with more digits than the grid
can accommodate, it may be either rounded or truncated, but it must fill the
entire grid.
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The following are four examples of how to record your answer in the spaces
provided. Keep in mind that there are four spaces provided to record each answer.
Examples 1 and 2
Beging skippable figure description.
Example 1: If your answer is a fraction such as seven-twelfths, it should be
recorded as follows. Enter seven in the first space, the fraction bar (a slash) in the
second space, one in the third space, and two in the fourth space. All four spaces
would be used in this example.
Example 2: If your answer is a decimal value such as two point five, it could be
recorded as follows. Enter two in the second space, the decimal point in the third
space, and five in the fourth space. Only three spaces would be used in
this example.
End skippable figure description.
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Example 3
Beging skippable figure description.
Example 3: Acceptable ways to record two-thirds are: two slash three, point
six six six, and point six six seven.
End skippable figure description.
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Example 4
Note: You may start your answers in any column, space permitting. Columns you
don’t need should be left blank.
Beging skippable figure description.
Example 4: It is not necessary to begin recording answers in the first space unless
all four spaces are needed. For example, if your answer is 201, you may record two
in the first space, zero in the second space, and one in the third space.
Alternatively, you may record two in the second space, zero in the third space, and
one in the fourth space. Spaces not needed should be left blank.
End skippable figure description.
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Mathematics Sample Questions
Question 1.
The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg).
One cup of milk contains 299 milligrams of calcium and one cup of juice contains
261 milligrams of calcium. Which of the following inequalities represents the
possible number of cups of milk m and cups of juice j a 20-year-old could drink
in a day to meet or exceed the recommended daily calcium intake from these
drinks alone?
A. 299m  261 j  1,000 299 m plus 261 j is greater than or equal to 1,000
B. 299m  261 j  1,000 299 m plus 261 j is greater than 1,000
C.
299 261

 the fraction 299 over m, plus the fraction 261 over j, is
m
j
greater than or equal to 1,000
299 261

 the fraction 299 over m, plus the fraction 261 over j, is
D.
m
j
greater than 1,000
Answer and Explanation. (Follow link to explanation of question 1.)
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Question 2.
A research assistant randomly selected 75 undergraduate students from the list of
all students enrolled in the psychology-degree program at a large university. She
asked each of the 75 students, “How many minutes per day do you typically spend
reading?” The mean reading time in the sample was 89 minutes, and the margin of
error for this estimate was 4.28 minutes. Another research assistant intends to
replicate the survey and will attempt to get a smaller margin of error. Which of the
following samples will most likely result in a smaller margin of error for the
estimated mean time students in the psychology-degree program read per day?
A. 40 randomly selected undergraduate psychology-degree program students
B. 40 randomly selected undergraduate students from all degree programs at the
college
C. 300 randomly selected undergraduate psychology-degree program students
D. 300 randomly selected undergraduate students from all degree programs at the
college
Answer and Explanation. (Follow link to explanation of question 2.)
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Questions 3 through 5 refer to the following information and figure.
The first metacarpal bone is located in the wrist. The following scatterplot shows
the relationship between the length of the first metacarpal bone and height for
9 people. The line of best fit is also shown.
Begin skippable figure description.
The figure presents a gridded graph titled “Height of Nine People and Length of
Their First Metacarpal Bone” and nine data points. The y-axis is labeled “Length
of first metacarpal bone,” in centimeters, and the x-axis is labeled “Height,” in
centimeters. The values 4, 4.5, and 5 are labeled on the x-axis with a vertical grid
line at every increment of 0.1. The values 155 through 185, in increments of 5, are
labeled on the y-axis with a horizontal grid line at every increment of one.
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The approximate values of the nine data points on the scatterplot are as follows.
4.0 comma 157.
4.1 comma 163.
4.3 comma 175.
4.5 comma 171.
4.6 comma 173.
4.7 comma 173.
4.8 comma 172.
4.9 comma 183.
5.0 comma 178.
A straight line of best fit is drawn for the data points. The approximate coordinates
of the line are as follows.
4.0 comma 161.5
4.1 comma 163.
4.2 comma 165.
4.3 comma 167.
4.4 comma 169.
4.5 comma 171.
4.6 comma 172.5.
4.7 comma 174.5.
4.8 comma 176.
4.9 comma 178.
5.0 comma 180.
End skippable figure description.
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Question 3 refers to the information and figure (follow link) provided on
pages 15 through 16.
Question 3.
How many of the nine people have an actual height that differs by more than
3 centimeters from the height predicted by the line of best fit?
A. 2
B. 4
C. 6
D. 9
Answer and Explanation. (Follow link to explanation of question 3.)
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Question 4 refers to the information and figure (follow link) provided on
pages 15 through 16.
Question 4.
Which of the following is the best interpretation of the slope of the line of best fit
in the context of this problem?
A. The predicted height increase in centimeters for one centimeter increase in
the first metacarpal bone
B. The predicted first metacarpal bone increase in centimeters for every
centimeter increase in height
C. The predicted height in centimeters of a person with a first metacarpal bone
length of 0 centimeters
D. The predicted first metacarpal bone length in centimeters for a person with a
height of 0 centimeters
Answer and Explanation. (Follow link to explanation of question 4.)
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Question 5 refers to the information and figure (follow link) provided on
pages 15 through 16.
Question 5.
Based on the line of best fit, what is the predicted height for someone with a first
metacarpal bone that has a length of 4.45 centimeters?
A. 168 centimeters
B. 169 centimeters
C. 170 centimeters
D. 171 centimeters
Answer and Explanation. (Follow link to explanation of question 5.)
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Question 6.
Aaron is staying at a hotel that charges $99.95 per night plus tax for a room. A tax
of 8% is applied to the room rate, and an additional one-time untaxed fee of $5.00
is charged by the hotel. Which of the following represents Aaron’s total charge, in
dollars, for staying x nights?
A.  99.95  0.08 x   5 parenthesis, 99.95 plus 0.08 x, close parenthesis, plus 5
B. 1.08  99.95 x   5 1.08, parenthesis, 99.95 x, close parenthesis, plus 5
C. 1.08  99.95 x  5 1.08, parenthesis, 99.95 x plus 5, close parenthesis
D. 1.08  99.95  5 x 1.08, parenthesis, 99.95 plus 5, close parenthesis, x
Answer and Explanation. (Follow link to explanation of question 6.)
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Question 7 refers to the following figure and system of
three equations.
Begin skippable figure description.
The figure presents the graph of a circle, a parabola, and a line in the x y-plane.
The horizontal axis is labeled x, the vertical axis is labeled y, and the origin is
labeled O. The integers negative 3 through 3 appear on both axes.
The circle has its center at the origin and radius of approximately 2.2.
The parabola has its vertex on the y-axis at negative 3 and opens upward.
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The circle and parabola intersect at four points, of which two are below the x-axis
and two are above the x-axis. Of the two points of intersection below the x-axis,
one is to the left of the y-axis and one is to the right of the y-axis. Of the two
points of intersection above the x-axis, one is to the left of the y-axis and one is to
the right of the y-axis.
The line slants upward and to the right, and passes through two of the four points
of intersection where the circle and parabola meet, one below the x-axis and to the
left of the y-axis, and one above the x-axis to the right of the y-axis. In other
words, the three graphs intersect at two points.
End skippable figure description.
The following system of three equations is given beneath the figure.
x 2  y 2  5 x squared plus y squared equals five.
y  x 2  3 y equals x squared minus three.
x  y  1 x minus y equals one.
Question 7.
A system of three equations and their graphs in the x y-plane are shown above.
How many solutions does the system have?
A. One
B. Two
C. Three
D. Four
Answer and Explanation. (Follow link to explanation of question 7.)
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Question 8 refers to the following information and table.
The following table classifies 103 elements as metal, metalloid, or nonmetal and as
solid, liquid, or gas at standard temperature and pressure.
Solids
Liquids Gases
Total
Metals
77
1
0
78
Metalloids
7
0
0
7
Nonmetals
6
1
11
18
Total
90
2
11
103
Question 8.
What fraction of all solids and liquids in the preceding table are metalloids?
Answer and Explanation. (Follow link to explanation of question 8.)
Question 9.
9
7
 2 3 t  1  2 , the fraction negative 9 over 5 is less than negative 3 t plus 1,
5
4
and negative 3 t plus 1 is less than the negative of the fraction 7 over 4, what is one
possible value of 9 t - 3? 9 t minus 3?
If 2
Answer and Explanation. (Follow link to explanation of question 9.)
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Questions 10 and 11 refer to the following information and table.
A survey was conducted among a randomly chosen sample of U. S. citizens about
U. S. voter participation in the November 2012 presidential election. The following
table displays a summary of the survey results.
Reported Voting by Age (in thousands)
Voted
Did Not Vote
No Response
Total
18- to 34-year-olds
30,329
23,211
9,468
63,008
35- to 54-year-olds
47,085
17,721
9,476
74,282
55- to 74-year-olds
43,075
10,092
6,831
59,998
People 75 years old
and over
Total
12,459
3,508
1,827
17,794
27,602
215,082
132,948 54,532
Question 10 refers to the preceding information and table (follow link).
Question 10.
According to the table (follow link), for which age group did the greatest
percentage of people report that they had voted?
A. 18- to 34-year-olds
B. 35- to 54-year-olds
C. 55- to 74-year-olds
D. People 75 years old and over
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Answer and Explanation. (Follow link to explanation of
question 10.)
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Question 11 refers to the information and table (follow link) provided on
page 24.
Question 11.
Of the 18- to 34-year-olds who reported voting, 500 people were selected at
random to do a follow-up survey where they were asked which candidate they
voted for. There were 287 people in this follow-up survey sample who said they
voted for Candidate A, and the other 213 people voted for someone else. Using the
data from both the follow-up survey and the initial survey, which of the following
is most likely to be an accurate statement?
A. About 123 million people 18 to 34 years old would report voting for
Candidate A in the November 2012 presidential election.
B. About 76 million people 18 to 34 years old would report voting for Candidate A
in the November 2012 presidential election.
C. About 36 million people 18 to 34 years old would report voting for Candidate A
in the November 2012 presidential election.
D. About 17 million people 18 to 34 years old would report voting for Candidate A
in the November 2012 presidential election.
Answer and Explanation. (Follow link to explanation of
question 11.)
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Question 12.
A company’s manager estimated that the cost C, in dollars, of producing n items
is C  7n  350. C equals 7 n plus 350. The company sells each item for $12. The
company makes a profit when total income from selling a quantity of items is
greater than the total cost of producing that quantity of items. Which of the
following inequalities gives all possible values of n for which the manager
estimates that the company will make a profit?
A.
B.
C.
D.
n  70
n  
n  
n  
n is less than 70.
n is less than 84.
n is greater than 70.
n is greater than 84.
Answer and Explanation. (Follow link to explanation of
question 12.)
Question 13.
At a primate reserve, the mean age of all the male primates is 15 years, and the
mean age of all female primates is 19 years. Which of the following must be true
about the mean age m of the combined group of male and female primates at the
primate reserve?
A.
B.
C.
D.
m  17 m equals 17.
m  17 m is greater than 17.
m  17 m is less than 17.
15  m  15 is less than m, and m is less than 19.
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Answer and Explanation. (Follow link to explanation of
question 13.)
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Question 14.
A researcher wanted to know if there is an association between exercise and sleep
for the population of 16-year-olds in the United States. She obtained survey
responses from a random sample of 2000 United States 16-year-olds and found
convincing evidence of a positive association between exercise and sleep. Which
of the following conclusions is well supported by the data?
A. There is a positive association between exercise and sleep for 16-year-olds in
the United States.
B. There is a positive association between exercise and sleep for 16-year-olds in
the world.
C. Using exercise and sleep as defined by the study, an increase in sleep is caused
by an increase of exercise for 16-year-olds in the United States.
D. Using exercise and sleep as defined by the study, an increase in sleep is caused
by an increase of exercise for 16-year-olds in the world.
Answer and Explanation. (Follow link to explanation of
question 14.)
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Question 15.
A biology class at Central High School predicted that a local population of animals
will double in size every 12 years. The population at the beginning of 2014 was
estimated to be 50 animals. If P represents the population n years after 2014, then
which of the following equations represents the class’s model of the population
over time?
A. P  12  50n P equals 12 plus 50 n.
B. P  50  12n P equals 50 plus 12 n.
C. P  50 2 12n P equals 50, parenthesis, 2, close parenthesis, to the power of
 
12 n.
n
P equals 50, parenthesis, 2, close parenthesis, to the power of the
12
fraction n over 12.
D. P  50  2 
Answer and Explanation. (Follow link to explanation of
question 15.)
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Question 16 is based on the following figure.
Begin skippable figure description.
The figure presents line segments A E and B D that intersect at point C. Line
segments A B and D E are drawn resulting in two triangles A B C and E D C.
A note under the figure says that the figure is not drawn to scale.
End skippable figure description.
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Question 16.
In the preceding figure (follow link), n ABC n EDC. triangle A B C is similar to
triangle E D C. Which of the following must be true?
A. AE BD Line segment A E is parallel to line segment B D.
B. AE  BD Line segment A E is perpendicular to line segment B D.
C. AB DE Line segment A B is parallel to line segment D E.
D. AB  DE Line segment A B is perpendicular to line segment D E.
Answer and Explanation. (Follow link to explanation of
question 16.)
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Question 17.
The gas mileage for Peter’s car is 21 miles per gallon when the car travels at an
average speed of 50 miles per hour. The car’s gas tank has 17 gallons of gas at the
beginning of a trip. If Peter’s car travels at an average speed of 50 miles per hour,
which of the following functions f models the number of gallons of gas remaining
in the tank t hours after the trip begins?
A. f  t   17 
21
f of t equals 17 minus the fraction 21 over 50 t.
50 t
50 t
f of t equals 17 minus the fraction 50 t over 21.
21
17  21t
C. f  t  
f of t equals the fraction whose numerator is 17 minus 21 t,
50
and whose denominator is 50.
17  50 t
D. f  t  
f of t equals the fraction whose numerator is 17 minus 50 t,
21
and whose denominator is 21.
B. f  t   17 
Answer and Explanation. (Follow link to explanation of
question 17.)
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Question 18.
The toll rates for crossing a bridge are $6.50 for a car and $10 for a truck. During a
two-hour period, a total of 187 cars and trucks crossed the bridge, and the total
collected in tolls was $1,338. Solving which of the following systems of equations
yields the number of cars, x, and the number of trucks, y, that crossed the bridge
during the two hours?
A. x  y  1,338 x plus y equals 1,338.
6.5 x  10 y  187 6.5 x plus 10 y equals 187.
B. x  y  187 x plus y equals 187.
6.5 x  10 y 
1,338
6.5 x, plus 10 y, equals the fraction 1,338 over 2.
2
C. x  y  187 x plus y equals 187.
6.5 x  10 y  1,338 6.5 x, plus 10 y equals 1,338.
D. x  y  187 x plus y equals 187.
6.5x  10 y  1,338  2 6.5 x, plus 10 y equals 1,338 times 2.
Answer and Explanation. (Follow link to explanation of
question 18.)
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Question 19.
When a scientist dives in salt water to a depth of 9 feet below the surface,
the pressure due to the atmosphere and surrounding water is
18.7 pounds per square inch. As the scientist descends, the pressure increases
linearly. At a depth of 14 feet, the pressure is 20.9 pounds per square inch. If the
pressure increases at a constant rate as the scientist’s depth below the surface
increases, which of the following linear models best describes the pressure p in
pounds per square inch at a depth of d feet below the surface?
A. p  0.44d  0.77 p equals 0.44 d plus 0.77.
B. p  0.44d  14.74 p equals 0.44 d plus 14.74.
C. p  2.2d  1.1 p equals 2.2 d minus 1.1.
D. p  2.2d  9.9 p equals 2.2 d minus 9.9.
Answer and Explanation. (Follow link to explanation of
question 19.)
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Question 20 is based on the following figure.
Begin skippable figure description.
The figure, titled “Count of Manatees,” presents the graph of a scatterplot with a
line. The horizontal axis is labeled “Year,” and the vertical axis is labeled “Number
of Manatees.” The years 1990 through 2015 are labeled on the horizontal axis, in
increments of 5 years. The numbers 1,000 through 6,000 are labeled on the vertical
axis, in increments of 1,000. Grid lines extend from the labeled increments of both
axes.
There are 24 data points on the graph. The data points range horizontally from
years 1991 to 2011 and vertically from approximately 1,300 manatees to
approximately 5,100 manatees.
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A line of best fit is drawn for the range of years represented by the data points. The
line begins at year 1991 with approximately 1,200 manatees and ends at year 2011
with approximately 4,200 manatees. The line of best fit intersects four vertical grid
lines, which represent 5-year increments, at the following approximate values.
Year 1995: 1,800 manatees.
Year 2000: 2,600 manatees.
Year 2005: 3,300 manatees.
Year 2010: 4,100 manatees.
End skippable figure description.
Question 20.
The preceding scatterplot (follow link) shows counts of Florida manatees, a type of
sea mammal, from 1991 to 2011. Based on the line of best fit to the data shown,
which of the following values is closest to the average yearly increase in the
number of manatees?
A. 0.75
B. 75
C. 150
D. 750
Answer and Explanation. (Follow link to explanation of
question 20.)
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Question 21 is based on the following figure.
Begin skippable figure description.
The figure, titled “Bacteria Growth,” presents a graph of two curved lines. The
horizontal axis is labeled “Time” in hours and the vertical axis is labeled “Area
covered” in square centimeters. Both axes are labeled from 0 to 10 in increments of
one with grid lines extending from each labeled increment.
The curved line labeled “Dish 1” begins on the vertical axis at 1 and curves steeply
up and to the right passing through the point with coordinates  2, 4  , 2 comma 4,
and the point with coordinates  3, 8. 3 comma 8. The curved line labeled “Dish 2”
begins on the vertical axis at 2 and moves to the right before curving gradually up
and to the right passing through the point with coordinates  3, 3 , 3 comma 3, and
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the point with coordinates  5, 6  . 5 comma 6. The two curved lines intersect at a
point with approximate coordinates 1.2, 2.1. 1.2 comma 2.1.
End skippable figure description.
Question 21.
A researcher places two colonies of bacteria into two petri dishes that each have
area 10 square centimeters. After the initial placement of the bacteria (t  0),
parenthesis, t equals zero, close parenthesis, the researcher measures and records
the area covered by the bacteria in each dish every ten minutes. The data for each
dish were fit by a smooth curve, as shown in the figure (follow link), where each
curve represents the area of a dish covered by bacteria as a function of time,
in hours. Which of the following is a correct statement about the preceding data?
A. At time t  0, t equals zero, both dishes are 100% covered by bacteria.
B. At time t  0, t equals zero, bacteria covers 10% of Dish 1 and 20% of Dish 2.
C. At time t  0, t equals zero, Dish 2 is covered with 50% more bacteria
than Dish 1.
D. For the first hour, the area covered in Dish 2 is increasing at a higher average
rate than the area covered in Dish 1.
Answer and Explanation. (Follow link to explanation of
question 21.)
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Question 22.
A typical image taken of the surface of Mars by a camera is 11.2 gigabits in size. A
tracking station on Earth can receive data from the spacecraft at a data rate of
3 megabits per second for a maximum of 11 hours each day. If 1 gigabit equals
1,024 megabits, what is the maximum number of typical images that the tracking
station could receive from the camera each day?
A. 3
B. 10
C. 56
D. 144
Answer and Explanation. (Follow link to explanation of
question 22.)
Question 23.
x 2  y 2  153 x squared plus y squared equals 153.
y  4 x y equals negative 4 x.
If ( x, y ) parenthesis, x comma y, close parenthesis, is a solution to the preceding
system of equations, what is the value of x 2 ? x squared?
A. 51 negative 51
B. 3
C. 9
D. 144
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Answer and Explanation. (Follow link to explanation of
question 23.)
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Question 24 is based on the following figure.
Begin skippable figure description.
The figure presents a metal nut with two hexagonal faces and six sides. The
thickness of one side, from one hexagonal face to the other hexagonal face of the
nut, is labeled as 1 centimeter.
End skippable figure description.
Question 24.
The preceding figure (follow link) shows a metal hex nut with two regular
hexagonal faces and a thickness of 1 centimeter. The length of each side of a
hexagonal face is 2 centimeters. A hole with a diameter of 2 centimeters is drilled
through the nut. The density of the metal is 7.9 grams per cubic centimeter. What
is the mass of this nut, to the nearest gram? (Density is mass divided by volume.)
Answer and Explanation. (Follow link to explanation of
question 24.)
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Questions 25 and 26 refer to the following information.
An international bank issues its Traveler credit cards worldwide. When a customer
makes a purchase using a Traveler card in a currency different from the customer’s
home currency, the bank converts the purchase price at the daily foreign exchange
rate and then charges a 4% fee on the converted cost.
Sara lives in the United States, but is on vacation in India. She used her Traveler
card for a purchase that cost 602 rupees (Indian currency). The bank posted a
charge of $9.88 to her account that included the 4% fee.
Question 25 refers to the information (follow link) provided at the top of
page 40.
Question 25.
What foreign exchange rate, in Indian rupees per one U. S. dollar, did the bank use
for Sara’s charge? Round your answer to the nearest whole number.
Answer and Explanation. (Follow link to explanation of
question 25.)
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Question 26 refers to the information (follow link) provided at the top of
page 40.
Question 26.
A bank in India sells a prepaid credit card worth 7,500 rupees. Sara can buy the
prepaid card using dollars at the daily exchange rate with no fee, but she will lose
any money left unspent on the prepaid card. What is the least number of the
7,500 rupees on the prepaid card Sara must spend for the prepaid card to be
cheaper than charging all her purchases on the Traveler card? Round your answer
to the nearest whole number of rupees.
Answer and Explanation. (Follow link to explanation of
question 26.)
Question 27.
If k is a positive constant different from 1, which of the following could be the
graph of y  x  k ( x  y) y minus x equals, k, parenthesis, x plus y, close
parenthesis, in the x y-plane?
Each of the four answer choices presents a graph in the x y-plane. The numbers
2 6 negative 6 through 6 appear along both axes, and the origin is labeled O.
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A.
Begin skippable figure description.
Choice A. The graph shows a line that goes up from left to right and intersects the
x-axis at negative 2, and the y-axis at 2.
End skippable figure description.
B.
Begin skippable figure description.
Choice B. The graph shows a line that goes down from left to right and goes
through the origin. The line appears to pass through the point with coordinates
negative 1 comma 2.
End skippable figure description.
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C.
Begin skippable figure description.
Choice C. The graph shows a line that goes up from left to right and intersects the
y-axis at negative 3, and the x-axis at 1.5.
End skippable figure description.
D.
Begin skippable figure description.
Choice D. The graph shows a smooth curve that appears to be a parabola. The
parabola has its vertex at the origin, opens upward, and passes through the point
with coordinates 1 comma 1.
End skippable figure description.
Answer and Explanation. (Follow link to explanation of
question 27.)
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Question 28.
The function f is defined by f ( x)  2 x3  3x 2  cx  8, f of x equals 2 x cubed,
plus 3 x squared, plus c x, plus 8, where c is a constant. In the x y-plane, the graph
of f intersects the x-axis at the three points 2 4, 0  , parenthesis, negative 4
1 
comma 0, close parenthesis,  , 0  , parenthesis, one-half comma 0, close
2 
parenthesis, and  p, 0  . parenthesis, p comma 0, close parenthesis. What is the
value of c?
A. 2 18 negative 18
B. 2 2 negative 2
C. 2
D. 10
Answer and Explanation. (Follow link to explanation of
question 28.)
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Question 29.
4 x2
the fraction whose numerator is 4 x squared, and whose
2x  1
denominator is 2 x minus one, is written in the equivalent
1
form
 A, the fraction whose numerator is one, and whose denominator is
2x  1
2 x minus one, plus A, what is A in terms of x?
If the expression
A. 2 x  1 2 x plus one.
B. 2 x  1 2 x minus one.
C. 4x 2 4 x squared.
D. 4 x 2  1 4 x squared minus one.
Answer and Explanation. (Follow link to explanation of
question 29.)
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Question 30 is based on the following information and figure.
An architect drew the following sketch while designing a house roof. The
dimensions shown are for the interior of the triangle.
Begin skippable figure description.
The figure presents a triangle with a horizontal base. Labels are given to two sides
and to two angles. The left side of the triangle is labeled 24 feet. The base of the
triangle is labeled 32 feet. The two lower interior angles are labeled x degrees. A
note under the figure says that the figure is not drawn to scale.
End skippable figure description.
Question 30.
What is the value of cos x ? cosine x?
Answer and Explanation. (Follow link to explanation of
question 30.)
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Mathematics Sample Question Answers and
Explanations
The following are explanations of answers to sample questions 1 through 30. The
heading of each explanation is hyperlinked to the actual question. In addition, each
explanation is followed by two hyperlinks: one to the question explained and one
to the next question.
There are two ways to follow a link. One is to move the flashing text cursor, or
caret, into the hyperlinked text and press the Enter key; the other is to place the
mouse cursor, or pointer, over the hyperlinked text and press Ctrl+left-click (that
is, press and release the left button on the mouse while holding down the Ctrl key
on the keyboard). After following a link in Microsoft Word, you can return to your
previous location (for example, the answer explanation) by pressing
Alt+left arrow.
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Explanation for question 1. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must identify the correct mathematical notation for an
inequality to represent a real-world situation.
Difficulty: Easy
Key: A
Choice A is correct. Multiplying the number of cups of milk by the amount of
calcium each cup contains and multiplying the number of cups of juice by the
amount of calcium each cup contains gives the total amount of calcium from each
source. The student must then find the sum of these two numbers to find the total
amount of calcium for the day. Because the question asks for the calcium from
these two sources to meet or exceed the recommended daily intake, the sum of
these two products must be greater than or equal to 1,000.
Choice B is not the correct answer. This answer may result from a
misunderstanding of the meaning of inequality symbols as they relate to real-life
situations. This answer does not allow for the daily intake to meet the
recommended daily amount.
Choice C is not the correct answer. This answer may result from a
misunderstanding of proportional relationships. Here the wrong operation is
applied, with the total amount of calcium per cup divided by the number of cups of
each type of drink. These values should be multiplied.
Choice D is not the correct answer. This answer may result from a combination of
mistakes. The inequality symbol used allows the option to exceed, but not to meet,
the recommended daily value, and the wrong operation may have been applied
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when calculating the total amount of calcium intake from each drink.
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Link back to question 1.
Link back to question 2.
Explanation for question 2. (Follow link back to original question.)
Program: S A T
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must first read and understand statistics calculated from a
survey that was conducted. Then, students must use apply their knowledge about
the relationship between sample size and subject selection on margin of error.
Difficulty: Easy
Key: C
Choice C is correct. Increasing the sample size while randomly selecting
participants from the original population of interest will most likely result in a
decrease in the margin of error.
Choice A is not the correct answer. This answer may result from a
misunderstanding of the importance of sample size to a margin of error. The
margin of error is likely to increase with a smaller sample size.
Choice B is not the correct answer. This answer may result from a
misunderstanding of the importance of sample size and participant selection to a
margin of error. The margin of error is likely to increase due to the smaller sample
size. Also undergraduate students from all degree programs at the college is a
different population than the original survey and therefore the impact to the mean
and margin of error cannot be predicted.
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Choice D is not the correct answer. This answer may result from a
misunderstanding of participant selection to a margin of error. Undergraduate
students from all degree programs at the college is a different population than the
original survey and therefore the impact to the mean and margin of error cannot be
predicted.
Link back to question 2.
Link back to question 3.
Explanation for question 3. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must read and interpret information from a data display.
Difficulty: Easy
Key: B
Choice B is correct. The people who have first metacarpal bones of length 4.0, 4.3,
4.8, and 4.9 centimeters have heights that are greater than 3 centimeters different
from the height predicted by the line of best fit.
Choice A is not the correct answer. There are 2 people whose actual heights are
more than 3 centimeters above the height predicted by the line of best fit.
However, there are also 2 people whose actual heights are farther than
3 centimeters below the line of best fit.
Choice C is not the correct answer. There are 6 data points in which the absolute
value of the between the actual height and the height predicted by the line of best
fit is greater than 1 centimeter.
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Choice D is not the correct answer. The data on the graph represents 9 different
people; however, the absolute value of the difference between actual height and
predicted height is not greater than 3 for all of the people.
Link back to question 3.
Link back to question 4.
Explanation for question 4. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must interpret the meaning of the slope of the line of best fit
in the context provided.
Difficulty: Easy
Key: A
Choice A is correct. The slope is the change in the vertical distance divided by the
change in the horizontal distance between any two points on a line. In this context,
the change in the vertical distance is the change in the predicted height of a person,
and the change in the horizontal distance is the change in the length of his or her
first metacarpal bone. The unit rate, or slope, is the increase in predicted height for
each increase of 1 centimeter of the first metacarpal bone.
Choice B is not the correct answer. Students who select this answer may have
interpreted slope incorrectly as run over rise.
Choice C is not the correct answer. Students who select this answer may have
mistaken slope for the y-intercept.
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Choice D is not the correct answer. Students who select this answer may have
mistaken slope for the x-intercept.
Link back to question 4.
Link back to question 5.
Explanation for question 5. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must use the line of best fit to make a prediction. Students
must also demonstrate fluency in reading graphs and decimal numbers.
Difficulty: Easy
Key: C
Choice C is correct. Students must see that the scale of the x-axis is 0.1, and
therefore the x-value of 4.45 is halfway between the unmarked value of 4.4 and the
marked value of 4.5. The student must then find the y-value on the line of best fit
that corresponds with an x-value of 4.45, which is 170.
Choice A is not the correct answer. A student who mistakenly finds the point on
the line between the x-values of 4.3 and 4.4 will find a predicted metacarpal bone
length of 168 centimeters.
Choice B is not the correct answer. A student who mistakenly finds the point on
the line that corresponds with an x-value of 4.4 centimeters will find a predicted
height of approximately 169 centimeters.
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Choice D is not the correct answer. A student who mistakenly finds the point on
the line that corresponds with an x-value of 4.5 centimeters will find a predicted
height of approximately 171 centimeters. Students might also choose this option if
they mistakenly use the data point which has an x-value closest to
4.45 centimeters.
Link back to question 5.
Link back to question 6.
Explanation for question 6. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must create a linear expression to represent a real-world
situation.
Difficulty: Easy
Key: B
Choice B is correct. The total charge that Aaron will pay is the room rate, the 8%
tax on the room rate, and a fixed fee. If Aaron stayed x nights, then the total
charge is  99.95x  0.08  99.95x   5, parenthesis, 99.95 x plus 0.08, times
99.95 x, close parenthesis, plus 5, which can be rewritten as 1.08  99.95 x   5.
1.08, parenthesis, 99.95 x, close parenthesis, plus 5.
Choice A is not the correct answer. The expression includes only one night stay in
the room and does not accurately account for tax on the room.
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Choice C is not the correct answer. The expression includes tax on the fee, and the
hotel does not charge tax on the $5.00 fee.
Choice D is not the correct answer. The expression includes tax on the fee and a
fee charge for each night.
Link back to question 6.
Link back to question 7.
Explanation for question 7. (Follow link back to original question.)
Program: S A T
Subscore: Passport to advanced math
Calculator usage: Yes
Objective: Students must demonstrate their conceptual understanding of graphical
representations of variable relationships.
Difficulty: Easy
Key: B
Choice B is correct. The solutions to the system of equations are the points where
the circle, parabola, and line all intersect. These points are (1, 2) parenthesis,
negative one comma two, close parenthesis, and (2,1) parenthesis,
two comma one, close parenthesis, and these are the only solutions to the system.
Choice A is not the correct answer. This answer may reflect the misconception that
a system of equations can have only one solution.
Choice C is not the correct answer. This answer may reflect the misconception that
a system of equations has as many solutions as the number of equations in the
system.
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Choice D is not the correct answer. This answer may reflect the misconception that
the solutions of the system are represented by the points where any two of the
curves intersect, rather than the correct concept that the solutions are represented
only by the points where all three curves intersect.
Link back to question 7.
Link back to question 8.
Explanation for question 8. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must read information from two-way tables and determine the
specific relationship between two categorical variables.
Difficulty: Easy
7
Key:
the fraction 7 over 92
92
There are 7 metalloids that are solid or liquid, and there are 92 total solids and
7
liquids. Therefore, the fraction of solids and liquids that are metalloids is
.
92
the fraction 7 over 92.
Link back to question 8.
Link back to question 9.
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Explanation for question 9. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must convert an existing compound inequality into a different
format and find a value that satisfies all conditions.
Difficulty: Easy
21
27
Key: Any value greater than
the fraction 21 over 4 and less than
the
5
4
fraction 27 over 5
Recognizing the structure of this inequality provides one solution strategy. With
this strategy, a student will look at the relationship between 3 t  1 negative 3 t
plus 1 and 9 t - 3 9 t minus 3 and recognize that the latter is - 3 negative 3
multiplied by the former.
Multiplying all parts of the inequality by - 3 negative 3 reverses the inequality
27
21
signs, resulting in
 9 t - 3  , the fraction 27 over 5 is greater than 9 t
5
4
minus 3, and 9 t minus 3 is greater than the fraction 21 over 4, or rather
21
27
the fraction 21 over 4 is less than 9 t minus 3, and 9 t minus 3 is
 9t - 3 
4
5
less than the fraction 27 over 5 when written with increasing values from left to
21
27
right. Any value greater than
the fraction 21 over 4 and less than
the
5
4
fraction 27 over 5 is correct.
Link back to question 9.
Link back to question 10.
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Explanation for question 10. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must conceptualize the context and retrieve relevant
information from a table, and then manipulate it to form or compare relevant
quantities.
Difficulty: Easy
Key: C
Choice C is the correct answer. The first question asks students to select the
relevant information from the table (follow link) to compute the percentage of
self-reported voters for each age group and then compare the percentages to
identify the largest one, choice C. Of the 55- to 74-year-old group’s total
population (59,998,000), 43,075,000 reported that they had voted, which represents
71.8% and is the highest percentage of reported voters from among the four
age groups.
Choice A is not the correct answer. The question is asking for the age group with
the largest percentage of self-reported voters. This answer reflects the age group
with the smallest percentage of self-reported voters. This group’s percentage of
30,329
self-reported voters is 48.1%, or
, the fraction 30,329 over 63,008, which is
63,008
less than that of the 55- to 74-year-old group.
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Choice B is not the correct answer. The question is asking for the age group with
the largest percentage of self-reported voters. This answer reflects the age group
with the largest number of self-reported voters, not the largest percentage. This
47,085
group’s percentage of self-reported voters is 63.4%, or
, the fraction
74,282
47,085 over 74,282, which is less than that of the 55- to 74-year-old group.
Choice D is not the correct answer. The question is asking for the age group with
the largest percentage of self-reported voters. This answer reflects the age group
with the smallest number of self-reported voters, not the largest percentage. This
12,459
group’s percentage of self-reported voters is 70.0%, or
, the fraction 12,459
17,794
over 17,794, which is less than that of the 55- to 74-year-old group.
Link back to question 10.
Link back to question 11.
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Explanation for question 11. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must extrapolate from a random sample to estimate a
population parameter.
Difficulty: Medium
Key: D
Choice D is the correct answer. This question asks students to extrapolate from a
random sample to estimate the number of 18- to 34-year-olds who voted for
Candidate A: this is done by multiplying the fraction of people in the random
sample who voted for Candidate A by the total population of voting 18- to
287
34-year-olds:
 30,329,000  17 million, the fraction 287 over 500, multiplied
500
by 30,329,000, is almost equal to 17 million, choice D.
Students without a clear grasp of the context and its representation in the table
might easily arrive at one of the other answers listed.
Choice A is not the correct answer. The student may not have multiplied the
fraction of the sample by the correct subgroup of people (18- to 34-year-olds who
voted). This answer may result from multiplying the fraction by the entire
population, which is an incorrect application of the information.
Choice B is not the correct answer. The student may not have multiplied the
fraction of the sample by the correct subgroup of people (18- to 34-year-olds who
voted). This answer may result from multiplying the fraction by the total number
of people who voted, which is an incorrect application of the information.
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Choice C is not the correct answer. The student may not have multiplied the
fraction of the sample by the correct subgroup of people (18- to 34-year-olds who
voted). This answer may result from multiplying the fraction by the total number
of 18- to 34-year-olds, which is an incorrect application of the information.
Link back to question 11.
Link back to question 12.
Explanation for question 12. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must interpret an expression or equation that models a
real-world situation and be able to interpret the whole expression (or specific parts)
in terms of its context.
Difficulty: Medium
Key: C
Choice C is correct. One method to find the correct answer is to create an
inequality. The income from sales of n items is 12 n. For the company to profit,
12n must be greater than the cost of producing n items; therefore, the inequality
12n  7n  350 12 n is greater than 7 n plus 350 can be used to model the context.
Solving this inequality yields n  70. n is greater than 70.
Choice A is not the correct answer. This answer may result from a
misunderstanding of the properties of inequalities. The student may have found the
number of items of the break-even point as 70 and used the incorrect notation to
express the answer, or the student may have incorrectly modeled the scenario when
setting up an inequality to solve.
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Choice B is not the correct answer. This answer may result from a
misunderstanding of how the cost equation models the scenario. A student who
uses the cost of $12 as the number of items n and evaluates the expression 7n will
find the value of 84. A student who does not understand how the inequality relates
to the scenario may think n should be less than this value.
Choice D is not the correct answer. This answer may result from a
misunderstanding of how the cost equation models the scenario. A student who
uses the cost of $12 as the number of items n and evaluates the expression 7n will
find the value of 84. A student who does not understand how the inequality relates
to the scenario may think n should be greater than this value.
Link back to question 12.
Link back to question 13.
Explanation for question 13. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must evaluate the means for two separate populations in order
to determine the constraints on the mean for the combined population.
Difficulty: Medium
Key: D
Choice D is correct. The student must reason that because the mean of the males is
lower than that of the females, the combined mean cannot be greater than or equal
to that of the females, while also reasoning that because the mean of the females is
greater than that of the males, the combined mean cannot be less than or equal to
the mean of the males. Therefore the combined mean must be between the two
separate means.
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Choice A is not the correct answer. This answer results from a student finding the
mean of the two means. This answer makes an unjustified assumption that there
are an equal number of male and female primates.
Choice B is not the correct answer. This answer results from a student finding the
mean of the two means and misapplying an inequality to the scenario. This answer
makes an unjustified assumption that there are more females than males.
Choice C is not the correct answer. This answer results from a student finding the
mean of the two means and misapplying an inequality to the scenario. This answer
makes an unjustified assumption that there are more males than females.
Link back to question 13.
Link back to question 14.
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Explanation for question 14. (Follow link back to original question.)
Program: S A T
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must use information from a research study to evaluate
whether the results can be generalized to the study population and whether a
cause-and-effect relationship exists. To conclude a cause-and-effect relationship
like the ones described in choice C and choice D, there must be random assignment
of participants to groups receiving different treatments. To conclude that the
relationship applies to a population, participants must be randomly selected from
that population.
Difficulty: Medium
Key: A
Choice A is correct. A relationship in the data can only be generalized to the
population that the sample was drawn from.
Choice B is not the correct answer. A relationship in the data can only be
generalized to the population that the sample was drawn from. The sample was
from high school students in the United States, not from high school students in the
entire world.
Choice C is not the correct answer. Evidence for a cause-and-effect relationship
can only be established when participants are randomly assigned to groups that
receive different treatments.
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Choice D is not the correct answer. Evidence for a cause-and-effect relationship
can only be established when participants are randomly assigned to groups that
receive different treatments. Also, a relationship in the data can only be
generalized to the population that the sample was drawn from. The sample was
from high school students in the United States, not from high school students in the
entire world.
Link back to question 14.
Link back to question 15.
Explanation for question 15. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Passport to advanced math
Calculator usage: Yes
Objective: Students must identify the correct mathematical notation for an
exponential relationship that represents a real-world situation.
Difficulty: Medium
Key: D
Choice D is correct. The student first recognizes that a population that doubles in
size over equal time periods is increasing at an exponential rate. An exponential
x
b
growth model can be written in the form y  a  2  y equals a, parenthesis 2,
close parenthesis, raised to the power the fraction x over b, where a is the
population at time 0, since 2 raised to the zeroth power is 1 and b is the doubling
time, since when x5 12, x equals 12, 2 is raised to the first power, and the
population will be 21 3 a 5 2a. 2 raised to the power of one, times a, equals 2 a.
From the way the variables were defined, the population at time n5 0 n equals 0
is 50 and the doubling time is 12.
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Choice A is not the correct answer. This answer may result from a
misunderstanding of exponential equations or of the context. A student who tries to
model the scenario with a linear equation or who misunderstands that the
y-intercept of a linear model should represent the initial number of animals may
produce this equation.
Choice B is not the correct answer. This answer may result from a
misunderstanding of exponential equations or of the scenario. A student who tries
to model the scenario with a linear equation may produce this equation.
Choice C is not the correct answer. A student who tries to produce an exponential
model, but does not understand how the 12 years affects the model, may
incorrectly write the exponent.
Link back to question 15.
Link back to question 16.
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Explanation for question 16. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: No subscore (additional topics in math)
Calculator usage: Yes
Objective: Students must use spatial reasoning and geometric logic to deduce
which relationship is possible based on the given information. Students must also
use mathematical notation to express the relationship between the line segments.
Difficulty: Medium
Key: C
Choice C is correct. Given that n ABC triangle A B C is similar to n EDC,
triangle E D C, students can determine that the corresponding angle B A C is
congruent to angle CE D. The converse of the alternate interior angle theorem tells
us that AB DE. line segment A B is parallel to line segment D E. (Students can
also use the fact that  ABC angle A B C and CDE angle C D E are congruent to
make a similar argument.)
Choice A is not the correct answer. This answer may result from multiple
misconceptions. The student may have visually identified the segments as
perpendicular and used the wrong notation to express this statement.
Choice B is not the correct answer. This answer may result from visual inspection
of the diagram. The line segments appear to be perpendicular, but need not be,
given the information provided.
Choice D is not the correct answer. This answer may result from misunderstanding
either the notation or the vocabulary of parallel and perpendicular lines. The
student has incorrectly identified or notated parallel lines as perpendicular.
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Link back to question 16.
Link back to question 17.
Explanation for question 17. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must create a function to represent a real-world situation.
Difficulty: Medium
Key: B
Choice B is correct. Since Peter’s car is traveling at an average speed of 50 miles
per hour and the car’s gas mileage is 21 miles per gallon, the number of gallons of
50 miles 1 gallon 50
gas used each hour can be found by

 . the fraction 50 miles
1 hour 21 miles 21
over 1 hour, times the fraction 1 gallon over 21 miles, equals the fraction 50
50
over 21. The car uses
the fraction 50 over 21 gallons of gas per hour, so it uses
21
50
t the fraction 50 over 21, times t, gallons of gas in t hours. The car’s gas tank
21
has 17 gallons of gas at the beginning of the trip. Therefore, the function that
models the number of gallons of gas remaining in the tank t hours after the trip
50 t
. f of t equals 17 minus the fraction 50 t over 21.
begins is f  t   17 
21
Choice A is not the correct answer. The number of gallons of gas used each hour is
determined by dividing the average speed by the car’s gas mileage.
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Choice C is not the correct answer. The number of gallons of gas used each hour is
21
misrepresented as
. the fraction 21 over 50. Also, the number of gallons used
50
each hour must be multiplied by time t before it is subtracted from the number of
gallons of gas in the tank at the beginning of the trip.
Choice D is not the correct answer. The number of gallons of gas used each hour
must be multiplied by time t before it is subtracted from the number of gallons of
gas in the tank at the beginning of the trip
Link back to question 17.
Link back to question 18.
Explanation for question 18. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must create a system of linear equations to represent a
real-world situation.
Difficulty: Medium
Key: C
Choice C is correct. If x is the number of cars that crossed the bridge during the
two hours and y is the number of trucks that crossed the bridge during the two
hours, then x  y x plus y represents the total number of cars and trucks that
crossed the bridge during the two hours, and 6.5 x  10 y 6.5 x plus 10 y represents
the total amount collected in the two hours. Therefore, the correct system of
equations is x  y  187 x plus y equals 187 and 6.5 x  10 y  1,338. 6.5 x plus
10 y equals 1,338.
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Choice A is not the correct answer. The student may have mismatched the
symbolic expressions for total cars and trucks and total tolls collected with the two
numerical values given. The expression x  y x plus y represents the total number
of cars and trucks that crossed the bridge, which is 187.
Choice B is not the correct answer. The student may have attempted to use the
information that the counts of cars, trucks, and tolls were taken over a period of
two hours, but this information is not needed in setting up the correct system of
equations. The expression 6.5 x  10 y 6.5 x plus 10 y represents the total amount
of tolls collected, which is $1,338, not
$1,338
. the fraction 1,338 dollars over 2.
2
Choice D is not the correct answer. The student may have attempted to use the
information that the counts of cars, trucks, and tolls were taken over a period of
two hours, but this information is not needed in setting up the correct system of
equations. The expression 6.5 x  10 y 6.5 x plus 10 y represents the total amount of
tolls collected, which is $1,338, not $1,338  2. 1,338 dollars times 2.
Link back to question 18.
Link back to question 19.
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Explanation for question 19. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must construct a linear equation to represent a real-world
situation.
Difficulty: Medium
Key: B
Choice B is correct. To determine the linear model, one can first determine the rate
at which the pressure due to the atmosphere and surrounding water is increasing as
20.9  18.7 2.2
the depth of the diver increases. Calculating this gives

, the
14  9
5
fraction whose numerator is 20.9 minus 18.7, and whose denominator is
14 minus 9, equals, the fraction 2.2 over 5, or 0.44. Then one needs to determine
the pressure due to the atmosphere or, in other words, the pressure when the diver
is at a depth of 0. Solving the equation 18.7  0.44(9)  b 18.7 equals, 0.44,
parenthesis, 9, close parenthesis, plus b gives b  14.74. b equals 14.74.
Therefore, the model that can be used to relate the pressure and the depth is
p  0.44d  14.74. p equals 0.44 d plus 14.74.
Choice A is not the correct answer. The rate is calculated correctly, but the student
may have incorrectly used the ordered pair (18.7, 9) parenthesis, 18.7 comma 9,
close parenthesis, rather than (9,18.7) parenthesis, 9 comma 18.7, close
parenthesis, to calculate the pressure at a depth of 0 feet.
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Choice C is not the correct answer. The rate here is incorrectly calculated by
subtracting 20.9 and 18.7 and not dividing by 5. The student then uses the
coordinate pair d  9 d equals 9 and p  18.7 p equals 18.7 in conjunction with the
incorrect slope of 2.2 to write the equation of the linear model.
Choice D is not the correct answer. The rate here is incorrectly calculated by
subtracting 20.9 and 18.7 and not dividing by 5. The student then uses the
coordinate pair d  14 d equals 14 and p  20.9 p equals 20.9 in conjunction with
the incorrect slope of 2.2 to write the equation of the linear model.
Link back to question 19.
Link back to question 20.
Explanation for question 20. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must interpret the slope of the line of best fit for the
scatterplot as the average increase in the number of manatees per year, while
taking the scales of the axes into consideration.
Difficulty: Medium
Key: C
Choice C is correct. The slope of the line of best fit is the value of the average
increase in manatees per year. Using approximate values found along the line of
best fit (1,200 manatees in 1991 and 4,200 manatees in 2011), the approximate
slope can be calculated as
3,000
5 150. the fraction 3,000 over 20 equals 150.
20
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Choice A is not the correct answer. This value may result from disregarding the
actual scale when approximating the slope and interpreting the scale as if each
square represents one unit.
Choice B is not the correct answer. This value may result from disregarding the
actual scale when approximating the slope, and interpreting the scale as if each
square along the x-axis represents one year and each tick mark along the y-axis
represents 100 manatees.
Choice D is not the correct answer. This value may result from disregarding the
actual scale along the x-axis when approximating the slope and interpreting each
square along the x-axis as one year.
Link back to question 20.
Link back to question 21.
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Explanation for question 21. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must synthesize graphical and textual information and
determine what information is relevant to the solution.
Difficulty: Medium
Key: B
Choice B is the correct answer. Each petri dish has area 10 square centimeters, and
 1 
so at time t 5 0, t equals zero, Dish 1 is 10% covered   parenthesis, one-tenth,
 10 
 2
close parenthesis, and Dish 2 is 20% covered   . parenthesis, two-tenths, close
 10 
parenthesis. Thus the statement in choice B is true.
Choice A is not the correct answer. At the end of the observations, both dishes are
100% covered with bacteria, but at time t 5 0, t equals zero, neither dish is
100% covered.
Choice C is not the correct answer. At time t 5 0, t equals zero, Dish 1 is covered
with 50% less bacteria than is Dish 2, but Dish 2 is covered with 100% more, not
50% more, bacteria than is Dish 1.
Choice D is not the correct answer. After the first hour, it is still true that more of
Dish 2 is covered by bacteria than is Dish 1, but for the first hour the area of Dish 1
that is covered has been increasing at a higher average rate (about
0.8 square centimeter per hour) than the area of Dish 2 (about
0.1 square centimeter per hour).
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Link back to question 21.
Link back to question 22.
Explanation for question 22. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must use the unit rate (data-transmission rate) and the
conversion between gigabits and megabits as well as conversions in units of time.
Difficulty: Medium
Key: B
Choice B is correct. The tracking station can receive 118,800 megabits each day
 3 megabits 60 seconds 60 minutes


´
´ 11 hours  , parenthesis, the fraction

1 minute
1 hour
 1second

3 megabits over 1 second, times, the fraction 60 seconds over 1 minute, times, the
fraction 60 minutes over 1 hour, times, 11 hours, close parenthesis, which is about
 118,800 
116 gigabits each day 
 . parenthesis, the fraction 118,800 over 1,024,
 1,024 
close parenthesis. If each image is 11.2 gigabits, then the number of images that
116
can be received each day is
 10.4. the fraction 116 over 11.2 is almost equal
11.2
to 10.4.
Since the question asks for the maximum number of typical images, rounding the
answer down to 10 is appropriate because the tracking station will not receive a
completed eleventh image in one day.
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Choice A is not the correct answer. The student may not have synthesized all of the
information. This answer may result from multiplying 3 (rate in
megabits per second) by 11 (hours receiving) and dividing by 11.2 (size of image
in gigabits), neglecting to convert 3 megabits per second into megabits per hour
and to utilize the information about 1 gigabit equaling 1,024 megabits.
Choice C is not the correct answer. The student may not have synthesized all of the
information. This answer may result from converting the number of gigabits in an
image to megabits (11,470), multiplying by the rate of 3 megabits per second
(34,410), and then converting 11 hours into minutes (660) instead of seconds.
Choice D is not the correct answer. The student may not have synthesized all of the
information. This answer may result from converting 11 hours into seconds
(39,600), then dividing the result by 3 gigabits converted into megabits (3,072),
and multiplying by the size of one typical image.
Link back to question 22.
Link back to question 23.
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Explanation for question 23. (Follow link back to original question.)
Program: S A T
Subscore: Passport to advanced math
Calculator usage: Yes
Objective: Students must manipulate one equation for use in another, in order to
solve for a given value, making use of familiar algebraic arrangements where
appropriate.
Difficulty: Medium
Key: C
Choice C is correct. The second equation gives y in terms of x, so a student can
use this to rewrite the first equation in terms of x. Substituting 2 4x negative 4 x
for y in the equation x 2  y 2  153 x squared plus y squared equals 153 gives
x 2  (4 x)2  153. x squared plus, parenthesis, negative 4 x, close parenthesis,
squared, equals 153. This can be simplified to x2  16 x2  153, x squared plus
16 x squared equals 153, or 17 x 2  153. 17 x squared equals 153. Since the
question asks for the value of x 2 , x squared, not x, dividing both sides of
17 x 2  153 17 x squared equals 153 by 17 gives the answer: x 2 
153
 9.
17
x squared equals the fraction 153 over 17, equals 9.
Choice A is not the correct answer. This answer may result from neglecting to
square the coefficient 2 4 negative 4 in y 5 2 4 x, y equals negative 4 x, which
would give y 2  4 x 2 . y squared equals negative 4 x squared. Then the first
equation would become x 2  4 x 2  3x 2  153, x squared minus 4 x squared,
equals negative 3 x squared, equals 153, which would give 2 51 negative 51 as the
value of x 2 . x squared.
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Choice B is not the correct answer. This answer may result from finding the value
for x, not the value of x 2 . x squared.
Choice D is not the correct answer. This answer may result from finding the value
of y 2 , y squared, not x 2 . x squared.
Link back to question 23.
Link back to question 24.
Explanation for question 24. (Follow link back to original question.)
Program: S A T
Subscore: No subscore (additional topics in math)
Calculator usage: Yes
Objective: Students must make connections between physical concepts such as
mass and density and essential geometric ideas such as the Pythagorean theorem
and volume formulas.
Difficulty: Medium
Key: 57
This question asks students to make connections between physical concepts such
as mass and density and essential geometric ideas such as the Pythagorean theorem
and volume formulas. There are multiple approaches to solving this problem, but
in any of them, the aim is to find the volume of the metal nut and then use the
density of the metal to calculate the mass of the nut (57 grams). This is a multistep
problem that requires students to devise a multistep strategy and carry out all the
algebraic and numerical steps without error.
To solve this problem, students need to find the volume of the hex nut and then use
the given fact that density is mass divided by volume.
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Finding the volume of the hex nut requires several steps. The first step is to
calculate the area of one of the hexagonal faces (without the drilled hole). Each
face is a regular hexagon, which can be divided into 6 equilateral triangles with
side lengths of 2 centimeters. Using 30-60-90 triangle properties, the height of
3 the square root of three centimeters. In turn, the
1
1
area of one equilateral triangle is bh5  2  3 5 3 one-half b h equals
2
2
one-half, parenthesis, 2, close parenthesis, parenthesis, the square root of 3, close
parenthesis, equals the square root of 3 square centimeters, so the area of the
each equilateral triangle is
 
hexagonal face is 6 3 6 times the square root of 3, square centimeters. The
volume of the hexagonal prism is the area of one face multiplied by the height (or


thickness in this case), 6 3 1 5 6 3 parenthesis, 6 times the square root of 3,
close parenthesis, parenthesis, 1, close parenthesis, equals 6 times the square root
of 3, cubic centimeters. Then, to account for the drilled hole, students need to
calculate the volume of a cylinder with diameter 2 centimeters (or radius
 
1 centimeter) and height one centimeter, V 5  r 2h 5  12 1 5  capital V
equals pi r squared, h, equals, pi, parenthesis, 1 squared, close parenthesis,
parenthesis, 1, close parenthesis, equals pi, cubic centimeter, and subtract it from
the volume of the hexagonal prism to yield 6 3 2  6 times the square root of 3,
minus pi, cubic centimeter.
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mass
. 7.9, equals the fraction
6 32 
whose numerator is mass and whose denominator is 6 times the square root of 3,
Finally, density is mass divided by volume, 7.9 5


minus pi. Multiplying both sides of the equation by 6 3 2  parenthesis, 6 times
the square root of 3, minus pi, close parenthesis, yields the mass of the hex nut as


7.9 6 3 2  7.9, parenthesis, 6 times the square root of 3, minus pi, close
parenthesis, grams. When the values for 3 the square root of 3 and  pi are
substituted and the result is rounded to the nearest gram, the answer is
approximately 57 grams. Note that it is critical for students to attend to the
precision of their calculations when solving this problem, and not apply any
intermediate rounding until the final answer is reached. Here, the use of a
calculator provides the ability to attend to precision more effectively, and thus is
highly encouraged.
Link back to question 24.
Link back to question 25.
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Explanation for question 25. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must use a linear equation to determine the conversion rate
between two currencies.
Difficulty: Medium
Key: 63
$9.88 represents the conversion of 602 rupees plus a 4% fee on the converted
cost. To calculate the original cost of the item in dollars, x:
1.04 x  9.88 1.04 x equals, 9.88.
x  9.5 x equals 9.5.
Since the original cost is $9.50, to calculate the exchange rate r, in Indian rupees
per one U. S. dollar:
9.50 dollars 
r rupees
 602 rupees 9 dollars and 50 cents, times the fraction
1 dollar
r rupees over one dollar, equals 602 rupees.
r
602
r equals, the fraction 602 over 9.50.
9.50
 63 rupees is almost equal to 63 rupees.
Link back to question 25.
Link back to question 26.
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Explanation for question 26. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Problem solving and data analysis
Calculator usage: Yes
Objective: Students must use the currency conversion information and the
constraints provided to determine a meaningful value in the context of the problem.
Difficulty: Hard
Key: 7,212
Let d dollars be the cost of the 7,500-rupee prepaid card. This implies that the
d
exchange rate on this particular day is
the fraction d over 7,500 dollars per
7,500
rupee. Suppose Sara’s total purchases on the prepaid card were r rupees. The
 d 
value of the r rupees in dollars is 
 r parenthesis, the fraction d over
 7,500 
7,500, close parenthesis, r, dollars. If Sara spent the r rupees on the Traveler card
 d 
instead, she would be charged 1.04  
 r parenthesis, 1.04, close
7,500


parenthesis, parenthesis, the fraction d over 7,500, close parenthesis, r, dollars. To
answer the question about how many rupees Sara must spend in order to make the
Traveler card a cheaper option (in dollars) for spending the r rupees, we set up the
 d 
inequality 1.04 
 r  d . 1.04, parenthesis, the fraction d over 7,500, close
 7,500 
parenthesis, r is greater than or equal to d. Rewriting both sides reveals
 r 
1.04 
 d  1 d , 1.04, parenthesis, the fraction r over 7,500, close
 7,500 
parenthesis, d is greater than or equal to, parenthesis, 1, close parenthesis, d, from
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 r 
which we can infer 1.04 
  1. 1.04, parenthesis, the fraction r over 7,500,
 7,500 
close parenthesis, is greater than or equal to 1.
Dividing on both sides by 1.04 and multiplying on both sides by 7,500 finally
yields r  7,212. r is greater than or equal to 7,212. Hence the least number of
rupees Sara must spend for the prepaid card to be cheaper than the Traveler card
is 7,212.
Link back to question 26.
Link back to question 27.
Explanation for question 27. (Follow link back to original question.)
Program: S A T, P S A T/N M S Q T, P S A T 10
Subscore: Heart of algebra
Calculator usage: Yes
Objective: Students must understand the relationship between an equation in two
variables and the characteristics of its graph (for example, shape, position,
intercepts, extreme points, or symmetry). In addition, students must transform the
given equation into a more suitable form and then make the connection between
the obtained equation and the graph.
Difficulty: Hard
Key: B
1 k 
Choice B is correct. Manipulating the equation to solve for y gives y  
 x,
1 k 
y equals, parenthesis, the fraction whose numerator is 1 plus k and whose
denominator is 1 minus k, close parenthesis, x, revealing that the graph of the
equation must be a line that passes through the origin. Of the choices given, only
the graph shown in choice B satisfies these conditions.
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Choice A is not the correct answer. The student may have seen that the term
k ( x  y) k, parenthesis, x plus y, close parenthesis, is a multiple of x  y x plus y
and wrongly concluded that this is the equation of a line with slope 1.
Choice C is not the correct answer. The student may have made incorrect steps
when simplifying the equation or may have not seen the advantage that putting the
equation in slope-intercept form would give in determining the graph, and thus
wrongly concluded the graph has a nonzero y-intercept.
Choice D is not the correct answer. The student may not have seen that term
k ( x  y) k, parenthesis, x plus y, close parenthesis, can be multiplied out and the
variables x and y isolated, and wrongly concluded that the graph of the equation
cannot be a line.
Link back to question 27.
Link back to question 28.
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Explanation for question 28. (Follow link back to original question.)
Program: S A T
Subscore: Passport to advanced math
Calculator usage: Yes
Objective: Students must understand the zeros of a polynomial function and how
they are used to construct algebraic representations of polynomials.
Difficulty: Hard
Key: A
Choice A is correct. The given zeros can be used to set up an equation to solve
for c. Substituting 2 4 negative 4 for x and 0 for y yields 4c  72, negative 4 c
equals 72, or c  18. c equals negative 18. Alternatively, since 2 4, negative 4,
1
, one-half, and p are zeros of the polynomial function
2
f ( x)  2 x3  3 x 2  cx  8, f of x equals 2 x cubed, plus 3 x squared, plus c x,
plus 8, it follows that f ( x)  (2 x  1)( x  4)( x  p). f of x equals, parenthesis,
2 x minus 1, close parenthesis, parenthesis, x plus 4, close parenthesis, parenthesis
x minus p, close parenthesis. Were this polynomial multiplied out, the constant
term would be (1)(4)( p)  4 p. parenthesis, negative 1, close parenthesis,
parenthesis, 4, close parenthesis, parenthesis negative p, close parenthesis, equals
4 p. (We can see this without performing the full expansion.) Since it is given that
this value is 8, it follows that 4 p  8 4 p equals 8 or rather, p  2. p equals 2.
Substituting 2 for p in the polynomial function yields
f ( x)  (2 x  1)( x  4)( x  2), f of x equals, parenthesis, 2 x minus 1, close
parenthesis, parenthesis, x plus 4, close parenthesis, parenthesis, x minus 2, close
parenthesis, and after multiplying the factors one finds that the coefficient of the x
term, or the value of c, is 2 18. negative 18.
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Choice B is not the correct answer. This value is a misunderstood version of the
value of p, not c, and the relationship between the zero and the factor (if a is the
zero of a polynomial, its corresponding factor is x  a) x minus a) has been
confused.
Choice C is not the correct answer. This is the value of p, not c. Using this value
as the third factor of the polynomial will reveal that the value of c is 2 18.
negative 18.
Choice D is not the correct answer. This represents a sign error in the final step in
determining the value of c.
Link back to question 28.
Link back to question 29.
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Explanation for question 29. (Follow link back to original question.)
Program: S A T
Subscore: Passport to advanced math
Calculator usage: Yes
Objective: Students must transform a given expression into a more useful form
(from improper to proper rational form).
Difficulty: Hard
Key: A
Choice A is correct. The form of the equation suggests performing long division on
4 x2
: the fraction whose numerator is 4 x squared, and whose denominator is
2 x2 1
2 x minus 1:
2x 1 1
2 x 2 1 4 x2
4 x2 2 2 x
2x
2x 2 1
1
Divide 2 x minus 1 into 4 x squared, place 2 x plus 1 on top of the long division
symbol. Multiply 2 x plus 1 by 2 x, yields 4 x squared minus 2 x.
Then divide 2 x minus 1 into 2 x, place positive 1 on top of the long division
symbol preceded by 2 x. Multiply 2 x minus 1 by positive 1, yields 2 x minus 1.
Subtract 2 x minus 1 from 2 x, yields the remainder 1.
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1
, the fraction whose
2 x2 1
numerator is 1 and whose denominator is 2 x minus 1, it is clear that A  2 x 1 1.
A equals 2 x plus 1.
Since the remainder 1 matches the numerator in
A short way to find the answer is to use the structure to rewrite the numerator of


the expression as 4 x 2 2 1 1 1, parenthesis, 4 x squared minus 1, close parenthesis,
plus 1, recognizing the term in parentheses as a difference of squares, making the
(2 x 2 1)(2 x 1 1) 1 1
1
expression equal to
 2 x 1 11
. the fraction whose
2x 2 1
2x 2 1
numerator is, parenthesis, 2 x minus 1, close parenthesis, parenthesis, 2 x plus 1,
close parenthesis, plus 1, and whose denominator is 2 x minus 1, equals, 2 x plus 1,
plus the fraction whose numerator is 1 and whose denominator is 2 x minus 1.
From this, the answer 2 x  1 2 x plus 1 is apparent. Another way to find the answer
4 x2
1
is to isolate A in the form A 
A equals the fraction whose
2
2x 2 1 2x 2 1
numerator is 4 x squared, and whose denominator is 2 x minus 1, minus, the
fraction whose numerator is 1, and whose denominator is 2 x minus 1, and
simplify. As with the first approach, this approach also requires students to
recognize 4 x 2 2 1 4 x squared minus 1 as a difference of squares that factors.
Choice B is not the correct answer. The student may have made a sign error while
subtracting partial quotients in the long division.
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Choice C is not the correct answer. The student may misunderstand how to work
with fractions and may have tried the incorrect calculation
 
1 4 x2
4 x2  
1


 4 x 2. the fraction whose numerator is 4 x squared, and
2x  1
2x  1
2x 1
whose denominator is 2 x minus 1, equals, the fraction whose numerator is,
parenthesis, 1, close parenthesis, parenthesis, 4 x squared, close parenthesis, and
whose denominator is 2 x minus 1, equals, the fraction whose numerator is 1, and
whose denominator is 2 x minus 1, plus 4 x squared.
Choice D is not the correct answer. The student may misunderstand how to work
with fractions and may have tried the incorrect calculation
4 x2 1  4 x2  1
1


 4 x 2  1. the fraction whose numerator is 4 x squared,
2x  1
2x 1
2x 1
and whose denominator is 2 x minus 1, equals, the fraction whose numerator is 1
plus 4 x squared minus ,, and whose denominator is 2 x minus 1, equals, the
fraction whose numerator is 1, and whose denominator is 2 x minus 1, plus,
4 x squared, minus 1.
Link back to question 29.
Link back to question 30.
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Explanation for question 30. (Follow link back to original question.)
Program: S A T
Subscore: No subscore (additional topics in math)
Calculator usage: Yes
Objective: Students must make use of properties of triangles to solve a problem.
Difficulty: Hard
2
Key: two-thirds
3
Because the triangle is isosceles, constructing a perpendicular from the top vertex
to the opposite side will bisect the base and create two smaller right triangles. In a
right triangle, the cosine of an acute angle is equal to the length of the side adjacent
16
to the angle divided by the length of the hypotenuse. This gives cos x 5
,
24
2
cosine x equals the fraction 16 over 24, which can be simplified to cos x 5 .
3
cosine x equals the fraction 2 over 3.
Link back to question 30.
This is the end of Mathematics Sample Question Answers and Explanations.
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