The SAT Suite of Assessments | College Board
The Redesigned S A T®
Mathematics Sample Sets
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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
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Mathematical Equations and Expressions
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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 [pic] f of x is a real number.
[pic]
Begin skippable figure descriptions.
The figure presents information for your reference in solving some of the problems.
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.
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.
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.
5. Mixed numbers such as [pic] three and one-half must be recorded as [pic] three point five or [pic] seven slash two. (If [pic] three and one-half is entered into the grid as [pic], three, one, slash, two, it will be interpreted as [pic] thirty-one halves, not [pic] three and one-half).
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.
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
[pic]
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.
Example 3
[pic]
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.
Example 4
[pic]
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.
Mathematics Sample Questions
Question 1.
The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (m g). 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. [pic] 299 m plus 261 j is greater than or equal to 1,000
B. [pic] 299 m plus 261 j is greater than 1,000
C. [pic] the fraction 299 over m, plus the fraction 261 over j, is greater than or equal to 1,000
D. [pic] the fraction 299 over m, plus the fraction 261 over j, is greater than 1,000
Answer and Explanation. (Follow link to explanation of question 1.)
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.)
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.
[pic]
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.
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.
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.)
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.)
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.)
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. [pic] parenthesis, 99.95 plus 0.08 x, close parenthesis, plus 5
B. [pic] 1.08, parenthesis, 99.95 x, close parenthesis, plus 5
C. [pic] 1.08, parenthesis, 99.95 x plus 5, close parenthesis
D. [pic] 1.08, parenthesis, 99.95 plus 5, close parenthesis, x
Answer and Explanation. (Follow link to explanation of question 6.)
Question 7 refers to the following figure and system of three equations.
[pic]
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.
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.
[pic] x squared plus y squared equals five.
[pic] y equals x squared minus three.
[pic] 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.)
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.
If [pic] the fraction negative 9 over 5 is less than negative 3 t plus 1, and negative 3 t plus 1 is less than the negative of the fraction 7 over 4, what is one possible value of [pic] 9 t minus 3?
Answer and Explanation. (Follow link to explanation of question 9.)
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 |12,459 |3,508 |1,827 |17,794 |
|Total |132,948 |54,532 |27,602 |215,082 |
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
Answer and Explanation. (Follow link to explanation of question 10.)
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.)
Question 12.
A company’s manager estimated that the cost C, in dollars, of producing n items is [pic] 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. [pic] n is less than 70.
B. [pic] n is less than 84.
C. [pic] n is greater than 70.
D. [pic] 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. [pic] m equals 17.
B. [pic] m is greater than 17.
C. [pic] m is less than 17.
D. [pic] 15 is less than m, and m is less than 19.
Answer and Explanation. (Follow link to explanation of question 13.)
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.)
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. [pic] P equals 12 plus 50 n.
B. [pic] P equals 50 plus 12 n.
C. [pic] P equals 50, parenthesis, 2, close parenthesis, to the power of 12 n.
D. [pic] P equals 50, parenthesis, 2, close parenthesis, to the power of the fraction n over 12.
Answer and Explanation. (Follow link to explanation of question 15.)
Question 16 is based on the following figure.
[pic]
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.
Question 16.
In the preceding figure (follow link), [pic]triangle A B C is similar to triangle E D C. Which of the following must be true?
A. [pic] Line segment A E is parallel to line segment B D.
B. [pic] Line segment A E is perpendicular to line segment B D.
C. [pic] Line segment A B is parallel to line segment D E.
D. [pic] Line segment A B is perpendicular to line segment D E.
Answer and Explanation. (Follow link to explanation of question 16.)
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. [pic] f of t equals 17 minus the fraction 21 over 50 t.
B. [pic] f of t equals 17 minus the fraction 50 t over 21.
C. [pic] f of t equals the fraction whose numerator is 17 minus 21 t, and whose denominator is 50.
D. [pic] f of t equals the fraction whose numerator is 17 minus 50 t, and whose denominator is 21.
Answer and Explanation. (Follow link to explanation of question 17.)
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. [pic] x plus y equals 1,338.
[pic] 6.5 x plus 10 y equals 187.
B. [pic] x plus y equals 187.
[pic] 6.5 x, plus 10 y, equals the fraction 1,338 over 2.
C. [pic] x plus y equals 187.
[pic] 6.5 x, plus 10 y equals 1,338.
D. [pic] x plus y equals 187.
[pic]6.5 x, plus 10 y equals 1,338 times 2.
Answer and Explanation. (Follow link to explanation of question 18.)
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. [pic] p equals 0.44 d plus 0.77.
B. [pic] p equals 0.44 d plus 14.74.
C. [pic] p equals 2.2 d minus 1.1.
D. [pic] p equals 2.2 d minus 9.9.
Answer and Explanation. (Follow link to explanation of question 19.)
Question 20 is based on the following figure.
[pic]
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.
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.)
Question 21 is based on the following figure.
[pic]
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 [pic] 2 comma 4, and the point with coordinates [pic] 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 [pic] 3 comma 3, and the point with coordinates [pic] 5 comma 6. The two curved lines intersect at a point with approximate coordinates [pic] 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 [pic] 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 [pic] t equals zero, both dishes are 100% covered by bacteria.
B. At time [pic] t equals zero, bacteria covers 10% of Dish 1 and 20% of Dish 2.
C. At time [pic] 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.)
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.
[pic] x squared plus y squared equals 153.
[pic] y equals negative 4 x.
If [pic] parenthesis, x comma y, close parenthesis, is a solution to the preceding system of equations, what is the value of [pic] x squared?
A. [pic] negative 51
B. 3
C. 9
D. 144
Answer and Explanation. (Follow link to explanation of question 23.)
Question 24 is based on the following figure.
[pic]
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.)
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.)
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 [pic] 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 [pic] negative 6 through 6 appear along both axes, and the origin is labeled O.
A. [pic]
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. [pic]
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.
C. [pic]
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. [pic]
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.)
Question 28.
The function f is defined by [pic] 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 [pic] parenthesis, negative 4 comma 0, close parenthesis, [pic] parenthesis, one-half comma 0, close parenthesis, and [pic] parenthesis, p comma 0, close parenthesis. What is the value of c?
A. [pic] negative 18
B. [pic] negative 2
C. 2
D. 10
Answer and Explanation. (Follow link to explanation of question 28.)
Question 29.
If the expression [pic] the fraction whose numerator is 4 x squared, and whose denominator is 2 x minus one, is written in the equivalent form [pic]the fraction whose numerator is one, and whose denominator is 2 x minus one, plus A, what is A in terms of x?
A. [pic] 2 x plus one.
B. [pic] 2 x minus one.
C. [pic] 4 x squared.
D. [pic] 4 x squared minus one.
Answer and Explanation. (Follow link to explanation of question 29.)
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.
[pic]
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 [pic]cosine x?
Answer and Explanation. (Follow link to explanation of question 30.)
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.
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 when calculating the total amount of calcium intake from each drink.
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.
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.
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.
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.
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 [pic] parenthesis, 99.95 x plus 0.08, times 99.95 x, close parenthesis, plus 5, which can be rewritten as [pic] 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.
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 [pic] parenthesis, negative one comma two, close parenthesis, and [pic] 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.
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
Key: [pic] the fraction 7 over 92
There are 7 metalloids that are solid or liquid, and there are 92 total solids and liquids. Therefore, the fraction of solids and liquids that are metalloids is [pic] the fraction 7 over 92.
Link back to question 8.
Link back to question 9.
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
Key: Any value greater than [pic] the fraction 21 over 4 and less than [pic] the 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 [pic] negative 3 t plus 1 and [pic] 9 t minus 3 and recognize that the latter is [pic] negative 3 multiplied by the former.
Multiplying all parts of the inequality by [pic] negative 3 reverses the inequality signs, resulting in [pic] the fraction 27 over 5 is greater than 9 t minus 3, and 9 t minus 3 is greater than the fraction 21 over 4, or rather [pic] the fraction 21 over 4 is less than 9 t minus 3, and 9 t minus 3 is less than the fraction 27 over 5 when written with increasing values from left to right. Any value greater than [pic] the fraction 21 over 4 and less than [pic] the fraction 27 over 5 is correct.
Link back to question 9.
Link back to question 10.
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 self-reported voters is 48.1%, or [pic] the fraction 30,329 over 63,008, which is less than that of the 55- to 74-year-old group.
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 group’s percentage of self-reported voters is 63.4%, or [pic] the fraction 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 group’s percentage of self-reported voters is 70.0%, or [pic]the fraction 12,459 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.
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 34-year-olds: [pic] the fraction 287 over 500, multiplied 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.
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, 12 n must be greater than the cost of producing n items; therefore, the inequality [pic] 12 n is greater than 7 n plus 350 can be used to model the context. Solving this inequality yields [pic] 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.
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 7 n 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 7 n 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.
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.
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.
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 growth model can be written in the form [pic] 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 [pic] x equals 12, 2 is raised to the first power, and the population will be [pic] 2 raised to the power of one, times a, equals 2 a. From the way the variables were defined, the population at time [pic] n equals 0 is 50 and the doubling time is 12.
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.
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 [pic] triangle A B C is similar to [pic] triangle E D C, students can determine that the corresponding angle B A C is congruent to angle C E D. The converse of the alternate interior angle theorem tells us that [pic] line segment A B is parallel to line segment D E. (Students can also use the fact that [pic] angle A B C and [pic] 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.
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 gas used each hour can be found by [pic] the fraction 50 miles over 1 hour, times the fraction 1 gallon over 21 miles, equals the fraction 50 over 21. The car uses [pic] the fraction 50 over 21 gallons of gas per hour, so it uses [pic] the fraction 50 over 21, times t, gallons of gas in t hours. The car’s gas tank 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 begins is [pic] f of t equals 17 minus the fraction 50 t over 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.
Choice C is not the correct answer. The number of gallons of gas used each hour is misrepresented as [pic] the fraction 21 over 50. Also, the number of gallons 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.
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 [pic] x plus y represents the total number of cars and trucks that crossed the bridge during the two hours, and [pic] 6.5 x plus 10 y represents the total amount collected in the two hours. Therefore, the correct system of equations is [pic] x plus y equals 187 and [pic] 6.5 x plus 10 y equals 1,338.
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 [pic] 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 [pic] 6.5 x plus 10 y represents the total amount of tolls collected, which is $1,338, not [pic] the fraction 1,338 dollars over 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 [pic] 6.5 x plus 10 y represents the total amount of tolls collected, which is $1,338, not [pic] 1,338 dollars times 2.
Link back to question 18.
Link back to question 19.
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 the depth of the diver increases. Calculating this gives [pic] the 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 [pic] 18.7 equals, 0.44, parenthesis, 9, close parenthesis, plus b gives [pic] b equals 14.74. Therefore, the model that can be used to relate the pressure and the depth is [pic] 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 [pic] parenthesis, 18.7 comma 9, close parenthesis, rather than [pic] parenthesis, 9 comma 18.7, close parenthesis, to calculate the pressure at a depth of 0 feet.
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 [pic] d equals 9 and [pic] 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 [pic] d equals 14 and [pic] 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 [pic] the fraction 3,000 over 20 equals 150.
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.
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 so at time [pic] t equals zero, Dish 1 is 10% covered [pic] parenthesis, one-tenth, close parenthesis, and Dish 2 is 20% covered [pic] parenthesis, two-tenths, close 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 [pic] t equals zero, neither dish is 100% covered.
Choice C is not the correct answer. At time [pic] 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).
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 [pic] parenthesis, the fraction 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 116 gigabits each day [pic] parenthesis, the fraction 118,800 over 1,024, close parenthesis. If each image is 11.2 gigabits, then the number of images that can be received each day is [pic] the fraction 116 over 11.2 is almost equal 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.
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.
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 [pic] negative 4 x for y in the equation [pic] x squared plus y squared equals 153 gives [pic] x squared plus, parenthesis, negative 4 x, close parenthesis, squared, equals 153. This can be simplified to [pic] x squared plus 16 x squared equals 153, or [pic] 17 x squared equals 153. Since the question asks for the value of [pic] x squared, not x, dividing both sides of [pic] 17 x squared equals 153 by 17 gives the answer: [pic] 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 [pic] negative 4 in [pic] y equals negative 4 x, which would give [pic] y squared equals negative 4 x squared. Then the first equation would become [pic] x squared minus 4 x squared, equals negative 3 x squared, equals 153, which would give [pic] negative 51 as the value of [pic] x squared.
Choice B is not the correct answer. This answer may result from finding the value for x, not the value of [pic] x squared.
Choice D is not the correct answer. This answer may result from finding the value of [pic] y squared, not [pic] 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.
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 each equilateral triangle is [pic] the square root of three centimeters. In turn, the area of one equilateral triangle is [pic] one-half b h equals 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 hexagonal face is [pic] 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), [pic] 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, [pic] 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 [pic] 6 times the square root of 3, minus pi, cubic centimeter.
Finally, density is mass divided by volume, [pic] 7.9, equals the fraction whose numerator is mass and whose denominator is 6 times the square root of 3, minus pi. Multiplying both sides of the equation by [pic] parenthesis, 6 times the square root of 3, minus pi, close parenthesis, yields the mass of the hex nut as [pic] 7.9, parenthesis, 6 times the square root of 3, minus pi, close parenthesis, grams. When the values for [pic] the square root of 3 and [pic] 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.
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:
[pic] 1.04 x equals, 9.88.
[pic] 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:
[pic] 9 dollars and 50 cents, times the fraction r rupees over one dollar, equals 602 rupees.
[pic] r equals, the fraction 602 over 9.50.
[pic] is almost equal to 63 rupees.
Link back to question 25.
Link back to question 26.
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 exchange rate on this particular day is [pic] the fraction d over 7,500 dollars per rupee. Suppose Sara’s total purchases on the prepaid card were r rupees. The value of the r rupees in dollars is [pic] parenthesis, the fraction d over 7,500, close parenthesis, r, dollars. If Sara spent the r rupees on the Traveler card instead, she would be charged [pic] parenthesis, 1.04, close 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 inequality [pic] 1.04, parenthesis, the fraction d over 7,500, close parenthesis, r is greater than or equal to d. Rewriting both sides reveals [pic] 1.04, parenthesis, the fraction r over 7,500, close parenthesis, d is greater than or equal to, parenthesis, 1, close parenthesis, d, from which we can infer [pic] 1.04, parenthesis, the fraction r over 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 [pic] 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
Choice B is correct. Manipulating the equation to solve for y gives [pic] 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.
Choice A is not the correct answer. The student may have seen that the term [pic] k, parenthesis, x plus y, close parenthesis, is a multiple of [pic] 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 [pic] 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.
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 [pic] negative 4 for x and 0 for y yields [pic] negative 4 c equals 72, or [pic] c equals negative 18. Alternatively, since [pic] negative 4, [pic] one-half, and p are zeros of the polynomial function [pic] f of x equals 2 x cubed, plus 3 x squared, plus c x, plus 8, it follows that [pic] 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 [pic] 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 [pic] 4 p equals 8 or rather, [pic] p equals 2. Substituting 2 for p in the polynomial function yields [pic] 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 [pic] negative 18.
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 [pic] 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 [pic] 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.
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 [pic] the fraction whose numerator is 4 x squared, and whose denominator is 2 x minus 1:
[pic]
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.
Since the remainder 1 matches the numerator in [pic] the fraction whose numerator is 1 and whose denominator is 2 x minus 1, it is clear that [pic] A equals 2 x plus 1.
A short way to find the answer is to use the structure to rewrite the numerator of the expression as [pic] parenthesis, 4 x squared minus 1, close parenthesis, plus 1, recognizing the term in parentheses as a difference of squares, making the expression equal to [pic] the fraction whose 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 [pic] 2 x plus 1 is apparent. Another way to find the answer is to isolate A in the form [pic] A equals the fraction whose 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 [pic] 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.
Choice C is not the correct answer. The student may misunderstand how to work with fractions and may have tried the incorrect calculation [pic]the fraction whose numerator is 4 x squared, and 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 [pic] the fraction whose numerator is 4 x squared, 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.
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
Key: [pic] two-thirds
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 to the angle divided by the length of the hypotenuse. This gives [pic] cosine x equals the fraction 16 over 24, which can be simplified to [pic] cosine x equals the fraction 2 over 3.
Link back to question 30.
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