PSAT/NMSQT Math Practice Tests

Question 1

1. A garden hose fills an empty water tank at a constant rate of 6 gallons per minute. At this rate, how many minutes will it take to fill a 150-gallon tank? (A) 25 (B) 144 (C) 156 (D) 900

Question 2

1. 4x - 7 = 21 What value of x is the solution to the given equation? (A) (7)/(2) (B) (21)/(4) (C) 7 (D) 28

Question 3

Line graph of temperature in degrees Celsius versus time. The points are: 6 a.m., 12; 8 a.m., 14; 10 a.m., 18; 12 p.m., 22; 2 p.m., 26; 4 p.m., 24; 6 p.m., 20. The vertical axis runs from 10 to 28 in steps of 2.

1. The line graph shows the air temperature, in degrees Celsius (°C), recorded at a weather station every 2 hours from 6 a.m. to 6 p.m. on one day. According to the graph, by how many degrees Celsius did the temperature increase from 8 a.m. to 2 p.m.? (A) 8 (B) 12 (C) 14 (D) 26

Question 4

A line on a grid. The horizontal axis is time in hours from 0 to 6; the vertical axis is total cost in dollars from 0 to 56 in steps of 4. The line starts at the labeled point (0, 12) on the vertical axis and passes through the labeled point (4, 44), rising 8 dollars for each hour.

1. The graph shows the total cost C, in dollars, of renting a kayak for h hours at a lake. The cost includes a charge for each hour plus a fixed amount. What is the best interpretation of the y-intercept of the graph in this context? (A) The rental costs $12 for each hour. (B) The fixed amount charged for the rental is $8. (C) The total cost of renting the kayak for 1 hour is $12. (D) A fixed charge of $12 is paid regardless of the number of hours.

Question 5

1. f(x) = x^2 - 3x + 5 For the function f defined above, what is the value of f(4)? (type your answer)

Question 6

1. The graph of 5x - 2y = 30 in the xy-plane crosses the x-axis at the point (a, 0). What is the value of a? (type your answer)

Question 7

1. The drama club sold 120 tickets to its spring show. Student tickets cost $4 each and adult tickets cost $7 each. The club collected a total of $606 from ticket sales. How many adult tickets did the club sell? (A) 42 (B) 60 (C) 78 (D) 126

Question 8

1. During a sale, a store sells a jacket for 0.72p dollars, where p is the jacket’s original price, in dollars. The sale price is a decrease of what percent from the original price? (A) 0.28% (B) 0.72% (C) 28% (D) 72%

Question 9

1. A soccer ball is kicked upward. Its height h, in feet, t seconds after it is kicked is modeled by h(t) = -16t^2 + 48t + 4. According to the model, what is the maximum height, in feet, that the ball reaches? (A) 1.5 (B) 4 (C) 40 (D) 52

Question 10

Right triangle ABC with the right angle at C. Side AC, along the bottom, is labeled 15. Side BC, the vertical side, is labeled 8. Side AB is the hypotenuse and is not labeled.

1. In the right triangle ABC shown, AC = 15 and BC = 8. What is the value of sin A? (A) (8)/(15) (B) (8)/(17) (C) (15)/(17) (D) (17)/(8)

Question 11

1. The table shows the number of 10th- and 11th-grade students in a school’s music program, by the type of instrument each student plays. If a student who plays a brass instrument is selected at random, what is the probability that the student is in 11th grade? (A) (27)/(112) (B) (2)/(5) (C) (27)/(59) (D) (3)/(5)

Question 12

1. 3x - 5y = 8 ax - 20y = 32 In the given system of equations, a is a constant. If the system has infinitely many solutions, what is the value of a? (type your answer)

Question 13

1. (2x + 3)(x - 5) = 2x^2 + bx - 15 In the given equation, b is a constant. If the equation is true for all values of x, what is the value of b? (type your answer)

Question 14

Rectangle ABCD with A at bottom left, B at bottom right, C at top right, and D at top left. Side AB is labeled 16 and side BC is labeled 10. Point E is on the top side CD, closer to D. Segments AE and BE are drawn, forming triangle ABE.

1. In the figure, ABCD is a rectangle with AB = 16 and BC = 10. Point E lies on side CD. What is the area of triangle ABE? (A) 26 (B) 40 (C) 52 (D) 80

Question 15

1. x^2 = 5x + 14 What are the solutions to the given equation? (A) -7 and 2 (B) -2 and 7 (C) 2 and 7 (D) -7 and -2

Question 16

1. A line in the xy-plane has a slope of (3)/(2) and passes through the points (-2, k) and (4, 2k - 1), where k is a constant. What is the value of k? (A) 4 (B) 5 (C) 8 (D) 10

Question 17

1. The mass m, in grams, of a radioactive sample t days after it was first measured is modeled by m(t) = 640((1)/(2))^t/5. According to the model, how many days after it was first measured will the sample have a mass of 40 grams? (A) 20 (B) 80 (C) 128 (D) 600

Question 18

1. y ≤ 2x - 3 x + y > 9 Which point (x, y) is a solution to the given system of inequalities? (A) (5, 5) (B) (4, 6) (C) (6, 3) (D) (3, 7)

Question 19

1. The function g is defined by g(x) = (x - 3)(x + k), where k is a constant. In the xy-plane, the vertex of the graph of y = g(x) has an x-coordinate of 4. What is the value of k? (type your answer)

Question 20

1. x^2 + y^2 - 6x + 10y = 2 The equation of a circle in the xy-plane is shown. What is the radius of the circle? (type your answer)

Question 21

1. In the xy-plane, line k is perpendicular to the line 3x - 4y = 12 and passes through the point (6, -1). What is the y-intercept of line k? (A) (0, -9) (B) (0, -(11)/(2)) (C) (0, 7) (D) (0, 9)

Question 22

1. y = x^2 - 4x + 7 y = 2x + c In the given system of equations, c is a constant. If the system has exactly one real solution, what is the value of c? (A) -29 (B) -2 (C) 2 (D) 3

Question 1

1. 5(x - 2) = 3x + 8 What is the solution to the given equation? (A) -1 (B) 5 (C) 9 (D) 18

Question 2

1. A car traveled 270 miles and used 7.5 gallons of gasoline. At the same rate of fuel use, how many gallons of gasoline will the car use to travel 450 miles? (A) 4.5 (B) 12.5 (C) 36 (D) 60

Question 3

1. For the linear function f, f(2) = 11 and f(6) = 23. What is the value of f(0)? (A) 3 (B) 5 (C) 8 (D) 17

Question 4

Scatterplot of test score versus hours of practice for 10 students, with the line of best fit y = 4x + 52. Points: (1, 55), (2, 62), (3, 63), (4, 66), (5, 74), (6, 75), (7, 88), (8, 83), (9, 90), (10, 91). The horizontal axis runs from 0 to 11 hours; the vertical axis runs from 48 to 100 in steps of 4.

1. The scatterplot shows the number of hours x that each of 10 students practiced for a typing test and each student’s score y on the test. A line of best fit for the data is also shown. An equation of the line is y = 4x + 52. For the student who practiced 7 hours, how many points greater is the student’s actual score than the score predicted by the line of best fit? (A) 8 (B) 12 (C) 80 (D) 88

Question 5

1. 4x^2 - 12x + 9 = (ax - b)^2 In the given equation, a and b are positive constants. If the equation is true for all values of x, what is the value of a + b? (type your answer)

Question 6

1. (2)/(3)(x - 4) = (x)/(2) + 1 What is the solution to the given equation? (type your answer)

Question 7

1. 2x + ky = 7 6x - 9y = 4 In the given system of equations, k is a constant. For what value of k does the system have no solution? (A) -27 (B) -3 (C) 3 (D) 27

Question 8

1. The function f is defined by f(x) = 2x^2 - 12x + 23. The function can be written in the form f(x) = 2(x - h)^2 + k, where h and k are constants. What is the value of k? (A) -13 (B) 3 (C) 5 (D) 14

Question 9

Right triangle PQR with the right angle at Q. Q is at the bottom left, P is at the bottom right, and R is above Q. The hypotenuse PR is labeled 39.

1. In triangle PQR shown, angle Q is a right angle, PR = 39, and cos P = (12)/(13). What is the length of QR? (A) 5 (B) 12 (C) 15 (D) 36

Question 10

1. The table summarizes the results of a survey of 200 commuters, classified by how they travel to work and how long their commute takes. If a commuter whose commute takes 30 minutes or more is selected at random, what is the probability that the commuter does not travel by train? (A) (27)/(100) (B) (9)/(20) (C) (5)/(11) (D) (6)/(11)

Question 11

1. √(2x + 7) = x + 2 Which of the following gives all the real solutions to the given equation? (A) 1 only (B) -3 only (C) -3 and 1 (D) There are no real solutions.

Question 12

1. 7 - 3x > 2(x - 9) What is the greatest integer value of x that satisfies the given inequality? (type your answer)

Question 13

1. The function h is defined by h(x) = a(x - 1)(x - 7), where a is a constant. The maximum value of h(x) is 27. What is the value of a? (type your answer)

Question 14

1. In the xy-plane, a circle has center (2, -3) and passes through the point (7, 9). Which equation represents this circle? (A) (x - 2)^2 + (y + 3)^2 = 169 (B) (x + 2)^2 + (y - 3)^2 = 169 (C) (x - 2)^2 + (y + 3)^2 = 13 (D) (x - 2)^2 + (y + 3)^2 = 61

Question 15

1. A water tank contains 540 liters of water when it begins to drain at a constant rate. After 12 minutes of draining, the tank contains 396 liters. How many minutes after the tank began to drain will it be empty? (A) 12 (B) 33 (C) 45 (D) 57

Question 16

1. (3)/(x - 2) - (2)/(x + 1) Which of the following is equivalent to the given expression for x > 2? (A) (1)/((x - 2)(x + 1)) (B) (x - 1)/((x - 2)(x + 1)) (C) (x + 5)/((x - 2)(x + 1)) (D) (x + 7)/((x - 2)(x + 1))

Question 17

Dot plot of hours worked by 15 volunteers on a number line from 0 to 16. Number of dots: 2 hours, 1; 3 hours, 2; 4 hours, 4; 5 hours, 3; 6 hours, 3; 13 hours, 1; 15 hours, 1.

1. The dot plot shows the number of hours each of 15 volunteers spent working at a food bank last month. The two volunteers who worked the greatest numbers of hours are removed from the data set. Which statement best describes how the mean and the median of the remaining 13 values compare with those of the original 15 values? (A) The mean and the median both decrease, and the mean decreases by more. (B) The mean and the median both decrease, and the median decreases by more. (C) The mean decreases, and the median does not change. (D) The mean decreases, and the median increases.

Question 18

1. A bakery spends exactly $60 on flour and sugar. Flour costs $3 per pound and sugar costs $4 per pound, so 3x + 4y = 60, where x is the number of pounds of flour and y is the number of pounds of sugar. If the bakery buys 3 more pounds of sugar and still spends exactly $60, how does the number of pounds of flour change? (A) It decreases by 3 pounds. (B) It decreases by (9)/(4) pounds. (C) It increases by 4 pounds. (D) It decreases by 4 pounds.

Question 19

1. The function f is defined by f(x) = a · b^x, where a and b are positive constants. In the xy-plane, the graph of y = f(x) passes through the points (1, 18) and (3, 162). What is the value of f(0)? (type your answer)

Question 20

Triangle ABC with A at the top, B at the bottom left, and C at the bottom right. Point D is on side AB and point E is on side AC, and segment DE is drawn parallel to BC (both marked with arrows). AD is labeled 6 and DB is labeled 4.

1. In the figure, DE is parallel to BC, AD = 6, and DB = 4. The area of triangle ADE is 27 square units. What is the area, in square units, of quadrilateral DBCE? (type your answer)

Question 21

A parabola opening upward on a coordinate grid with x from -4 to 6 and y from -6 to 8. The vertex is at (1, -4). The graph crosses the x-axis at (-1, 0) and (3, 0) and crosses the y-axis at (0, -3). It passes through (-2, 5) and (4, 5).

1. The graph of y = f(x) is shown. The function g is defined by g(x) = f(x + 3) - 2. What is the value of g(0)? (A) -5 (B) -2 (C) 2 (D) 8

Question 22

1. A chemistry teacher mixes a 10% saline solution with a 40% saline solution to make 60 milliliters of a 20% saline solution. How many milliliters of the 10% solution does the teacher use? (A) 12 (B) 20 (C) 30 (D) 40