PSAT 10 Math Practice: Official Tests and 48 Skill Questions

Start with current official PSAT 10 practice

For a full-length digital PSAT 10 session, use College Board’s free official practice options. Choose PSAT 10 and follow the Bluebook instructions. Downloadable paper practice is also available, with guidance for paper-based accommodations.

The current Math section has 44 questions in two 35-minute modules, with an acceptable calculator allowed throughout. See College Board’s Math overview.

The 48 questions below are corrected legacy Effortless Math skill practice. They are independent materials, not an official test or an adaptive digital simulation. Use them untimed to review individual skills. Do not convert a raw total into a PSAT score. Question 41 is a secant extension exercise.

Need a lesson first? The free PSAT 10 Math hub connects worksheets, formulas and topic review. Check each answer after working, then revisit the skill behind any mistake.

Question 1

Which of the following is the solution of the following inequality? \(12x-14>10x+5-3x+1\)

• A. \(x<3\)
• B. \(x>3\)
• C. \(x≤4\)
• D. \(x>4\)

Question 2

For every positive value of \(x\), \(f(x^2)=10-5 x\). What is \(f(169)\)?

• A. \(-55\)
• B. \(-20\)
• C. \(20\)
• D. \(55\)

Question 3

If a and b are solutions of the following equation, which of the following is the ratio \(\frac{a}{b}\)? \((a>b)\) \(8x^2-5x+8=7x^2+x+15\)

• A. \(-\frac{1}{7}\)
• B. \(-\frac{1}{5}\)
• C. \(-5\)
• D. \(-7\)

Question 4

A line in the \(xy\)-plane passes through the origin and has slope \(-\frac{2}{3}\). Which point lies on the line?

• A. \((2,1)\)
• B. \((6,4)\)
• C. \((6,-4)\)
• D. \((9,3)\)

Question 5

John earns a monthly salary of $3,600 plus a bonus equal to 4% of the dollar value of his monthly sales. If \(x\) is his monthly sales in dollars, which expression gives his monthly earnings in dollars?

• A. \(0.04x\)
• B. \(0.96x-3,800\)
• C. \(0.4x+3,800\)
• D. \(0.04x+3,600\)

Question 6

What is the value of \(f(-3)\) for the following function \(f\)? \(f(x)=3x^2+3x+2\)

• A. 5
• B. 10
• C. 15
• D. 20

Question 7

For \(x\ne-2,2,6\), which expression is equivalent to \(\frac{1}{\frac{1}{x-6}+\frac{1}{x+2}}\)?

• A. \(\frac{(x-6)(x+2)}{2x-4}\)
• B. \(\frac{2x-4}{(x-6)(x+2)}\)
• C. \(\frac{2x+4}{(x-6)(x+2)}\)
• D. \(\frac{(x-6)(x+2)}{2x+4}\)

Question 8

The table shows the ages and genders of 39 employees. One employee is selected at random. What is the probability of selecting a female employee under 45 or a male employee age 45 or older?

• A. \(\frac{5}{39}\)
• B. \(\frac{6}{39}\)
• C. \(\frac{6}{13}\)
• D. \(\frac{5}{6}\)

Question 9

If a parabola with equation \(y=ax^2+6x+16\), where a is constant, passes through point \((-2, 8)\), what is the value of \(a^2\)?

• A. 1
• B. 2
• C. 3
• D. 4

Question 10

If \(a, b\) and c are positive integers and \(4a=6b=10c\), then the value of \(2a+3b-4c\) is how many times the value of a?

• A. 2.4
• B. 2.6
• C. 3.5
• D. 4.2

Question 11

In the diagram, A, B and C are collinear, and E, B and D are collinear. EA and CD are perpendicular to AC. If EA=10, CD=16 and BC=20, what is AB?

• A. 3.8
• B. 8.8
• C. 12.5
• D. 24.6

Question 12

What is the solution of \(\begin{cases}\frac{x}{3}+\frac{4 y}{6}=1\\y-x=6\end{cases}\)?

• A. \(x=5,y=-3\)
• B. \(x=-3,y=5\)
• C. \(x=-3,y=3\)
• D. \(x=3,y=3\)

Question 13

A pepper plant is 4 inches tall when John buys it and grows 8 inches per year. If height is a function of years since purchase, what does the y-intercept represent?

• A. The rate of growth, 8 inches per year.
• B. The initial height, 4 inches.
• C. The rate of growth, 4 inches per year.
• D. There is no y-intercept.

Question 14

A rectangle is 8 meters longer than twice its width. Its perimeter is 88 meters. What is its area, in square meters?

Your answer: ____________________

Question 15

In the following equation, what is the value of \(y-4x\)? ___________ \(\frac{2y}{9}=x-\frac{1}{9}x+4\)

Your answer: ____________________

Question 16

For \(x\ne-2\), solve \(\frac{x^2-4}{x+2}+4(x+3)=20\).

Your answer: ____________________

Question 17

For nonzero \(x\) and \(y\), simplify \(\frac{(6 xy^3)^2}{x^2(3 y^2)^3}\).

Your answer: ____________________

Question 18

A basketball team won exactly 72% of its games last season. Which could be the total number of games it played?

• A. 40
• B. 50
• C. 55
• D. 60

Question 19

If a gas tank can hold 30 gallons, how many gallons does it contain when it is \(\frac{5}{6}\) full?

• A. 20
• B. 25
• C. 62.5
• D. 100

Question 20

The points \((-2,4)\) and \((6,20)\) lie on line A in the xy-plane. Which equation represents a line parallel to line A?

• A. \(y= \frac{x}{2}\)
• B. \(y=x\)
• C. \(y=2x\)
• D. \(y=3x\)

Question 21

If \(y=nx+4\), where n is a constant, and when \(x=3\), \(y=22\), what is the value of y when \(x=4\)?

• A. 10
• B. 24
• C. 28
• D. 32

Question 22

If \(4+4 x\) is 8 more than 12, what is \(8 x\)?

• A. 32
• B. 38
• C. 62
• D. 84

Question 23

If \(-1<x<1\) and \(x\ne 0\), which inequality is true?

• A. \(x>\sqrt{x^2+1}>\sqrt{x^2}+1\)
• B. \(x<\sqrt{x^2}+1<\sqrt{x^2+1}\)
• C. \(\sqrt{x^2+1}<x<\sqrt{x^2}+1\)
• D. \(\sqrt{x^2}+1>\sqrt{x^2+1}>x\)

Question 24

Eight different negative integers have sum -100. The smallest is -16. What is the largest possible value of one of the other seven integers?

• A. -14
• B. -9
• C. -5
• D. -1

Question 25

A red box has 40% more capacity than a blue box. The red box holds exactly 35 equal-sized books with no unused space. How many of the same books fill the blue box?

• A. 20
• B. 22
• C. 25
• D. 30

Question 26

The pollution data table gives five counts: 8, 6, 7, 9, 5. Let a be their mean and c their median. Which statement is true?

• A. \(a<c\)
• B. \(c<a\)
• C. \(a=c\)
• D. \(a+c=0\)

Question 27

Of the 10 cities in the pollution table, what percentages have pollution types A, C and E, respectively?

• A. \(60\%, 40\%, 90\%\)
• B. \(30\%, 40\%, 90\%\)
• C. \(30\%, 40\%, 60\%\)
• D. \(80\%, 70\%, 50\%\)

Question 28

Initially 5 cities have pollution type E and 9 have type D. Additional cities are added to type E only. The type D count stays 9. What is the minimum whole number of additional type E cities needed for the E-to-D ratio to be at least 0.8?

• A. 1
• B. 2
• C. 3
• D. 4

Question 29

What is the ratio of the minimum value to the maximum value of the following function? \(-4≤x≤5\) \(f(x)=-2x+2\)

• A. \(-1\)
• B. \(\frac{-4}{5}\)
• C. \(\frac{6}{10}\)
• D. \(\frac87\)

Question 30

A right triangle has legs AB=15 centimeters and BC=36 centimeters. A similar triangle has every side length equal to one-third of the corresponding original length. What is the numerical value of its perimeter in centimeters divided by its area in square centimeters?

• A. \(\frac{1}{2}\)
• B. 1
• C. 2
• D. 3

Question 31

The ratio of boys and girls in a class is 5: 9. If there are 84 students in the class, how many more boys should be enrolled to make the ratio 1: 1?

• A. 12
• B. 18
• C. 24
• D. 30

Question 32

Using the city population table, what is the greatest ratio of women to men among the four cities, rounded to the nearest hundredth?

• A. 0.88
• B. 0.91
• C. 0.96
• D. 0.98

Question 33

Using the population table, divide the percentage of city A residents who are men by the percentage of city C residents who are women. What is the result, rounded to the nearest hundredth?

• A. 0.90
• B. 0.95
• C. 1.15
• D. 1.38

Question 34

Using the table counts as numbers of people, how many women must be added to city D so that its women-to-men ratio is 1.4, with the number of men unchanged?

• A. 470
• B. 480
• C. 495
• D. 500

Question 35

If \(f(x)=-2x+2(2x+3)+1\) then \(f(-3x)=\)?

• A. \(5x\)
• B. \(-6x+4\)
• C. \(-6x+7\)
• D. \(6x-7\)

Question 36

A car starts with 60 liters of fuel and uses 7.5 liters per hour at a constant rate. Which expression gives the fuel remaining after \(x\) hours, for \(0\le x\le 8\)?

• A. \(60+7.5x\)
• B. \(60-7.5x\)
• C. \(60×7.5x\)
• D. \(\frac{60}{7.5x}\)

Question 37

In the triangle, angle A is \((9 x-2)^\circ\), angle B is \(12 x^\circ\), and angle C is \((y-12)^\circ\). If angle A is 43 degrees, what is y?

• A. 89
• B. 90
• C. 91
• D. 96

Question 38

The six marks are 8, 12, 15, 15, 8, 14, as shown in the table. What is their mean?

• A. 10
• B. 12
• C. 14
• D. 15

Question 39

A rectangle has side lengths x and y centimeters, with y>10. Its area is 96 square centimeters and its perimeter is 40 centimeters. What are x and y, respectively?

• A. 6, 12
• B. 8, 10
• C. 8, 11
• D. 8, 12

Question 40

If \(\frac{a-b}{2b}=\frac{5}{9}\), then which of the following must be true?

• A. \(\frac{a}{b}=\frac{11}{21}\)
• B. \(\frac{a}{b}=\frac{21}{11}\)
• C. \(\frac{a}{b}=\frac{21}{10}\)
• D. \(\frac{a}{b}=\frac{19}{9}\)

Question 41

Extension: In the right triangle, angle beta has adjacent leg a and opposite leg b, with a, b>0. What is sec (beta), the reciprocal of cos (beta)?

• A. \(\frac{a}{b}\)
• B. \(\frac{a}{\sqrt{a^2+b^2}}\)
• C. \(\frac{\sqrt{a^2+b^2}}{a}\)
• D. \(\frac{b}{\sqrt{a^2+b^2}}\)

Question 42

Solve \(\left|\frac{2 x}{3}-4 x-6\right|<4\).

• A. \(x<-3\)
• B. \(-3<x<-\frac35\)
• C. \(\frac{10}{3}<x<10\)
• D. \(-10<x<-\frac{10}{3}\)

Question 43

For positive x and y, x is directly proportional to the square of y. If x=18 when y=3, what is y when x=288?

• A. \(\frac{1}{5}\)
• B. \(1\)
• C. \(5\)
• D. \(12\)

Question 44

Which pair solves \(\begin{cases}2 x+4 y=6\\x+3 y=-20\end{cases}\)?

• A. \(x=33,y=-18\)
• B. \(x=49,y=-23\)
• C. \(x=72,y=-23\)
• D. \(x=83,y=-18\)

Question 45

If \(x+4 y=\frac{-8 y^2+12}{2 x}\), with \(x\ne 0\), what is \((x+2 y)^2\)?

Your answer: ____________________

Question 46

A ladder makes a 45 degree angle with level ground and rests against a vertical wall. Its foot is 10 feet from the wall. What is its exact length, in feet?

Your answer: ____________________

Question 47

A cube has edge length s units. Its volume is numerically one-fourth of its surface area when measured in cubic units and square units, respectively. What is s?

Your answer: ____________________

Question 48

The graph of \(f(x)=ax^2+bx+c\) has vertex (-3, 11). One intersection with \(g(x)=x+2\) has x-coordinate 3. What is abc?

Your answer: ____________________