Full-Length PSAT Math Practice Test

Full-Length PSAT Math Practice Test

Taking a Full-length PSAT Math practice test is the best way to help you get familiar with the test format and feel more confident. Not only will this help you measure your exam readiness and solidify the concepts you’ve learned, but it is the best way to simulate test day.
To help you get the best out of this complete and realistic PSAT Math practice test and prepare your mind and body for the actual test, we recommend that you treat this practice test as a real test. Prepare scratch papers, pencil, a timer, and a calculator and take the test in one sitting and follow the time limits to the minute.
Take the following full-length PSAT Math practice test to simulate the test day experience. After you’ve finished, score your tests using the answer keys.
Good luck!

The Absolute Best Book to Ace the PSAT Math Test

Time to refine your Math skill with a practice test

Take practice PSAT/ NMSQT Math Tests to simulate the test day experience. After you’ve
finished, score your tests using the answer keys.

Before You Start

  • You’ll need a pencil, a timer and a calculator to take the tests.
  • There are two types of questions:
    Multiple choice questions: for each of these questions, there are four or more possible
    answers. Choose which one is best.
    Grid-ins questions: for these questions, you have to fill your answer into a grid.
  • It’s okay to guess. You won’t lose any points if you’re wrong.
  • The PSAT Mathematics test contains a formula sheet, which displays formulas relating to
    geometric measurement and certain algebra concepts. Formulas are provided to test-
    takers so that they may focus on application, rather than the memorization, of formulas.
  • After you’ve finished the test, review the answer key to see where you went wrong and
    what areas you need to improve.

Good Luck!

Best PSAT Math Prep Resource for 2020

PSAT Math Practice Test

Two Sections
Total number of questions: 48
Section 1 (Non-Calculator): 17 questions
Section 2 (Calculator): 31 questions
Total time for two parts: 70 Minutes

PSAT Math Practice Test

Section 1 (Non-Calculator)
17 questions

Total time for two parts: 70 Minutes

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. \(x>4\)

2- If \(f(x^2 )=10-5x\), for all positive value of \(x\), what is the value of \(f(169)\)?

☐A. \(-55\)

☐B. \(-20\)

☐C. \(20\)

☐D. \(55\)

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. \(-7\)

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

☐A. \((2,1)\)

☐B. \((6, 4)\)

☐C. \((6,3)\)

☐D. \((9,3)\)

5- John works for an electric company. He receives a monthly salary of \($3,600\) plus \(4\%\) of all his monthly sales as bonus. If \(𝑥\) is the number of all John’s sales per month, which of the following represents John’s monthly revenue in dollars?

☐A. \(0.04x\)

☐B. \(0.96x-3,800\)

☐C. \(0.4x+3,800\)

☐D. \(0.04x+3,600\)

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

7- If \(x≠-2\) and \(x≠6\), which of the following 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. \(\frac{(x-6)(x+2)}{2x+4}\)

8- The table above shows the distribution of age and gender for 39 employees in a company. If one employee is selected at random, what is the probability that the employee selected be either a female under age 45 or a male age 45 or older?

☐A. \(\frac{5}{39}\)

☐B. \(\frac{6}{39}\)

☐C. \(\frac{6}{13}\)

☐D. \(\frac{5}{6}\)

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

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

11- What is the length of AB in the following figure if AE=10, CD=16 and BC=20?

☐A. 3.8

☐B. 8.8

☐C. 12.5

☐D. 24.6

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

☐A. \(x=5,y=-3\)

☐B. \(x=-3,y=5\)

☐C. \(x=-3,y=3\)

☐D. \(x=3,y=3\)

13- John buys a pepper plant that is 4 inches tall. With regular watering the plant grows 8 inches a year. Writing John’s plant’s height as a function of time, what does the intercept represent?

☐A. The y-intercept represents the rate of grows of the plant which is 8 inches

☐B. The y-intercept represents the starting height of 6 inches

☐C. The y-intercept represents the rate of growth of plant which is 4 inches per year

☐D. There is no y-intercept

14- The length of a rectangle is 8 meters greater than 2 times its width.  The perimeter of the rectangle is 88 meters.  What is the area of the rectangle in meters? __________


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


16- What is the value of  in the following equation?
\(\frac{x^2-4}{x+2}+4(x+3)=20\)


17- If\(x≠0\), what is the value of \(\frac{(6(x)(y^3 ))^2}{x^2 (3y^2)^3}\)? _____________


PSAT Math Practice Test

Section 2 (Calculator)
31 questions

Total time for two parts: 70 Minutes

18- A football team won exactly 72% of the games it played during last session. Which of the following could be the total number of games the team played last season?

☐A. 40

☐B. 50

☐C. 55

☐D. 60

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

20- In the 𝑥𝑦-plane, the point \((-2,4)\) and \((6,20)\) are on line A. Which of the following equations of lines is parallel to line A?

☐A. \(y= \frac{x}{2}\)

☐B. \(y=x\)

☐C. \(y=2x\)

☐D. \(y=3x\)

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

22- If \(4+4x\) is\(8\) more than \(12\), what is the value of \(8x\)?

☐A. 32

☐B.38

☐C. 62

☐D. 84

23- If \(x\) is greater than\(-1\) and less than \(1\) and\(x≠0\), which of the following 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.\(\sqrt{x^2}+1>sqrt{x^2+1}>x\)

24- The sum of eight different negative integers is \(-100\). If the smallest of these integers is \(-16\), what is the largest possible value of one of the other five integers?

☐A. \(-14\)

☐B. \(-9\)

☐C. \(–5\)

☐D. \(-1\)

25- The capacity of a red box is \(40\%\) bigger than the capacity of a blue box. If the red box can hold \(35\) equal sized books, how many of the same books can the blue box hold?

☐A. 20

☐B.22

☐C. 25

☐D.30

Questions 9 and 11 are based on the following data

26- If a is the mean (average) of the number of cities in each pollution type category, b is the mode, and c is the median of the number of cities in each pollution type category, then which of the following must be true?

☐A. \(a<b<c\)

☐B. \(b<a<c\)

☐C. \(a=c\)

☐D. \(b<c=a\)

27- What percent of cities are in the type of pollution A, C, and E respectively?

☐A. \(60\%, 40\%, 90\%\)

☐B.\(30\%, 40\%, 90\%\)

☐C. \(30\%, 40\%, 60\%\)

☐D. \(80\%, 70\%, 50\%\)

28- How many cities should be added to type of pollutions E until the ratio of cities in type of pollution E to cities in type of pollution D will be 0.8?

☐A. 2.2

☐B.3.5

☐C. 4.2

☐D.5.1

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.\(frac{8}{7}\)

30- In the following right triangle, if the sides AB and AC become one third shorter, what will be the ratio of the perimeter of the triangle to its area?

☐A. \(\frac{1}{2}\)

☐B. 1

☐C. 2

☐D. 3

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

Questions 15 to 17 are based on the following data

32- What’s the maximum ratio of women to men in the four cities?

☐A. 0.88

☐B. 0.91

☐C. 0.96

☐D.0.98

33- What’s the ratio of percentage of men in city A to percentage of women in city C?

☐A. 0.9

☐B. 0.95

☐C. 1.23

☐D. 1.38

34- How many women should be added to city D until the ratio of women to men will be 1.4?

☐A. 470

☐B. 480

☐C. 495

☐D. 500

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

☐A. \(5x\)

☐B. \(-6x+4\)

☐C. \(-6x+7\)

☐D. \(6x-7\)

36- If a car has 60-liter petrol and after one hour driving the car use 7.5-liter petrol, how much petrol will remain after -hours driving?

☐A. \(60+7.5x\)

☐B. \(60-7.5x\)

☐C. \(60×7.5x\)

☐D. \(60÷7.5x\)

37- In the triangle below, if the measure of angle  is 43 degrees, then what is the value of ? (figure is NOT drawn to scale)

☐A. 89

☐B. 90

☐C. 91

☐D. 96

38- The following graph shows the mark of six students in mathematics. What is the mean (average) of the marks?

☐A. 10

☐B. 12

☐C. 14

☐D. 15

39- In the rectangle below if  \(y > 10\) cm and the area of rectangle is 96 cm\(^2\) and the perimeter of the rectangle is 40 cm, what is the value of  and  respectively?

☐A. 6, 12

☐B. 8, 10

☐C. 8, 11

☐D. 8, 12

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. \(\frac{a}{b}=\frac{19}{9}\)

41- Given the right triangle ABC bellow, sec\((β)\) is equal to?

☐A. \(\frac{a}{b}\)

☐B. \(\frac{a}{\sqrt{a^2+b^2}}\)

☐C. \(\frac{\sqrt{a^2+b^2}}{a}\)

☐D. \(\frac{b}{\sqrt{a^2+b^2}}\)

42- Solve the following inequality.

☐A. \(|\frac{2x}{3}-4x-6|<4\)

☐B. \(\frac{6}{-10}>x>-3\)

☐C. \(\frac{10}{3}<x<10\)

☐D. \(-10<x<-\frac{10}{3}\)

43- If \(x) is directly proportional to the square of \(y\), and \(y=3\) when \(x=18\), then when \(x=288 , y= \)?

☐A. \(\frac{1}{5}\)

☐B. \(1\)

☐C. \(5\)

☐D. \(12\)

27- Which of the following values for \(x\) and \(y\) satisfy the following system of equations?
\(\begin{cases}2x+4y=6 \\x+3y=-20 \end{cases}\)

☐A. \(x=33,y=-18\)

☐B. \(x=49,y=-23\)

☐C. \(x=72,y=-23\)

☐D. \(x=83,y=-18\)

28- If \(x+4y=\frac{-8y^2+12}{2x}\), what is the value of \((x+2y)^2\)? \((x≠0)\)


29- A ladder leans against a wall forming a 45ᵒ angle between the ground and the ladder. If the bottom of the ladder is 10 feet away from the wall, how many feet is the ladder?   


      

30- The volume of cube A is \(\frac{1}{4}\) of its surface area. What is the length of an edge of cube A?


        

31- \(f(x)=ax^2+bx+c\) is a quadratic function where a, b and c are constant. The value of x of the point of intersection of this quadratic function and linear function \(g(x)=x+2\) is \(3\). The vertex of \(f(x)\) is at \((-3, 11)\). What is the product of a, b and c?


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