FREE SHSAT Math Practice Test

Welcome to our FREE SHSAT Math practice test, with answer key and answer explanations. This practice test’s realistic format and high-quality practice questions can help you succeed on the SHSAT Math test. Not only does the test closely match what you will see on the real SHSAT, but it also comes with detailed answer explanations. For additional educational resources, see our SHSAT Math Formula Cheat Sheet.

For this practice test, we’ve selected 20 real questions from past exams for your SHSAT Practice test. You will have the chance to try out the most common SHSAT Math questions. For every question, there is an in-depth explanation of how to solve the question and how to avoid mistakes next time.

Use our free SHSAT Math practice tests and study resources (updated for 2026) to ace the SHSAT Math test! Make sure to follow some of the related links at the bottom of this post to get a better idea of what kind of mathematics questions you need to practice.

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1- Out of 55,000 voters in a city, 2,500 voted for a particular candidate. What fraction of voters voted for this candidate?

A. \(\frac{1}{22}\)

B. \(\frac{1}{20}\)

C. \(\frac{1}{18}\)

D. \(\frac{1}{15}\)

2- What is the value of \(4 \div \frac{1}{6} \)?

A. \(\frac{2}{3}\)

B. 6

C. 18

D. 24

3- Simplify \((6x^3-8x^2+2x^4 )-(4x^2-2x^4+2x^3 )\).

A. \(2x^4+4x^3-4x^2\)

B. \(4x^4+4x^3-12x^2\)

C. \(4x^4-4x^3-12x^2\)

D. \(8x^4+8x^3-12x^2\)

4- The population of a city increased by \(15\%\) in the first year and by \(20\%\) in the second year. What is the total percent increase in population over the two years?

A. \(35\%\)

B. \(37\%\)

C. \(38\%\)

D. \(40\%\)

5- What is the surface area of a cylinder with a diameter of 6 inches and a height of 8 inches?

A. \(48\pi \space in^2\)

B. \(54\pi \space in^2\)

C. \(66\pi \space in^2\)

D. \(132\pi \space in^2\)

6- A cruise line ship left Port A and traveled 80 miles due west and then 150 miles due north. At this point, what is the shortest distance from the cruise to port A in miles? ____________

7- A shirt costing $200 is discounted \(15\%\). After a month, the shirt is discounted another \(15\%\). Which of the following expressions can be used to find the selling price of the shirt?

A. \((200) (0.70)\)

B. \((200) – 200 (0.30)\)

C. \((200) (0.15) – (200) (0.15)\)

D. \((200) (0.85) (0.85)\)

8- \(5 + 8 \times (- 2) – [4 + 22 \times 5] \div 6 = ? \)

A. \(-30\)

B. \(-20\)

C. \(-10\)

D. 0

9- Which of the following points lies on the line \(x+2y=4\)?

A. \((-2, 3)\)

B. \((1, 2)\)

C. \((-1, 3)\)

D. \((-3, 4)\)

10- 5 less than twice a positive integer is 83. What is the integer?

A. 39

B. 41

C. 42

D. 44

11- 11 yards 6 feet and 4 inches equal to how many inches?

A. 388

B. 468

C. 472

D. 476

12- Mr. Carlos’s family is choosing a menu for their reception. They have 3 choices of appetizers, 5 choices of entrees, and 4 choices of cake. How many different menu combinations are possible for them to choose from?

A. 12

B. 32

C. 60

D. 120

13- The average of five consecutive numbers is 38. What is the smallest number?

A. 38

B. 36

C. 34

D. 12

14- What is the difference between the smallest 4-digit number and the biggest 4-digit number?

A. 6666

B. 6789

C. 8888

D. 8999

15- How many tiles of 8 cm\(^2 \) are needed to cover a floor of dimension 6 cm by 24 cm?

A. 6

B. 12

C. 18

D. 24

16- A ladder leans against a wall forming a \(60^ \circ \) angle between the ground and the ladder. If the bottom of the ladder is 30 feet away from the wall, how long is the ladder?

A. 30 feet

B. 40 feet

C. 50 feet

D. 60 feet

17- The average weight of 18 girls in a class is 60 kg, and the average weight of 32 boys in the same class is 62 kg. What is the average weight of all the 50 students in that class?

A. 60

B. 61.28

C. 61.68

D. 62.90

18- An angle is equal to one-fifth of its supplement. What is the measure of that angle?

A. 20

B. 30

C. 45

D. 60

19- In a stadium, the ratio of home fans to visiting fans in a crowd is 5:7. Which of the following could be the total number of fans in the stadium?

A. 12,324

B. 42,326

C. 44,566

D. 66,812

20- If \(40\%\) of a class are girls, and \(25\%\) of girls play tennis, what percent of the class play tennis?

A. \(10\%\)

B. \(15\%\)

C. \(20\%\)

D. \(40\%\)

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Answers:

2- D
\(4 \div \frac{1}{6} = 24 \)

3- B
Simplify and combine like terms.
\((6x^3-8x^2+2x^4 )-(4x^2-2x^4+2x^3 ) {\Rightarrow} (6x^3-8x^2+2x^4 )-4x^2+2x^4-2x^3 {\Rightarrow}
4x^4+4x^3-12x^2\)

4- C
the population is increased by \(15\%\) and \(20\%\).
\(15\%\) increase changes the population to \(115\%\) of original population.
For the second increase, multiply the result by \(120\%\).
\((1.15) \times (1.20) = 1.38 = 138\%\)
38 percent of the population is increased after two years.

5- C
Surface Area of a cylinder \(= 2\pi\) r (r + h),
The radius of the cylinder is \(3(6 \div 2)\) inches and its height is 8 inches. Therefore,
Surface Area of a cylinder \(= 2\pi (3) (3 + 8) = 66\pi\)

6- 170
Use the information provided in the question to draw the shape.
\( \img {https://appmanager.effortlessmath.com/public/images/questions/22222222222222222222222222222.JPG
} \)
Use Pythagorean Theorem: \(a^2 + b^2 = c^2 \)
\(80^2 + 150^2 = c^2 {\Rightarrow} 6400 + 22500 = c^2 {\Rightarrow} 28900 = c^2 {\Rightarrow} c = 170\).

7- D
To find the discount, multiply the number by (\(100{\%} – \)rate of discount).
Therefore, for the first discount we get: \((200) (100{\%} – 15{\%}) = (200) (0.85) = 170\)
For the next \(15{\%}\) discount: \((200) (0.85) (0.85)\)

8- A
Use PEMDAS (order of operation):
\(5 + 8 \times (-2) – [4 + 22 \times 5] \div 6 = 5 + 8 \times (-2) – [4 + 110] \div 6 = 5 + 8 \times (-2) – [114] \div 6 = 5 + (-16) – 19
5 + (-16) – 19 = -11 – 19 = -30\)

9- A
\(x+2y=4\). Plugin the values of \(x\) and \(y\) from the choices provided. Then:
A. \((-2,3) x+2y=4(\rightarrow )-2+2(3)=4(\rightarrow )-2+6=4\) This is true!
B. \((1,2) x+2y=4(\rightarrow )1+2(2)=4(\rightarrow )1+4=4\) (This is NOT true!)
C. \((-1,3) x+2y=4(\rightarrow )-1+2(3)=4(\rightarrow )-1+6=4\) (This is NOT true!)
D. \((-3,4) x+2y=4(\rightarrow )-3+2(4)=4(\rightarrow )-3+8=4\) (This is NOT true!)

10- D
Let \(x\) be the integer. Then:
\(2x – 5 = 83\)
Add 5 both sides: \(2x = 88\)
Divide both sides by \(2: x = 44\)

11- C
\(11 \times 36 + 6 \times 12 + 4 = 472\)

12- C
To find the number of possible outfit combinations, multiply the number of options for each factor:
\(3 \times 5 \times 4 = 60\)

13- B
Let \(x\) be the smallest number. Then, these are the numbers:
\(x, x+1, x+2, x+3, x+4 \)
average \(= \frac{sum of terms}{number of terms} \Rightarrow 38 = \frac{x+(x+1)+(x+2)+(x+3)+(x+4)}{5} \Rightarrow 38= \frac{5x+10}{5} \Rightarrow 190 = 5x+10 \Rightarrow 180 = 5x \Rightarrow x=36 \)

14- D
The smallest 4–digit number is 1000, and the biggest 4–digit number is 9999. The difference is: 8999

15- C
The area of the floor is: 6 cm \(\times\) 24 cm = 144 cm\(^2 \)
The number of tiles needed \(= 144 \div 8 = 18\)

16- D
The relationship among all sides of a special right triangle
\(30^ \circ -60^ \circ – 90^ \circ \) is provided in this triangle:
\( \img {https://appmanager.effortlessmath.com/public/images/questions/10-explain-1.png} \)
In this triangle, the opposite side of the \(30^ \circ \) angle is half of the hypotenuse.
Draw the shape for this question:
The latter is the hypotenuse. Therefore, the latter is 60 ft.
\( \img {https://appmanager.effortlessmath.com/public/images/questions/10-explain-2.png} \)

17- B
average \(= \frac{sum of terms}{number of terms}\)
The sum of the weight of all girls is: \(18 \times 60 = 1080\) kg
The sum of the weight of all boys is:\( 32 \times 62 = 1984\) kg
The sum of the weight of all students is: \(1080 + 1984 = 3064\) kg
average \(= \frac{3064}{50}=61.28 \)

18- B
The sum of supplement angles is 180. Let \(x\) be that angle. Therefore,
\(x + 5x = 180\)
\(6x = 180\), divide both sides by 6: \(x = 30\)

19- A
In the stadium the ratio of home fans to visiting fans in a crowd is \(5:7\). Therefore, total number of fans must be divisible by \(12: 5 + 7 = 12.\)
Let’s review the choices:
A. \(12,324: 12,324 \div 12 = 1027\)
B. \(42,326: 42,326 \div 12 = 3,527.166\)
C. \(44,566: 44,566 \div 12 = 3,713.833\)
D. \(66,812: 66,812 \div 12 = 5,567.666\)
Only choice A when divided by 12 results in a whole number.

20- A
The percentage of girls playing tennis is:
\(40 \% \times 25\% = 0.40 \times 0.25 = 0.10 = 10\%\)

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Recommended EffortlessMath Books

For a step-by-step workbook on every SHSAT math topic, see the SHSAT Math for Beginners. For complete specialized high school prep with multiple practice tests, the SHSAT Math Test Prep Bundle combines the workbook with timed practice tests.

Frequently Asked Questions

How many math questions are on the SHSAT?

57 math questions total. The test also has 57 ELA questions. You get 180 minutes for the whole test and can split your time between math and ELA however you choose.

Is a calculator allowed on the SHSAT?

No. No calculators are permitted on any portion of the SHSAT. Practice mental arithmetic, fraction work, and quick estimation strategies before test day.

How is the SHSAT scored?

Your raw score (number correct on math + ELA) converts to a scaled composite score. The composite is roughly out of 700. There’s no penalty for wrong answers — guess on every question you don’t answer.

What composite score do I need for Stuyvesant or Bronx Science?

Cutoffs change each year based on test difficulty and applicant pool. As a rough historical guide: Stuyvesant ~560+; Bronx Science ~520+; Brooklyn Tech ~500+; the other specialized high schools ~470-490+. Aim for 560+ to keep all 8 options open.

What math topics are on the SHSAT?

Number sense, ratios and proportions, linear equations, exponents, polynomials, factoring, coordinate geometry, plane geometry (area, perimeter, angles, triangles, circles), volume, and basic probability and statistics. Roughly Algebra I level with some geometry.

What’s the difference between multiple-choice and grid-in questions?

Multiple-choice questions have 4 answer choices labeled A-D. Grid-in questions have no choices — you compute the answer and bubble in each digit on a grid. There are typically 5 grid-in math questions on the test. They’re worth the same as multiple-choice.

How long is the SHSAT?

180 minutes total for both sections combined, plus a short instruction period. Most successful test-takers spend roughly 90 minutes on math and 90 on ELA, but you can flex either direction based on your strengths.

Who takes the SHSAT?

NYC public, parochial, and private school 8th graders applying for 9th grade admission, and 9th graders applying for 10th grade admission, at NYC’s specialized high schools. Students must be NYC residents. The test is the only admissions criterion for 8 of the 9 specialized high schools.

Can I retake the SHSAT?

You can take the SHSAT once as an 8th grader for 9th-grade admission and once as a 9th grader for 10th-grade admission. There are no in-year retakes — each grade gets one shot.

How long should I study for the SHSAT math?

Most successful applicants put in 3 to 6 months of preparation, including 30-60 minutes a day plus weekend practice tests. Start with a diagnostic, identify weak topics, and build a study calendar that revisits each topic every 7-10 days.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• SHSAT math formula cheat sheet
• How to find the slope of a line
• How to use the Pythagorean theorem
• How to use the quadratic formula
• Take more SHSAT math practice tests