# FREE ACT Math Practice Test

Welcome to our FREE ACT 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 ACT Math test. Not only does the test closely match what you will see on the real ACT Math, but it also comes with detailed answer explanations.

For this practice test, we’ve selected 20 real questions from past exams for your ACT Math Practice test. You will have the chance to try out the most common ACT 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 ACT Math practice tests and study resources (updated for 2022) to ace the ACT 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 math questions you need to practice.

## The Absolute Best Book to Ace the ACT Math Test

## 10 Sample **ACT Mathematics **Math Practice Questions

1- If the area of a trapezoid is 100, what is the perimeter of the trapezoid?

\( \img {https://appmanager.effortlessmath.com/public/images/questions/q2222222222222222222222222222.JPG

}\)

A. 35

B. 45

C. 55

D. 65

2- If f(x)=\(2x^3+5x^2+2 \) and \(g(x)=-2\), what is the value of f(g(x))?

A. 36

B. 32

C. 24

D. 4

E. 0

3- In the following equation when z is divided by 3, what is the effect on x?

\(x = \frac{8y + \frac{r}{r + 1}}{\frac{6}{z}}\)

A. x is divided by 2

B. x is divided by 3

C. x does not change

D. x is multiplied by 3

E. x is multiplied by 2

4- A boat sails 40 miles south and then 30 miles east. How far is the boat from its start point?

A. 45 miles

B. 50 miles

C. 55 miles

D. 60 miles

E. 70 miles

5- If x \(\begin{bmatrix}a & b \\c & d \end{bmatrix}\) = \(\begin{bmatrix}x+3y-5 & 0 \\0 & 2y+10 \end{bmatrix}\), what is the product of \(x\) and \(y\)?

A. 1

B. 2

C. 10

D. 11

E. 12

6- If \(f(x)=3^x \) and \(g(x)=log_3 x\), which of the following expressions is equal to f\(3g(p)\)?

A. 3p

B. \(3^p\)

C. \(p^3\)

D. \(p^9\)

E. \(\frac{p}{3}\)

7- \(x\) is \(y\%\) of what number?

A. \(\frac{100x}{y}\)

B. \(\frac{100y}{x}\)

C. \(\frac{100y}{x}\)

D. \(\frac{x}{100y}\)

E. \(\frac{xy}{100}\)

8- 120 is equal to?

A. \( 20-(4×10)+(6×30)\)

B. \((\frac{11}{8}×72)+ (\frac{125}{5})\)

C. \(((\frac{30}{4} + \frac{13}{2})×7) – \frac{11}{2}+ \frac{110}{4}\)

D. \((2×10)+(50×1.5)+15\)

E. \(\frac{481}{6}+ \frac{121}{3}\)

9- If the cotangent of an angle β is 1, then the tangent of angle β is?

A. \(-1\)

B. 0

C. 1

D. 2

E. 3

10- 6 liters of water are poured into an aquarium that’s 15cm long, 5cm wide, and 60cm high. How many cm will the water level in the aquarium rise due to this added water? (1 liter of water = 1000 \(cm^3)\)

A. 80

B. 40

C. 20

D. 10

E. 5

11- If a box contains red and blue balls in the ratio of 2 : 3, how many red balls are there if 90 blue balls are in the box?

A. 90

B. 60

C. 30

D. 10

E. 8

12- If |a|<1, then which of the following is true? (b>0)?

\(I. –b < ba < b\)

\(II. -a < a^2 < a \ if \ a<0\)

\(III. -5 < 2a-3 < -1\)

A. I only

B. II only

C. I and III only

D. III only

E. I, II, and III

13- The diameter of a cylinder is 6 inches and its height is 8 inches. What is the surface area of the cylinder?

A. \(48 \ π \in^2\)

B. \(57\ π \ in^2\)

C. \(66\ π \ in^2\)

D. \(288\ π \in^2\)

E.\(400\ π \ in^2\)

14- Five years ago, Amy was three times as old as Mike was. If Mike is 10 years old now, how old is Amy?

A. 4

B. 8

C. 12

D. 14

E. 20

15- A chemical solution contains \(4\%\) alcohol. If there are 24 ml of alcohol, what is the volume of the solution?

A. 240 ml

B. 480 ml

C. 600 ml

D. 1,200 ml

E. 2,400 ml

16- Which of the following points lies on the line \(2x + 4y = 10\)?

A. (2, 1)

B. (–1, 3)

C. (–2, 2)

D. (2, 2)

E. (2, 8)

17- In the following figure, ABCD is a rectangle, and E and F are points on AD and DC, respectively. The area of ∆BED is 16, and the area of ∆BDF is 18. What is the perimeter of the rectangle?

\( \img {https://appmanager.effortlessmath.com/public/images/questions/test37373737373773737373737373.JPG

}\)

A. 20

B. 22

C. 32

D. 40

D. 44

18- What is the solution to the following inequality?

\(|x-10|≤3\)

A. \( x≥13 ∪ x≤7\)

B. \(7≤x≤13\)

C. \( x≥13 \)

D. \( x≤7\)

E. Set of real numbers

19- A bag contains 18 balls: two green, five black, eight blue, a brown, a red, and one white. If 17 balls are removed from the bag at random, what is the probability that a brown ball has been removed?

A. \(\frac{1}{9}\)

B. \(\frac{1}{6}\)

C. \(\frac{16}{17}\)

D. \(\frac{17}{18}\)

E. \(\frac{1}{2}\)

20- If a tree casts a 24–foot shadow at the same time that a 3 feet yardstick casts a 2–foot shadow, what is the height of the tree?

A. 24 ft

B. 28 ft

C. 36 ft

D. 48 ft

E. 52 ft

## Best ACT Math Prep Resource for 2022

## Answers:

1- **A**

The area of trapezoid is:

\((\frac{8+12}{2})×x=100→10x=100→x=10\)

\(y=\sqrt{(3^2+4^2 )}=5\)

\(Perimeter \space is: 12+10+8+5=35\)

2- **D**

\(g(x)=-2, then f(g(x))= f(-2)=2 (-2)^3+5(-2)^2+2(-2)= -16+20-4=0\)

3-** B**

\(x = \frac{8y + \frac{r}{r + 1}}{\frac{6}{\frac{z}{3}}}\)

\(x = \frac{8y + \frac{r}{r + 1}}{\frac{3×6}{z}}\)

\(\frac{1}{3}×\frac{8y +\frac{r}{r + 1}}{\frac{6}{z}}=\frac{x}{3}\)

4- **B**

Use the information provided in the question to draw the shape.

Use Pythagorean Theorem: (a^2 + b^2 = c^2)

\(402 + 302 = c2 ⇒ 1600 + 900 = c2 ⇒ 2500 = c2 ⇒ c = 50\)

\( \img {https://appmanager.effortlessmath.com/public/images/questions/test28282828282828282828282.JPG

} \)

5- **D**

\(2x=x+3y-5\)

\(4x=2y-10\)

\(\Downarrow\)

\(x-3y=-5\)

\(4x-2y=10\)

Multiply first equation by -4.

\(-4x+12y=20\)

\(4x-2y=10\)

add two equations.

\(10y=30→y=3→x=4→ x×y=12\)

6- **C**

To solve for f(3g(P)), first, find 3g(p)

\(g(x)=log_3 x\)

\(g(p)=log_3 p\)

\(3g(p)=3log_3 p=log_3 p^3\)

Now, find f(3g(p)):

\(f(x)=3^x\)

\(f(log_3 p^3 )=3^{log_3 p^3 }\)

Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operations. Then:

\(f(log_3 p^3 )=3^{log_3 p^3 }=p^3\)

7- **A**

Let the number be A. Then:

\(x=y\% ×A\)

Solve for A.

\(x = \frac{y}{100}×A\)

Multiply both sides by 100/y:

\(x×\frac{100}{y}=\frac{y}{100}×\frac{100}{y}×A\)

\(A=\frac{100x}{y}\)

8- **C**

\(((\frac{30}{4} + \frac{13}{2})×7) – \frac{11}{2}+ \frac{110}{4}= ((\frac{30+26}{4})×7)- \frac{11}{2}+ \frac{55}{2}= ((\frac{56}{4})×7)+ \frac{55-11}{2}=(14×7)+\frac{44}{2}=98+22=120\)

9- **C**

tangen β\(=\frac{1}{cotangent \ β}=\frac{1}{1}=1\)

10- **A**

One liter=1000 \(cm^3\)→ 6 liters=6000 \(cm^3\)

\(6000=15×5×h→h=\frac{6000}{75}=80 cm\)

11- **B**

\(2/3×90=60\)

12- **C**

I. |a|<1→-10→-ba^2>a (plug-in -1/2, and check!) (It’s false)

III. -1<a<1,multiply all sides by 2,then: -2<2a<2

Subtract 3 from all sides. Then:

\(-2-3<2a-3<2-3→-5<2a-3<-1 (It \space is \ true!)\)

13- **C**

Surface Area of a cylinder = 2πr (r + h),

The radius of the cylinder is 3 (6 ÷ 2) inches and its height is 8 inches. Therefore,

Surface Area of a cylinder = 2π (3) (3 + 8) = 66 π

14- **E**

Five years ago, Amy was three times as old as Mike. Mike is 10 years now. Therefore, 5 years ago Mike was 5 years.

Five years ago, Amy was: A=3×5=15

Now Amy is 20 years old: 15 + 5 = 20

15- **C**

\(4\%\) of the volume of the solution is alcohol. Let x be the volume of the solution.

Then: \(4\%\) of x = 24 ml ⇒ 0.04 x = 24 ⇒x = 24 ÷ 0.04 = 600

16- **B**

Plug in each pair of number in the equation:

A. (2, 1): 2 (2) + 4 (1) = 8 Nope!

B. (–1, 3): 2 (–1) + 4 (3) = 10 Bingo!

C. (–2, 2): 2 (–2) + 4 (2) = 4 Nope!

D. (2, 2): 2 (2) + 4 (2) = 12 Nope!

E. (2, 8): 2 (2) + 4 (8) = 36 Nope!

17- **D**

The area of ∆BED is 16, then:

\(\frac{4×AB}{2}\)=16→4×AB=32→AB=8

The area of ∆BDF is 18, then:

\(\frac{3×BC}{2}\) =18→3×BC=36→BC=12

The perimeter of the rectangle is = 2×(8+12)=40

18- **B**

\(|x-10|≤3→-3≤x-10≤3→-3+10≤x-10+10≤3+10→7≤x≤13\)

19- **D**

If 17 balls are removed from the bag at random, there will be one ball in the bag. The probability of choosing a brown ball is 1 out of 18. Therefore, the probability of not choosing a brown ball is 17 out of 18 and the probability of having not a brown ball after removing 17 balls is the same.

20- **C**

Write a proportion and solve for \(x\).

\(\frac{3}{2}=\frac{x}{24} ⇒ 2x=3 ×24 ⇒ x=36 ft\)

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