Reasonable Rates

Common Easy Math ACT practice test — Problems Part 1

This comprehensive guide helps students master the Math ACT practice test with proven strategies and clear step-by-step solutions. Whether you’re aiming for a perfect score or looking to improve your performance, this resource focuses on advanced techniques, real ACT math practice questions, and expert insights.

Unlike other resources, this guide is built specifically for high-achieving students who want to tackle hard ACT math questions with confidence. While the guide is still under development, it already provides valuable content covering the most challenging topics in the ACT math test, helping students strengthen their problem-solving skills and boost their scores.

Let’s start .

1. What is the difference between the mean and the median of the squares of the integers from 3 to 6? 3

$ 0 $
$ 0.5 $
$ 0.75 $
$ 1 $

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D

2. What is 8% of $3.46\times 10^6$? 3

$ 276 $
$ 2,768 $
$ 27,680 $
$ 276,800 $

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D

3. What number is halfway between $\dfrac{3}{5}$ and $\dfrac{2}{3}$? 3

$ \dfrac{5}{16} $
$ \dfrac{19}{30} $
$ \dfrac{5}{8} $
$ \dfrac{7}{10} $

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B

4. Point $M$ is the midpoint of $AB$. $A$ is $(4,-2)$ and $M$ is $(-3,-4)$. What are the coordinates of point $B$? 3

$ \left(\dfrac{1}{2},-3\right) $
$ (11,0) $
$ (-9,-7) $
$ (-10,-6) $

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D

5. In the standard $xy$-coordinate plane, what is the midpoint of the line segment between $(-5,-2)$ and $(-1,7)$? 3

$ (-9, -3)$
$ \left(-3, \dfrac{5}{2}\right)$
$ (3, 16) $
$ \left(\dfrac{5}{2}, 2\right) $

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B

6. What is the distance between $(-2,3)$ and $(4,10)$? 3

$ 9 $
$\dfrac{25}{3} $
$ \sqrt{85} $
$\sqrt{87} $

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C

7.Bob and Julio were painting a house. They started with 8 gallons of paint. On the first day Bob used $1\dfrac{7}{8}$ gallons and Julio used $1\dfrac{3}{4}$ gallons. How many gallons were left? 3

$ 3\dfrac{3}{8} $
$ 4\dfrac{3}{8}$
$ 5\dfrac{3}{8} $
$ 4\dfrac{5}{8}$

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B

8. If $f(x)={(3x+4)}^2$, what is $f(1)$? 2

$ 7 $
$ 14 $
$ 16 $
$ 49 $

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D

9. If $a=6$, $b=2$, and $c=5$, what is\\ $(a+b-c)(b-c)$? 2

$ -39 $
$ 19-9 $
$ 0 $
$ 21 $

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B

10. If $f(x,y)=4xy^2-x^2$, what is $f(2,5)$? 2

$ 36 $
$ 76 $
$ 196 $
$ 200 $

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C

11. What is the least common denominator of $\dfrac{5}{8}$, $\dfrac{1}{12}$, and $\dfrac{3}{10}$? 3

$24 $
$ 60 $
$ 120 $
$ 240 $

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C

12. One sign flashes every 9 seconds and another every 12 seconds. At a certain instant, they flash at the same time. How many seconds until they flash at the same time again? 3

$ 24$
$ 36 $
$ 48 $
$ 72 $

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B

13. What is the least common multiple of 40, 60, and 80? 3

$ 120 $
$ 240 $
$ 720 $
$ 1440 $

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B

14. Which of the following expressions is equivalent to $(2x+7)(5x-2)$? 2

$ 2x^2+31x-14 $
$ 5x^2+31x-14 $
$ 10x^2+31x-14 $
$ 10x^2+31x+14 $

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C

15. Which of the following expressions is equivalent to $6(x+3)-4(2x-5)$? 1

$ -22x+28 $
$ -2x+38 $
$ 2x-2 $
$ 10x-2 $

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B

16. Which of the following expressions is equivalent to $(2x^4y^5)(4x^2y^3)$? 2

$4x^4y^5 $
$6x^6y^8 $
$ 8x^6y^8 $
$ x^2-5x+30 = 0 $
$ 6x^8y^{15} $

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C

17. Which of the following is equivalent to $(x^3)^{18}$? 2

$x^{21} $
$ x^{24} $
$ x^{36} $
$ x^{54} $

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D

18. What are the vertices of the ellipse $\dfrac{(x + 5)^2}{25} + \dfrac{(y - 8)^2}{49} = 1$?2

$ 9x^6$
$ 9x^5 $
$ 27x^5$
$ 27x^6 $

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D

19. Which of the following is equivalent to $(5x^3)\cdot(4x^8)$?2

$ 5x^{11/4}$
$ 20x^{11} $
$ 20x^{24}$
$ 100x^{11} $

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B

20. A train left Atlanta at 8:54 PM and arrived in New York at 10:12 AM the next day in the same time zone. How many hours and minutes did the trip take?4

12 hours and 18 minutes
13 hours and 18 minutes
13 hours and 22 minutes
13 hours and 28 minutes

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B

21. What is the smallest positive integer greater than $\sqrt{210}$?2

$ 13 $
$ 14 $
$ 15 $
$ 16 $

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C

22. If $8x-7=3x+8$, then $x=$? 1

$ \dfrac{1}{5} $
$ \dfrac{15}{11} $
$ \dfrac{8}{5} $
$ 3 $

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D

23. If $\dfrac{3x}{4}-2=13$, then $x=?$3

$ \dfrac{33}{4} $
$ \dfrac{45}{4} $
$ \dfrac{44}{3} $
$ 20 $

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D

24. Which of the following inequalities is equivalent to $6x-5>2x+7$? 3

$ x>\dfrac{1}{2}$
$ x>\dfrac{3}{2} $
$ x<3 $
$ x>3$

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D

25. A board which is 9 feet 4 inches long is cut into two equal parts. How long is each part in feet and inches? 3

4 feet 2 inches
4 feet 4 inches
4 feet 8 inches
4 feet 9 inches

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C

26. In the standard $xy$-coordinate plane, what is the slope of a line containing $(3,2)$ with $y$-intercept of 4? 3

$ -2 $
$ \dfrac{-3}{2} $
$ \dfrac{-2}{3} $
$ \dfrac{2}{3} $

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C

27. In the standard $xy$-coordinate plane, what is the slope of the line $4x+7y=-11$? 3

$ \dfrac{-7}{4} $
$ \dfrac{-11}{7} $
$ \dfrac{-4}{7} $
$ \dfrac{4}{7} $

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C

28. What is the slope of a line through $(-4,-2)$ and $(-1,5)$ in the standard $xy$-coordinate plane? 3

$ \dfrac{3}{7} $
$ 1 $
$ \dfrac{3}{2} $
$ \dfrac{7}{3} $

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D

29. What is the slope of $y-11=\dfrac{2}{3}(5x+4)$? 3

$ \dfrac{3}{10} $
$ \dfrac{8}{3} $
$ \dfrac{10}{3} $
$ 10 $

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C

Answer key: 1.D, 2.D, 3.B, 4.D, 5.B, 6.C, 7.B, 8.D, 9.B, 10.C, 11.C, 12.B, 13.B, 14.C, 15.B, 16.C, 17.D, 18.D, 19.B, 20.B, 21.C, 22.D, 23.D, 24.D, 25.C, 26.C, 27.C, 28.D, 29.C.

Solutions

1. (D) The squares of the integers from 3 to 6 are 9, 16, 25, and 36. Calculate the mean: $ \dfrac{9+16+25+36}{4} = \dfrac{86}{4} = 21.5. $ Calculate the median by averaging the two middle terms: $\dfrac{16+25}{2} = 20.5$. So, the difference is $21.5 - 20.5 = 1$.

2. (D) Convert 8% to a decimal and multiply: $ 0.08 \cdot 3.46 \times 10^6 = 0.2768 \times 10^6 = 276,800. $

3. (B) Calculate the average of the two fractions: $ \dfrac{\dfrac{3}{5} + \dfrac{2}{3}}{2} = \dfrac{\dfrac{19}{15}}{2} = \dfrac{19}{30}. $

4. (D) Substitute the known coordinates into the midpoint formula $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$. Set the average of the $x$-coordinates equal to $-3$: $$ -3=\dfrac{4+x}{2}\implies -6=4+x \implies x=-10. $$ Set the average of the $y$-coordinates equal to $-4$: $$ -4=\dfrac{-2+y}{2}\implies -8=-2+y\implies y=-6. $$ Thus, point $B$ has coordinates $(-10,-6)$.Alternatively, determine the change in values from $A$ to $M$. From $A$ to $M$, the $x$-value decreases by 7 and the $y$-value decreases by 2. Apply these changes to $M$ to find $B$: $$ B(-3-7,-4-2)=(-10,-6). $$

5. (B) Apply the midpoint formula, $\left( \dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$, to the endpoints $(-5,-2)$ and $(-1,7)$: $$ \left( \dfrac{-5+(-1)}{2},\dfrac{-2+7}{2}\right)=\left( \dfrac{-6}{2},\dfrac{5}{2}\right)=\left(-3,\dfrac{5}{2}\right). $$ If the formula is unfamiliar, understand or derive it by taking the average of the $x$-coordinates and the average of the $y$-coordinates.

6. (C) Apply the distance formula, $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$: $$ \sqrt{(4-(-2))^2+(10-3)^2}=\sqrt{6^2+7^2}=\sqrt{36+49}=\sqrt{85}. $$ The formula derives from the Pythagorean theorem. If the formula is forgotten, apply the Pythagorean theorem directly by using the difference in $x$-coordinates and the difference in $y$-coordinates as the legs of a right triangle. This pattern is common on the ACT; while memorizing the formula is most efficient, the answer may also be reasoned or the formula derived during the exam.

7. (B) First, determine the total amount of paint used by summing the amounts used by Bob and Julio: $$ 1\dfrac{7}{8}+1\dfrac{3}{4}=1\dfrac{7}{8}+1\dfrac{6}{8}=2\dfrac{13}{8}=3\dfrac{5}{8}. $$ Next, subtract this amount from the initial 8 gallons to find the remaining paint: $$ 8-3\dfrac{5}{8}=7\dfrac{8}{8}-3\dfrac{5}{8}=4\dfrac{3}{8}. $$

8. (D) Substitute $1$ for $x$ in the function: $$ f(1)={(3(1)+4)}^2=7^2=49. $$

9. (B) Substitute the given values $a=6$, $b=2$, and $c=5$ into the expression: $$ (6+2-5)(2-5)=(3)(-3)=-9. $$

10. (C) Substitute $x=2$ and $y=5$ into the function: $$ f(2,5)=4(2)(5)^2-2^2=8(25)-4=200-4=196. $$

11. (C) Find the least common multiple (LCM) of the denominators 8, 12, and 10 by prime factorizing each denominator: $$ 8=2^3, \quad 12=2^2 \cdot 3, \quad 10=2 \cdot 5. $$ Then, select the highest power of each prime factor present to calculate the LCM: $2^3\cdot 3\cdot 5=120.$

12. (B) Find the least common multiple (LCM) of 9 and 12 by prime factorizing each number: $$ 9=3^2, \quad 12=2^2 \cdot 3. $$ Then, select the highest power of each prime factor present to calculate the LCM: $2^2\cdot 3^2=36.$

13. (B) Find the least common multiple (LCM) of 40, 60, and 80 by prime factorizing each number: $$ 40=2^3\cdot 5, \quad 60=2^2\cdot 3\cdot 5, \quad 80=2^4\cdot 5. $$ Then, select the highest power of each prime factor present to calculate the LCM: $2^4\cdot 3\cdot 5=240.$Alternatively, find the LCM of 4, 6, and 8, which is 24, and multiply the result by 10.

14. (C) Expand the expression by distributing the terms: $$ (2x+7)(5x-2) = 10x^2-4x+35x-14 = 10x^2+31x-14. $$

15. (B) Expand the expression by distributing the terms and combining like terms: $$ 6(x+3)-4(2x-5) = 6x+18-8x+20 = -2x+38. $$

16. (C) Multiply the coefficients and combine the variables by adding their exponents: $$ (2x^4y^5)(4x^2y^3) = (2\cdot 4)(x^{4+2})(y^{5+3}) = 8x^6y^8. $$

17. (D) Apply the power of a power property by multiplying the exponents: $$ (x^3)^{18}=x^{3\cdot 18}=x^{54}. $$

18. (D) Raise the coefficient to the third power and apply the power of a power property to the variable: $$ (3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^{2\cdot 3} = 27x^6. $$

19. (B) Multiply the coefficients and combine the variables by adding their exponents: $$ (5x^3)(4x^8) = (5\cdot 4)(x^{3+8}) = 20x^{11}. $$

20. (B) Break the trip into hour and minute intervals. First, calculate the time from 8:54 PM to 9:54 AM the next day: $$ 8:54 \text{ PM} \to 9:54 \text{ AM} = 13 \text{ hours}. $$ Next, calculate the remaining minutes from 9:54 AM to 10:12 AM: $$ 10:12 \text{ AM} - 9:54 \text{ AM} = 18 \text{ minutes}. $$ Thus, the total duration is 13 hours and 18 minutes.

21. (C) Estimate the value of $\sqrt{210}$ to determine its location between integers:$\sqrt{210} \approx 14.49$The smallest integer greater than 14.49 is 15.Avoid the common error of simply rounding to 14.$14^2 = 196$, $15^2 = 225$So $14 < \sqrt{210} < 15$.

22. (D) Group like terms to isolate $x$ : $$ 8x-7=3x+8 \implies 8x-3x=8+7 \implies 5x=15 \implies x=3. $$

23. (D) Isolate the variable term by adding 2 to both sides, then multiply by the reciprocal of the fraction: $$ \dfrac{3x}{4}-2=13 \implies \dfrac{3x}{4}=15 \implies x=15 \left(\dfrac{4}{3}\right) = 20. $$

24. (D) Group like terms to isolate $x$: $$ 6x-5>2x+7 \implies 6x-2x>7+5 \implies 4x>12 \implies x>3. $$

25. (C) Divide the length into two equal parts. First, halve the feet and inches separately: $$ \frac{1}{2}(9 \text{ feet}) = 4 \text{ feet } 6 \text{ inches}, \quad \frac{1}{2}(4 \text{ inches}) = 2 \text{ inches}. $$ Combine the results to find the length of each part: $4 \text{ feet } 6 \text{ inches} + 2 \text{ inches} = 4 \text{ feet } 8 \text{ inches}.$ Alternatively, convert the total length to inches, divide by 2, and convert back: $ 9 \text{ feet } 4 \text{ inches} = (9 \times 12) + 4 = 112 \text{ inches} $ $$\frac{112 \text{ inches}}{2} = 56 \text{ inches} = 4 \text{ feet } 8 \text{ inches}. $$

26.(C) Identify the coordinates of the $y$-intercept as $(0,4)$. Then, calculate the slope between $(3,2)$ and $(0,4)$ using the slope formula $m = \dfrac{y_2-y_1}{x_2-x_1}$: $$ m = \dfrac{4-2}{0-3}=\dfrac{2}{-3}=-\dfrac{2}{3}. $$

27. (C) Rearrange the equation into slope-intercept form ($y=mx+b$) by solving for $y$: $$ 4x+7y=-11 \implies 7y=-4x-11 \implies y=-\dfrac{4}{7}x-\dfrac{11}{7}. $$ Identify the slope $m$ as the coefficient of $x$: $m = -\dfrac{4}{7}.$

28. (D) Apply the slope formula $m = \dfrac{y_2-y_1}{x_2-x_1}$ to the points $(-4,-2)$ and $(-1,5)$: $$ m = \dfrac{5-(-2)}{-1-(-4)} = \dfrac{7}{3}. $$

29. (C) Distribute the fraction $\dfrac{2}{3}$ on the right side to identify the coefficient of $x$: $$ y-11 = \dfrac{2}{3}(5x+4) \implies y-11 = \dfrac{10}{3}x + \dfrac{8}{3}. $$ The slope is the coefficient of the $x$ term: $m = \dfrac{10}{3}.$

Difficulty Key

1 Easiest
2 – 4 Intermediate
5 Most Difficult

Linear Equations in One Variable

In addition to basic linear equations, these problems often involve combining fractions using least common denominators and isolating the variable to find the solution.

A typical problem asks what value must be added to both the numerator and the denominator of $\dfrac{2}{7}$ to obtain $\dfrac{3}{4}$. We write: \begin{gather*} \dfrac{2+x}{7+x}=\dfrac{3}{4}, \end{gather*} Cross multiplying yields \begin{align*} 4(2+x) & = 3(7+x) \implies \\ 8+4x & = 21+3x \implies \\ x & = 13. \end{align*}