Common Easy ACT Math Problems Practice

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Common Mostly Easy ACT math problems practice — Part 2

This comprehensive guide is designed to help students master the hardest ACT math problems using proven strategies and step-by-step explanations. 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. Which of the following lines is perpendicular to $5x+4y=7$?3

$ -5x-4y=3 $
$ -5x+4y=3 $
$ 4x-5y=9 $
$ 4x+5y=12 $

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C

2. What is the $342^{nd}$ digit to the right of the decimal point in $.\overline{8245}$?4

$ 2 $
$ 3 $
$ 4 $
$ 5 $

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A

3. If $\dfrac{2x-y}{x+y}=\dfrac{3}{5}$, what does $\dfrac{x}{y}$ equal?4

$ \dfrac{-1}{2}$
$ \dfrac{1}{2}$
$ \dfrac{7}{8} $
$ \dfrac{8}{7} $

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D

4. If $\dfrac{1}{4}$ inch represents 30 miles, how far apart are two towns which are $\dfrac{7}{8}$ inch apart on the map?3

$ 35 $
$ 100 $
$ 105 $
$ 210 $

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C

5. What is $|3-x|$ when $x=11$?2

$ -8 $
$ 7 $
$ 8 $
$ 14 $

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C

6. What is $|-18|-|17-33|$?2

$ -2 $
$2 $
$ 18 $
$ 34$

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B

7. What is $|15-3|-|2-9|$?2

$ -1 $
$ 1 $
$ 5 $
$ 15 $

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C

8. What is the sum of the solutions of $x^2-5x-18=0$? 3

$ -5 $
$ \dfrac{5}{2} $
$ \dfrac{18}{5} $
$ 5 $

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D

9. Which of the following is a factor of $x^2+4x-21$? 3

$x-7 $
$x-4 $
$ x-3 $
$ x+3 $

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C

10. What is the area of a circle with circumference $8\sqrt{5}\pi$? 3

$ 40\pi $
$ 60\pi $
$ 72\pi $
$ 80\pi $

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D

11. A square has sides 24. A rectangle with the same area has a width 8. What is the length of that rectangle?2

$ 24 $
$ 48 $
$ 72 $
$ 81 $

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C

12. Two adjacent sides of a rectangle have lengths $2x-5$ and $x+3$. What is the area of that rectangle?2

$x^2+x+15$
$ x^2+9x-15 $
$ x^2+9x+15 $
$ 2x^2+x-15 $

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D

13. A square and a rectangle have the same area. The length of the rectangle is 75 and its width is 12. What is the side length of the square?3

$ 24 $
$ 30 $
$ 32 $
$ 36 $

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B

14. A triangle has sides of length $x$, $2x-1$, and $3x-7$ and a perimeter of 34. What is its longest side length? 3

$ 7 $
$ 13 $
$14 $
$ 15 $

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C

15. In $\triangle ABC$, $\angle A$ is a right angle, $AB=3$ and $AC=2$. What is $\sin\angle B$?3

$ \dfrac{2}{5} $
$ \dfrac{2}{\sqrt{13}} $
$ \dfrac{3}{\sqrt{13}} $
$ \dfrac{2}{3} $

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B

16. For an angle with measure $\theta$ in a right triangle, $\sin \theta=\dfrac{5}{7}$. What is $\tan\theta$?3

$ \dfrac{5}{2\sqrt{6}} $
$ \dfrac{5}{\sqrt{74}}$
$ \dfrac{5}{4\sqrt{6}} $
$ \dfrac{7}{5} $

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A

17. If $\sin x=\dfrac{7}{25}$ and $\tan x=\dfrac{7}{24}$, what is $\cos x$?3

$ \dfrac{7}{26} $
$ \dfrac{17}{24} $
$ \dfrac{18}{25} $
$ \dfrac{24}{25} $

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D

18. What is the amplitude of the function $y=3\sin\left(\dfrac{x}{2}+\dfrac{\pi}{8}\right)$?3

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

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D

19. In triangle $ABC$, what is $\cos\angle A$? 3

$ \dfrac{3}{5} $
$ \dfrac{\sqrt{7}}{4}$
$ \dfrac{3}{4} $
$ \dfrac{4}{5} $

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B

20. For $i=\sqrt{-1}$, $(2+5i)^2$ equals which of the following?3

$ -1 $
$ 4+45i $
$ 20i-21 $
$ 20i-29 $

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C

21. When $a$ is rational and $i=\sqrt{-1}$, the product of $3a-2i$ and which of the following numbers must be a rational number?2

$ i $
$ 3a-2i $
$ 3a+2i $
$ 3a-i $

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C

22. The first 5 terms of a geometric sequence are 6, $-12$, 24, $-48$ and 96. What is the $6^{th}$ term?3

$ -192 $
$ -144 $
$-96 $
$ 192 $

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A

23. There are 9 players in a baseball starting lineup. The center fielder will bat first, the shortstop will bat $8^{th}$, and the catcher will bat last. How many different possible lineups of those 9 players are there?3

$ 21 $
$ 120 $
$ 720 $
$ 2520 $

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C

24. Given the functions $f(x)=5x+3$ and $g(x)=x^2-7$, what is $f(g(-3))$?3

$ -77 $
$ -12 $
$ 3 $
$ 13 $

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D

25. If $f(x)=3x-2$ and $g(x)=x^2+5$, what is $f(g(x))$?3

$ x^2-3 $
$ 3x^2+13 $
$ 3x^2-13 $
$ 3x^2+17$

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B

26. Which of the following matrices is equivalent to $\left[ \begin{array}{cc} -3 & 2 \\ 5 & -4 \end{array}\right]+\left[ \begin{array}{cc} 3 & 5 \\ 6 & -7 \end{array}\right]$?

$ \left[ \begin{array}{cc} -9 & 10 \\ 30 & 28 \end{array}\right] $
$ \left[ \begin{array}{cc} -6 & -3 \\ -1 & 3 \end{array}\right] $
$ \left[ \begin{array}{cc} 0 & 7 \\ 11 & -11 \end{array}\right] $
$ \left[ \begin{array} {cc} 0 & 7 \\ 16 & -14 \end{array}\right] $

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C

27. Which of the following is the result of reflecting the point $(-7,3)$ about the $y$-axis?3

$ (-14,6) $
$ (-7,-3) $
$ (-7,3) $
$ (7,3) $

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D

28. The point $(7,-5)$ is translated right 3 units and down 8 units. What are the coordinates of the new point?3

$ (-1,-2) $
$ (-1,2) $
$ (7,-13) $
$ (10,-13) $

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D

29. Which of the following statements is equivalent to "All cats have fur"?

"If it does not have fur, then it is not a cat"
"If it does not have fur, then it is a cat"
"If it has fur, then it is not a cat"
"If it has fur, then it is a cat"

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A

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

Solutions

1. (C) Find the slope of the given line $5x+4y=7$ by rearranging it into slope-intercept form: $$ 5x+4y=7 \implies 4y=-5x+7 \implies y=-\dfrac{5}{4}x+\dfrac{7}{4}. $$ The slope of the given line is $-\dfrac{5}{4}$. A perpendicular line must have a slope that is the negative reciprocal, which is $\dfrac{4}{5}$. Rearrange option (C) to verify its slope: $$ 4x-5y=9 \implies -5y=-4x+9 \implies y=\dfrac{4}{5}x-\dfrac{9}{5}. $$ Since the slope is $\dfrac{4}{5}$, this line is perpendicular to the given line.

2. (A) Determine the length of the repeating block $8245$, which is 4. Divide the position number 342 by 4 to find the remainder: $342 \div 4 = 85 \text{ R } 2.$ Since the remainder is 2, the $342^{nd}$ digit corresponds to the $2^{nd}$ digit of the sequence. Thus, the $342^{nd}$ digit in $8245$ is 2.

3. (D) Cross-multiply to eliminate the fractions and distribute the constants, then group the $x$ terms on one side and the $y$ terms on the other to solve for the ratio $\dfrac{x}{y}$: $$ 5(2x-y)=3(x+y) \implies 10x-3x=3y+5y \implies 7x=8y \implies \dfrac{x}{y}=\dfrac{8}{7}. $$

4. (C) Set up a proportion relating the map distance to the actual distance, then Solve for $x$: $$ \dfrac{1/4}{30} = \dfrac{7/8}{x} \implies x = 30 \cdot \dfrac{7/8}{1/4} = 30 \cdot \left(\dfrac{7}{8} \cdot \dfrac{4}{1}\right) = 30 \cdot \dfrac{7}{2} = 105 \text{ miles}. $$

5. (C) Substituting $x=11$ into the expression yields: $|3-x| = |3-11| = |-8| = 8.$

6. (B) Evaluate each absolute value term separately, then find the difference: $$ |-18|-|17-33| = 18 - |-16| = 18 - 16 = 2. $$

7. (C) Evaluate each absolute value term separately, then find the difference: $$ |15-3|-|2-9| = |12| - |-7| = 12 - 7 = 5. $$

8. (D) The formula for the sum of the solutions is $\dfrac{-b}{a}$. With $a=1$ and $b=-5$: $\text{Sum} = \dfrac{-(-5)}{1} = 5.$ Alternatively, you can find the individual solutions using the quadratic formula and add them.

9. (C) Factor the expression by finding two numbers that multiply to $-21$ and add to $4$. These numbers are $7$ and $-3$: $x^2+4x-21 = (x+7)(x-3)$. Therefore, $x-3$ is a factor.Alternatively, you can find the roots using the quadratic formula, or plug the values derived from the answer choices into the expression to see which equals zero.

10. (D) Set the circumference formula $2\pi r$ equal to the given value to solve for the radius $r$: $$ 2\pi r = 8\sqrt{5}\pi \implies r = 4\sqrt{5}. $$ Substitute the radius into the area formula $A = \pi r^2$: $A = \pi(4\sqrt{5})^2 = \pi(16 \cdot 5) = 80\pi.$

11. Calculate the area of the square: $24^2=576$. Set the area of the rectangle ($A=lw$) equal to 576 and solve for the length $l$ given the width $w=8$: $$ 576 = l \cdot 8 \implies l = \dfrac{576}{8} = 72. $$

12. (D) Calculate the area by multiplying the adjacent side lengths (the length and width): $$ \text{Area} = (2x-5)(x+3) = 2x^2+6x-5x-15 = 2x^2+x-15. $$

13. (B) The area of the rectangle is $l\cdot w=75\cdot 12=900$. The side length of the square is $\sqrt{900}=30$.

14. (C) Set the sum of the side lengths equal to the perimeter and solve for $x$: $$ x + (2x-1) + (3x-7) = 34 \implies 6x - 8 = 34 \implies 6x = 42 \implies x = 7. $$ Substitute $x=7$ into the expressions for the sides to find their lengths: $7$, $2(7)-1=13$, and $3(7)-7=14$. The longest side is $14$.

15. (B) Find the length of the hypotenuse $BC$ using the Pythagorean theorem: $BC = \sqrt{3^2 + 2^2} = \sqrt{9+4} = \sqrt{13}$. Then, determine $\sin \angle B$ using the ratio of the opposite side to the hypotenuse: $\sin \angle B = \dfrac{AC}{BC} = \dfrac{2}{\sqrt{13}}.$

16. (A) Identify the sides of the right triangle: opposite $= 5$ and hypotenuse $= 7$. Use the Pythagorean theorem to find the adjacent side: $$ \text{adjacent} = \sqrt{7^2 - 5^2} = \sqrt{49 - 25} = \sqrt{24} = 2\sqrt{6}. $$ Then, find the tangent ratio: $$ \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{5}{2\sqrt{6}}. $$ Alternatively, use the identity $1+\cot^2\theta = \csc^2\theta$. Since $\csc \theta = \frac{7}{5}$: $$ 1+\cot^2 \theta = \left(\dfrac{7}{5}\right)^2 \implies \cot^2 \theta = \dfrac{49}{25} - 1 = \dfrac{24}{25} \implies \cot\theta = \dfrac{2\sqrt{6}}{5}. $$ Taking the reciprocal gives $\tan\theta = \dfrac{5}{2\sqrt{6}}$. You can also calculate $\sin^{-1}(5/7)$ and find the tangent of that angle.

17. (D) Apply the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, then solve for $\cos x$ by isolating the squared term: $$ \left(\dfrac{7}{25}\right)^2 + \cos^2 x = 1 \implies \dfrac{49}{625} + \cos^2 x = 1. \implies \cos^2 x = 1 - \dfrac{49}{625} = \dfrac{576}{625} \implies \cos x = \dfrac{24}{25}. $$ Alternatively, use a calculator to find $x = \sin^{-1}(7/25)$ and then compute $\cos x$, or draw a right triangle to determine the adjacent side using the Pythagorean theorem.

18. (D) The amplitude corresponds to the absolute value of the coefficient $A$ in the standard form $y = A\sin(Bx+C)$. Here, $A=3$, thus $\text{Amplitude} = |3| = 3$.

19. (B) By the Pythagorean theorem, the adjacent $AC = \sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7}$. Thus, $$ \cos \angle A = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{AC}{AB} = \dfrac{\sqrt{7}}{4}. $$

20. (C) Expand the binomial and substitute $i^2 = -1$: $$ (2+5i)^2 = 4 + 20i + 25i^2 = 4 + 20i - 25 = 20i - 21. $$

21. (C) Multiplying $3a-2i$ by the complex conjugate, $3a+2i$, eliminates the imaginary unit: $$ (3a-2i)(3a+2i) = (3a)^2 - (2i)^2 = 9a^2 - 4(-1) = 9a^2 + 4. $$ Since $a$ is rational, the result $9a^2+4$ is a rational number.

22. (A) The sequence is geometric with a common ratio of $r = \dfrac{-12}{6} = -2$. Multiply the fifth term by the ratio to find the sixth term: $96 \cdot (-2) = -192$.

23. (C) Three positions are fixed (1st, 8th, and 9th), so we only need to arrange the remaining $9-3=6$ players. The number of possible lineups is the factorial of $6$: $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$.

24. (D) First, evaluate the inner function $g(x)$ at $x=-3$: $g(-3) = (-3)^2 - 7 = 9 - 7 = 2$. Next, substitute this result into the outer function $f(x)$: $f(g(-3)) = f(2) = 5(2) + 3 = 10 + 3 = 13$.Alternatively, you could determine the composite function $f(g(x))$ first and then substitute $x=-3$, though evaluating the numbers directly is often faster.

25. (B) Substitute the expression for $g(x)$ into $f(x)$, then simplify: $$ f(g(x)) = 3(x^2+5)-2 = 3x^2 + 15 - 2 = 3x^2 + 13. $$

26. (C) Add the corresponding elements of the two matrices: $$ \begin{bmatrix} -3 & 2 \\ 5 & -4 \end{bmatrix} + \begin{bmatrix} 3 & 5 \\ 6 & -7 \end{bmatrix} = \begin{bmatrix} -3+3 & 2+5 \\ 5+6 & -4-7 \end{bmatrix} = \begin{bmatrix} 0 & 7 \\ 11 & -11 \end{bmatrix}. $$

27. (D) To reflect a point about the $y$-axis, negate the $x$-coordinate while keeping the $y$-coordinate unchanged: $(-x, y) \implies (-(-7), 3) = (7, 3)$.

28. (D) Apply the translation to the coordinates by adding 3 to the $x$-coordinate (right 3) and subtracting 8 from the $y$-coordinate (down 8): $(7+3, -5-8) = (10, -13).$

29. (A) The contrapositive is logically equivalent to the original conditional statement. Form the contrapositive by negating both the hypothesis and the conclusion and reversing their order: "If it does not have fur, then it is not a cat."

Difficulty Key

1 Easiest
2 – 4 Intermediate
5 Most Difficult