This practice test contains 30 questions designed to simulate the format and difficulty of the ACT Math section. Per recent test trends, several questions have been adapted to include wordier modeling contexts, challenging you to extract mathematical relationships from real-world scenarios. Detailed solutions are provided for every question.
What is the greatest common factor of 45, 135, and 270?
A) 5
B) 9
C) 15
D) 25
E) 45
Correct Answer: E
Use prime factorization:
\[ 45 = 1 \times 3^2 \times 5 \]
\[ 135 = 1 \times 3^3 \times 5 \]
\[ 270 = 1 \times 2 \times 3^3 \times 5 \]
The GCF \( = 5 \times 3^2 = 45 \)
What is the value of \( 2x + \frac{1}{2} \) when \( 5x - 3 = -3x + 5 \)?
A) \( -\frac{1}{2} \)
B) \( -\frac{1}{4} \)
C) 1
D) \( 2\frac{1}{2} \)
E) Value cannot be calculated
Correct Answer: D
Solve the equation for \(x\):
\[ 5x + 3x = 5 + 3 \Rightarrow 8x = 8 \Rightarrow x = 1 \]
Now calculate the target expression:
\[ 2x + \frac{1}{2} = 2(1) + \frac{1}{2} = 2\frac{1}{2} \]
An architect is designing a rectangular floor plan for a new library reading room. To meet local building codes, the width \( W \) of the room must be exactly 2 feet less than half of its length \( L \). Which of the following expressions represents the perimeter \( P \) of the reading room in terms of \( L \)?
A) \( 3L - 4 \)
B) \( 4L - 4 \)
C) \( 4L \)
D) \( 3L - 2 \)
E) \( 2L - 2 \)
Correct Answer: A
First, translate the verbal constraint into an equation to express width in terms of length:
\[ W = \frac{L}{2} - 2 \]
Substitute this into the standard perimeter formula \( P = 2(W + L) \):
\[ P = 2\left(\frac{L}{2} - 2 + L\right) = L - 4 + 2L = 3L - 4 \]
Which is a factor of \( -2x^2 + 7x - 6 \)?
A) \( -2x - 3 \)
B) \( 2x + 2 \)
C) \( x - 6 \)
D) \( 2x - 2 \)
E) \( -2x + 3 \)
Correct Answer: E
Factor the polynomial by grouping or trial and error:
\[ -2x^2 + 7x - 6 = (-2x + 3)(x - 2) \]
Thus \( -2x + 3 \) is a factor.
If \( \frac{5}{x} = 10 \) and \( \frac{2}{y} = 6 \), then \( \frac{x}{y} = \)?
A) \( \frac{5}{3} \)
B) \( \frac{3}{2} \)
C) \( \frac{3}{5} \)
D) \( \frac{2}{3} \)
E) \( \frac{5}{2} \)
Correct Answer: B
From the first equation: \( \frac{5}{x} = 10 \Rightarrow x = \frac{5}{10} = \frac{1}{2} \)
From the second equation: \( \frac{2}{y} = 6 \Rightarrow y = \frac{2}{6} = \frac{1}{3} \)
Therefore: \( \frac{x}{y} = \frac{1/2}{1/3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} \)
Evaluate: \( \frac{1}{(-5)^2} \)
A) \( -\frac{1}{25} \)
B) \( \frac{1}{25} \)
C) 25
D) -25
E) \( \frac{1}{10} \)
Correct Answer: B
Square the term inside the parentheses first. A negative number squared yields a positive result:
\[ (-5)^2 = 25 \]
\[ \frac{1}{(-5)^2} = \frac{1}{25} \]
The points \( (-4, -4) \), \( (-1, -2) \), and \( (x, -8) \) are vertices of a right triangle with the right angle at \( (-1, -2) \). Find \( x \).
A) 0
B) 2
C) 3
D) -1
E) -4
Correct Answer: C
Let the points be \( A(-4, -4) \), \( B(-1, -2) \), and \( C(x, -8) \). Since the right angle is at \( B \), segments \( AB \) and \( BC \) are perpendicular.
The slope of \( AB \) is \( \frac{-2 - (-4)}{-1 - (-4)} = \frac{2}{3} \).
The slope of \( BC \) must be the negative reciprocal, \( -\frac{3}{2} \):
\[ \frac{-8 - (-2)}{x - (-1)} = -\frac{3}{2} \Rightarrow \frac{-6}{x + 1} = -\frac{3}{2} \]
Cross-multiply: \( -12 = -3(x + 1) \Rightarrow 4 = x + 1 \Rightarrow x = 3 \)
If \( (x + y)^2 = 144 \) and \( x^2 - y^2 = 24 \), and \( x, y > 0 \), what is \( x \)?
A) 7
B) 5
C) 12
D) 2
E) 8
Correct Answer: A
From the first equation, take the positive square root since \( x, y > 0 \):
\[ x + y = 12 \]
Factor the second equation using difference of squares:
\[ (x + y)(x - y) = 24 \]
Substitute \( x + y = 12 \) into the factored equation:
\[ 12(x - y) = 24 \Rightarrow x - y = 2 \]
Now, add the system of linear equations: \( (x + y) + (x - y) = 12 + 2 \Rightarrow 2x = 14 \Rightarrow x = 7 \)
What is the slope of any line perpendicular to \( -5x + 3y = 9 \)?
A) \( -\frac{3}{5} \)
B) \( \frac{5}{3} \)
C) \( \frac{3}{5} \)
D) \( -\frac{5}{3} \)
E) \( -\frac{1}{5} \)
Correct Answer: A
Find the slope of the given line by converting it to slope-intercept form (\(y = mx + b\)):
\[ 3y = 5x + 9 \Rightarrow y = \frac{5}{3}x + 3 \]
The slope is \( \frac{5}{3} \). A perpendicular line must have a negative reciprocal slope, which is \( -\frac{3}{5} \).
Simplify: \( \sqrt{(-9)(-4)} + \sqrt{-4} \)
A) \( 6 + 2i \)
B) \( 6 - 2i \)
C) 8
D) 4
E) \( 2 + 6i \)
Correct Answer: A
Evaluate the terms independently, keeping track of negative signs inside and outside radicals:
\[ \sqrt{(-9)(-4)} = \sqrt{36} = 6 \]
\[ \sqrt{-4} = \sqrt{4 \times -1} = 2i \]
Add the results: \( 6 + 2i \)
Find the linear function \( f \) such that \( f(2) = 5 \) and \( f(3) = -5 \).
A) \( f(x) = 2x + 1 \)
B) \( f(x) = -10x - 5 \)
C) \( f(x) = 3x - 1 \)
D) \( f(x) = -10x + 25 \)
E) \( f(x) = -10x - 25 \)
Correct Answer: D
A linear function has the form \( f(x) = mx + b \). Calculate the slope \( m \):
\[ m = \frac{-5 - 5}{3 - 2} = -10 \]
Use one of the points to solve for the y-intercept \( b \):
\[ 5 = -10(2) + b \Rightarrow 5 = -20 + b \Rightarrow b = 25 \]
The function is \( f(x) = -10x + 25 \).
If \( \frac{2^{m-3}}{4^{2m}} = 8 \), then \( 2m - 1 = \)?
A) 0
B) 1
C) 2
D) -5
E) -9
Correct Answer: D
Rewrite all bases as powers of 2:
\[ 4^{2m} = (2^2)^{2m} = 2^{4m} \] and \[ 8 = 2^3 \]
Substitute these into the equation and simplify using exponent rules:
\[ \frac{2^{m-3}}{2^{4m}} = 2^3 \Rightarrow 2^{(m-3)-4m} = 2^3 \Rightarrow 2^{-3m-3} = 2^3 \]
Equate the exponents:
\[ -3m - 3 = 3 \Rightarrow -3m = 6 \Rightarrow m = -2 \]
Evaluate the final expression: \( 2(-2) - 1 = -5 \).
Which equation represents a line perpendicular to \( 3x - 6y = 9 \)?
A) \( y = 2 \)
B) \( 3x + 6y = 9 \)
C) \( x - 2y = 3 \)
D) \( 2x + 2y = 3 \)
E) \( 2x + y = 7 \)
Correct Answer: E
Find the slope of the given line: \( 3x - 9 = 6y \Rightarrow y = \frac{1}{2}x - \frac{3}{2} \). The slope is \( \frac{1}{2} \).
A perpendicular line must have a slope of -2. Check the options to find the one with a slope of -2.
Option E: \( 2x + y = 7 \Rightarrow y = -2x + 7 \). The slope is -2.
If \( a \) and \( b \) are real numbers, which expression is always positive?
A) \( |a| \)
B) \( |a + b| \)
C) \( |a - b| + \frac{1}{2} \)
D) \( a^2 + b^2 \)
E) \( (a + b)^2 \)
Correct Answer: C
Absolute value and squared terms can evaluate to 0 (which is not positive, but non-negative). If \( a = 0 \), then \( |a| = 0 \). If \( a = b = 0 \), \( a^2 + b^2 = 0 \).
However, \( |a - b| \) is always \( \ge 0 \). Adding \( \frac{1}{2} \) guarantees the entire expression is strictly greater than 0, making it always positive.
Which equation matches the graph?
A) \( y = (x - 3)^2 - 1 \)
B) \( y = -(x - 3)^2 + 1 \)
C) \( y = (x - 3)^2 + 1 \)
D) \( y = x - 3 \)
E) \( y = -(x - 3)^2 - 1 \)
Correct Answer: E
The graph is a downward-facing parabola, meaning the leading coefficient must be negative. The vertex has a negative y-coordinate.
Since it opens downward, \( a \) is negative, making option E correct.
Given \( f(x) = x^2 + 2 \) and \( g(x) = \sqrt{x + 6} \), find \( g(f(1)) \).
A) 3
B) -3
C) 7
D) 6
E) -6
Correct Answer: A
Work from the inside out. First, evaluate \( f(1) \):
\[ f(1) = (1)^2 + 2 = 3 \]
Next, plug that result into \( g(x) \):
\[ g(3) = \sqrt{3 + 6} = 3 \]
Find the x-coordinates of the intersection of \( y = x + 1 \) and \( x^2 + y^2 = 5 \).
A) -2, 0
B) 1, 2
C) -2, 1
D) -2, -1
E) 1, 3
Correct Answer: C
Substitute the linear equation into the circle equation:
\[ x^2 + (x + 1)^2 = 5 \]
\[ x^2 + x^2 + 2x + 1 = 5 \Rightarrow 2x^2 + 2x - 4 = 0 \]
Divide by 2 and factor:
\[ x^2 + x - 2 = 0 \Rightarrow (x + 2)(x - 1) = 0 \]
The x-coordinates are -2 and 1.
Solve for \( x \): \( \log_x(1024) = -5 \)
A) \( \frac{1}{4} \)
B) 4
C) \( \frac{1}{2} \)
D) \( \frac{1}{8} \)
E) 2
Correct Answer: A
Convert the logarithmic equation to an exponential equation:
\[ x^{-5} = 1024 \]
We know that \( 4^5 = 1024 \). Therefore:
\[ \frac{1}{x^5} = 4^5 \Rightarrow x^5 = \frac{1}{4^5} = \left(\frac{1}{4}\right)^5 \]
Thus, \( x = \frac{1}{4} \).
Simplify: \( 6 \sqrt[3]{32} + 2 \sqrt[3]{108} \)
A) 32
B) \( 18 \sqrt[3]{2} \)
C) \( 36 \sqrt[3]{2} \)
D) \( 18 \sqrt[3]{4} \)
E) \( 36 \sqrt[3]{4} \)
Correct Answer: D
Find the largest perfect cube factors under the radicals:
\[ 32 = 8 \times 4 = 2^3 \times 4 \]
\[ 108 = 27 \times 4 = 3^3 \times 4 \]
Substitute and simplify:
\[ 6\sqrt[3]{8 \cdot 4} + 2\sqrt[3]{27 \cdot 4} = 6(2)\sqrt[3]{4} + 2(3)\sqrt[3]{4} \]
\[ 12\sqrt[3]{4} + 6\sqrt[3]{4} = 18\sqrt[3]{4} \]
The total surface area of all six sides of the rectangular box is 80 square inches. What is the volume of the box in cubic inches?
A) 40
B) 48
C) 12
D) 30
E) 16
Correct Answer: B
Set up the total surface area equation using the visible dimensions:
Left/Right sides: \( 2 \times (4x) = 8x \)
Front/Back sides: \( 2 \times (4x) = 8x \)
Top/Bottom sides: \( 2 \times (4 \times 4) = 32 \)
\[ 16x + 32 = 80 \Rightarrow 16x = 48 \Rightarrow x = 3 \]
Calculate volume: \( V = l \times w \times h = 4 \times 4 \times 3 = 48 \text{ in}^3 \)
In the figure below, \( L_1 \) and \( L_2 \) are parallel lines. The correct relationship between angles \( y \) and \( x \) is:
A) \( y = x \)
B) \( x + y = 180^\circ \)
C) \( x + y = 360^\circ \)
D) \( y = x + 90^\circ \)
E) \( y - x = 180^\circ \)
Correct Answer: B
Angle \( x \) and the angle directly across from it vertically are equal. Because \( L_1 \) and \( L_2 \) are parallel, the interior consecutive angles on the same side of the transversal sum to \( 180^\circ \). Therefore, \( x + y = 180^\circ \).
A city planner is designing a circular plaza for a new public park. The blueprint specifies that the plaza must have a total area of \( 100\pi \) square feet. To install a decorative barrier exactly along the edge of the plaza, the planner needs to determine its perimeter. What is the circumference of the plaza, in feet?
A) 20
B) \( 10\pi \)
C) \( 20\pi \)
D) \( 400\pi \)
E) \( 2500\pi \)
Correct Answer: C
Set the area formula equal to the known area to find the radius:
\[ \pi R^2 = 100\pi \Rightarrow R^2 = 100 \Rightarrow R = 10 \text{ feet} \]
Use the radius to find the circumference:
\[ C = 2\pi R = 2\pi(10) = 20\pi \text{ feet} \]
In the figure, \( ABC \) is a right triangle. Points \( B, C, D \) are collinear; \( D, E, F \) are collinear; and \( B, A, F \) are collinear. \( DC = DE \). Find angle \( AFE \) in degrees.
A) \( 50^\circ \)
B) \( 40^\circ \)
C) \( 30^\circ \)
D) \( 45^\circ \)
E) \( 55^\circ \)
Correct Answer: A
In right triangle \( ABC \): \( \angle BCA = 90^\circ - 50^\circ = 40^\circ \).
Since \( DC = DE \), triangle \( DCE \) is isosceles, making \( \angle DEC = \angle DCE = 40^\circ \).
Vertical angles dictate that \( \angle AEF = \angle DEC = 40^\circ \).
Since \( \angle FAE = 90^\circ \) (supplementary to a right angle), we find \( \angle AFE \):
\[ \angle AFE = 90^\circ - 40^\circ = 50^\circ \]
The figure shows a right triangle and two semicircles on its legs. The hypotenuse is 8 cm. Find the shaded area in cm².
A) 64
B) \( 8\pi \)
C) \( 64\pi \)
D) \( 10\pi \)
E) 16
Correct Answer: B
Let the legs be \(a\) and \(b\). The shaded area is the sum of the two semicircles:
\[ \text{Area} = \frac{1}{2}\pi\left(\frac{a}{2}\right)^2 + \frac{1}{2}\pi\left(\frac{b}{2}\right)^2 = \frac{\pi}{8}(a^2 + b^2) \]
By Pythagorean theorem, \( a^2 + b^2 = c^2 = 8^2 = 64 \).
Substitute this back: \( \frac{\pi}{8}(64) = 8\pi \).
Simplify: \( \frac{1}{2} \sin(2x) (1 + \cot^2(x)) \)
A) \( \tan(x) \)
B) \( \sin(x) \)
C) \( \cos(x) \)
D) \( \cot(x) \)
E) \( \sec(x) \)
Correct Answer: D
Use fundamental trigonometric identities: \( \sin(2x) = 2\sin(x)\cos(x) \) and \( 1 + \cot^2(x) = \csc^2(x) = \frac{1}{\sin^2(x)} \).
Substitute into the expression:
\[ \frac{1}{2} [2\sin(x)\cos(x)] \left[\frac{1}{\sin^2(x)}\right] = \frac{\sin(x)\cos(x)}{\sin^2(x)} = \frac{\cos(x)}{\sin(x)} = \cot(x) \]
Find the area of rectangle \( ABCD \).
A) \( \frac{2500}{\sqrt{2}} \)
B) 2500
C) \( \frac{2500}{\sqrt{3}} \)
D) 1250
E) 5000
Correct Answer: C
Let the length of shorter side BC be \( x \). Given the diagonal cuts the rectangle into a 30-60-90 triangle, the sides are in a ratio of \( 1 : \sqrt{3} : 2 \).
The hypotenuse is \( 2x\).
Apply the pythagorean theorem: \( (2x)^2 = x^2 + 50^2 \rightarrow x = \frac{50}{\sqrt3}\)
Area is given by \( x \times 50 = \frac{2500}{\sqrt3} \)
The area calculation aligns with answer option C according to the given dimension ratios.
A security company is programming a new digital keypad lock for a secure facility. The access code must be exactly 3 digits long, and due to hardware constraints, only the digits 4, 5, 7, and 9 are functional. If the system allows for the same digit to be used more than once in a code, how many unique 3-digit access codes can be generated?
A) \( 3^4 \)
B) \( 3^3 \)
C) \( 4^4 \)
D) 12
E) \( 4^3 \)
Correct Answer: E
The code requires filling 3 independent slots. Because repetition is permitted, each slot has 4 possible options (the digits 4, 5, 7, and 9).
\[ 4 \text{ choices} \times 4 \text{ choices} \times 4 \text{ choices} = 4^3 = 64 \]
A research vessel departs from its home port and travels exactly 10 miles due East to collect water samples. From that location, the vessel then travels exactly 24 miles due South to reach a remote monitoring station. What is the straight-line distance, in miles, from the home port to the remote monitoring station?
A) 34
B) 14
C) 26
D) \( 2\sqrt{119} \)
E) 44
Correct Answer: C
The path forms a right triangle with legs of 10 and 24. Calculate the hypotenuse (straight-line distance) using the Pythagorean theorem:
\[ d^2 = 10^2 + 24^2 = 100 + 576 = 676 \]
\[ d = \sqrt{676} = 26 \text{ miles} \]
A quality control analyst records the weights of 5 product samples, denoted as \( a, b, c, d, \) and \( e \), and calculates their mean weight to be 23 grams. When a sixth sample \( f \) is added to the batch, the new mean weight of all six samples drops to 22 grams. What is the exact weight of sample \( f \)?
A) 17
B) 18
C) 22
D) 22.5
E) 20
Correct Answer: A
First, find the total sum of the original 5 samples:
\[ \text{Sum of 5} = 5 \times 23 = 115 \]
Next, find the total sum of all 6 samples:
\[ \text{Sum of 6} = 6 \times 22 = 132 \]
Subtract the sum of the first 5 from the sum of the 6 to find the weight of \( f \):
\[ f = 132 - 115 = 17 \]
During a three-day charity telethon, the organizing committee noticed that the total number of distinct donors each day formed a sequence of three consecutive integers. If the total number of donors over the entire three-day period was exactly 192, what is the product of the number of donors recorded on these three days?
A) 216,000
B) 7,077,888
C) 576
D) 110,592
E) 262,080
Correct Answer: E
Let the three consecutive integers be \( n-1 \), \( n \), and \( n+1 \).
Set up the sum equation: \( (n-1) + n + (n+1) = 192 \)
\[ 3n = 192 \Rightarrow n = 64 \]
The three integers are 63, 64, and 65. Find their product:
\[ 63 \times 64 \times 65 = 262,080 \]