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ACT Math

ACT Math Practice Test: Practice Test 46

Practice Test 46 for ACT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

Apply exponent rules to simplify: 23⋅252^3\cdot 2^523⋅25

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Question 1

Apply exponent rules to simplify: 23⋅252^3\cdot 2^523⋅25

  1. 2152^{15}215
  2. 282^828 (correct answer)
  3. 484^848
  4. 222^222

Explanation: To simplify 2³ · 2⁵, we apply the product rule for exponents: a^m · a^n = a^(m+n). Therefore, 2³ · 2⁵ = 2^(3+5) = 2⁸. The bases are the same, so we add the exponents. Choice A incorrectly multiplies the exponents (3 × 5 = 15), while choice C changes the base to 4.

Question 2

A taxi fare is modeled by y=2.25x+4.50y = 2.25x + 4.50y=2.25x+4.50, where xxx is the number of miles traveled and yyy is the total fare in dollars.

What does the slope represent in this context?

  1. The taxi charges a $4.50 starting fee.
  2. The taxi charges $4.50 per mile.
  3. The taxi charges a $2.25 starting fee.
  4. The taxi charges $2.25 per mile. (correct answer)

Explanation: This is a linear model y = 2.25x + 4.50 where y represents total taxi fare and x represents miles traveled. The slope 2.25 represents the rate of change in fare per unit change in miles, meaning the taxi charges 2.25permiledriven.Theintercept4.50representsthefarewhenx=0miles,whichisthestartingfeeof2.25 per mile driven. The intercept 4.50 represents the fare when x = 0 miles, which is the starting fee of 2.25permiledriven.Theintercept4.50representsthefarewhenx=0miles,whichisthestartingfeeof4.50. For each additional mile traveled, the total fare increases by exactly $2.25\$2.25$2.25. Choice B incorrectly doubles the per-mile rate, while choices C and D confuse the slope with the intercept values.

Question 3

What is the mode of the data set [7, 8, 10, 9, 10, 7, 7]?

  1. 7 (correct answer)
  2. 8
  3. 10
  4. 9

Explanation: The mode is the value that appears most frequently in the data set. Looking at [7, 8, 10, 9, 10, 7, 7], the value 7 appears three times. The value 10 appears twice, and values 8 and 9 each appear once. Therefore, the mode is 7.

Question 4

What is the determinant of the matrix [−4231]?\begin{bmatrix}-4 & 2\\ 3 & 1\end{bmatrix}?[−43​21​]?

  1. −10-10−10 (correct answer)
  2. 222
  3. 101010
  4. −2-2−2

Explanation: To find the determinant of the 2×2 matrix [[-4,2],[3,1]], we use the formula ad - bc. Here, the determinant is (-4)(1) - (2)(3) = -4 - 6 = -10. The result is -10. Remember that the determinant formula is ad - bc, not ac - bd or other variations.

Question 5

What angle has sin⁡(θ)=12\sin(\theta) = \frac{1}{2}sin(θ)=21​?

  1. 30∘30^\circ30∘ (correct answer)
  2. 45∘45^\circ45∘
  3. 60∘60^\circ60∘
  4. 90∘90^\circ90∘

Explanation: This asks for the angle where sin⁡(θ)=12\sin(\theta) = \frac{1}{2}sin(θ)=21​. From the unit circle, sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}sin(30∘)=21​. Using SOH-CAH-TOA, sine equals oppositehypotenuse\frac{\text{opposite}}{\text{hypotenuse}}hypotenuseopposite​, and 30∘30^\circ30∘ is the standard angle with this ratio. Choice C (60∘60^\circ60∘) has sin⁡(60∘)=32\sin(60^\circ) = \frac{\sqrt{3}}{2}sin(60∘)=23​​, not 12\frac{1}{2}21​.

Question 6

A regular polygon has 6 sides. What is the sum of its interior angles?

  1. 720° (correct answer)
  2. 540°
  3. 1080°
  4. 900°

Explanation: This question asks for the sum of interior angles of a 6-sided polygon (hexagon). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 6: 180(6-2) = 180(4) = 720°. Choice B would result from using a pentagon (5 sides) with sum 180(3) = 540°.

Question 7

What is the slope of the line through points (0, 0) and (4, 8)?

  1. 0
  2. 4
  3. 1
  4. 2 (correct answer)

Explanation: To find the slope of the line through points (0, 0) and (4, 8), we use the slope formula m = (y₂ - y₁)/(x₂ - x₁). Substituting the coordinates: m = (8 - 0)/(4 - 0) = 8/4 = 2. The slope is 2, meaning the line rises 2 units vertically for every 1 unit horizontally. This line passes through the origin and has a constant rate of change of 2. Choice B (4) incorrectly uses just the y-coordinate of the second point.

Question 8

Two sensors produce readings stored in matrices. What is A+BA+BA+B if A=[03−52]A=\begin{bmatrix}0 & 3\\ -5 & 2\end{bmatrix}A=[0−5​32​] and B=[1−421]B=\begin{bmatrix}1 & -4\\ 2 & 1\end{bmatrix}B=[12​−41​]?

  1. [17−33]\begin{bmatrix}1 & 7\\ -3 & 3\end{bmatrix}[1−3​73​]
  2. [−17−71]\begin{bmatrix}-1 & 7\\ -7 & 1\end{bmatrix}[−1−7​71​]
  3. [1−1−33]\begin{bmatrix}1 & -1\\ -3 & 3\end{bmatrix}[1−3​−13​] (correct answer)
  4. [0−12−102]\begin{bmatrix}0 & -12\\ -10 & 2\end{bmatrix}[0−10​−122​]

Explanation: This problem requires matrix addition, where we add corresponding entries of matrices A and B. For matrices A = [[0, 3], [-5, 2]] and B = [[1, -4], [2, 1]], we compute: entry (1,1): 0 + 1 = 1, entry (1,2): 3 + (-4) = -1, entry (2,1): -5 + 2 = -3, entry (2,2): 2 + 1 = 3. The result is [[1, -1], [-3, 3]].

Question 9

What is the range of xxx if ∣x−3∣<4|x-3|<4∣x−3∣<4? (Write your answer as an inequality.)

  1. x<−1x<-1x<−1 or x>7x>7x>7
  2. −1<x<7-1<x<7−1<x<7 (correct answer)
  3. −1≤x≤7-1\le x\le 7−1≤x≤7
  4. −7<x<1-7<x<1−7<x<1

Explanation: This absolute value inequality represents distance and splits into two cases. |x - 3| < 4 means the distance from x to 3 is less than 4. This gives us -4 < x - 3 < 4. Adding 3 to all parts: -1 < x < 7. The solution is all values strictly between -1 and 7. Choice A shows the exterior solution (for ≥), while choices C and D have incorrect boundaries. For |expression| < constant, always get the interior solution between two bounds.

Question 10

What is the sum of the interior angles of a pentagon?

  1. 450°
  2. 540° (correct answer)
  3. 360°
  4. 720°

Explanation: This question asks for the sum of interior angles of a pentagon (5 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 5: 180(5-2) = 180(3) = 540°. Choice A would result from incorrectly using 180(2.5) for some fractional calculation.

Question 11

A train covers a distance of 180 miles in 3 hours. What is the train's speed in miles per hour?

  1. 50 mph
  2. 60 mph (correct answer)
  3. 70 mph
  4. 80 mph

Explanation: This problem asks for the train's speed. We need to use the formula: speed = distance ÷ time. Train's speed = 180 miles ÷ 3 hours = 60 mph. Choice A (50 mph) might result from dividing incorrectly or confusing the calculation.

Question 12

What percent of 500 is 50?

  1. 5%
  2. 10% (correct answer)
  3. 15%
  4. 20%

Explanation: This question asks what percent 50 is of 500. To find what percent one number is of another, use: (part/whole) × 100. Calculate: (50/500) × 100 = 0.10 × 100 = 10%. Choice A (5%) represents a common calculation error, possibly from dividing incorrectly.

Question 13

A student has 5 different stickers and wants to place all 5 in a row on a notebook. Since the order left-to-right matters, how many different arrangements are possible?

  1. 25
  2. 120 (correct answer)
  3. 60
  4. 20

Explanation: Since the stickers are placed in a row and left-to-right order matters, this is a permutation of all 5 items. We use the factorial formula for arrangements: 5! = 5 × 4 × 3 × 2 × 1. Calculating step by step: 5 × 4 = 20, 20 × 3 = 60, 60 × 2 = 120, 120 × 1 = 120. A key distractor is choice C (60), which is P(5,3) or half of 120, perhaps from mistakenly taking only partial arrangements.

Question 14

A rectangular prism has a surface area of 94 square units and dimensions 5 units by 4 units. What is the height of the prism?

  1. 5 units
  2. 4 units
  3. 3 units (correct answer)
  4. 2 units

Explanation: We need to find the height of a rectangular prism with surface area 94 square units and dimensions 5 by 4 units. Using SA = 2(lw + lh + wh) where l=5, w=4, and h is unknown: 94 = 2(5×4 + 5h + 4h) = 2(20 + 9h). Solving: 94 = 40 + 18h, so 54 = 18h, therefore h = 3 units.

Question 15

A teacher wants to test whether listening to instrumental music improves quiz scores. She randomly assigns half the class to take a quiz with instrumental music playing and the other half to take the same quiz in silence. Students are tested at the same time of day. Which study design would avoid bias from students choosing their condition?

  1. Let students choose music or silence, since they know what helps them focus best
  2. Assign conditions randomly, so preference is balanced between the two groups (correct answer)
  3. Use only students who already listen to music while studying, to keep groups similar
  4. Give the quiz only to students with high grades, to reduce score variability

Explanation: Random assignment of conditions prevents bias by ensuring that student preferences and abilities are balanced between the music and silence groups. Random assignment eliminates confounding variables by distributing both known and unknown factors equally across treatment groups. Without random assignment, students might self-select conditions based on their usual study habits, making it impossible to isolate the effect of music. Choice A would introduce selection bias because students' choices might correlate with factors that independently affect quiz performance.

Question 16

A coin is flipped three times. What is the probability of getting exactly two heads?

  1. ( \frac{1}{2} )
  2. ( \frac{3}{8} ) (correct answer)
  3. ( \frac{1}{4} )
  4. ( \frac{1}{8} )

Explanation: When flipping a coin three times, there are 2³ = 8 total possible outcomes. The favorable outcomes for exactly two heads are: {HHT, HTH, THH}, giving us 3 favorable outcomes. Using P(exactly 2 heads) = favorable/total = 3/8. This can also be calculated using the binomial formula: C(3,2) × (1/2)² × (1/2)¹ = 3 × 1/4 × 1/2 = 3/8. Choice A (1/2) incorrectly assumes equal probability for all numbers of heads.

Question 17

For the piecewise function f(x)={3x+2if x<−1x2−1if −1≤x<24x−3if x≥2f(x) = \begin{cases} 3x + 2 & \text{if } x < -1 \\ x^2 - 1 & \text{if } -1 \leq x < 2 \\ 4x - 3 & \text{if } x \geq 2 \end{cases}f(x)=⎩⎨⎧​3x+2x2−14x−3​if x<−1if −1≤x<2if x≥2​, what is f(1)?

  1. 0 (correct answer)
  2. 1
  3. 2
  4. 3

Explanation: For x = 1, check intervals: 1 < -1? No. -1 ≤ 1 < 2? Yes. So use the second piece f(x) = x2−1x^2 - 1x2−1. Substitute x = 1: f(1) = 12−1=1−1=01^2 - 1 = 1 - 1 = 012−1=1−1=0. The value x = 1 falls within the middle interval boundaries.

Question 18

A jar contains 10 candies: 4 red, 3 blue, and 3 green. If one candy is chosen at random, what is the probability that it is not blue?

  1. ( \frac{2}{5} )
  2. ( \frac{3}{10} )
  3. ( \frac{7}{10} ) (correct answer)
  4. ( \frac{1}{10} )

Explanation: The total number of candies is 4 red + 3 blue + 3 green = 10. The favorable outcomes for "not blue" include all red and green candies: 4 + 3 = 7 candies. Using the probability rule P(not blue) = favorable/total = 7/10. Alternatively, P(not blue) = 1 - P(blue) = 1 - 3/10 = 7/10. Choice B (3/10) gives the probability of selecting blue instead of not blue.

Question 19

Ray OBOBOB is perpendicular to ray OAOAOA (right angle marked with a square). Ray OCOCOC lies between them, forming ∠AOC=25∘\angle AOC = 25^\circ∠AOC=25∘ and ∠COB=x\angle COB = x∠COB=x.

B
|\
| \ x
|  \
O---A
 25°

What is the value of xxx?

  1. 90∘90^\circ90∘
  2. 115∘115^\circ115∘
  3. 25∘25^\circ25∘
  4. 65∘65^\circ65∘ (correct answer)

Explanation: Ray OB is perpendicular to ray OA, creating a 90∘90^\circ90∘ angle between them. Ray OC divides this right angle into two parts: ∠AOC=25∘\angle AOC = 25^\circ∠AOC=25∘ and ∠COB=x\angle COB = x∠COB=x. Since these two angles must add up to the total right angle, we have 25∘+x=90∘25^\circ + x = 90^\circ25∘+x=90∘. Therefore, x=90∘−25∘=65∘x = 90^\circ - 25^\circ = 65^\circx=90∘−25∘=65∘. Choice B (115∘115^\circ115∘) incorrectly adds 25∘25^\circ25∘ to 90∘90^\circ90∘ instead of subtracting.

Question 20

What is f(−3)f(-3)f(−3) for the function defined as f(x)={5x+1if x<−23x2−xif x≥−2f(x) = \begin{cases} 5x + 1 & \text{if } x < -2 \\ 3x^2 - x & \text{if } x \geq -2 \end{cases}f(x)={5x+13x2−x​if x<−2if x≥−2​?

  1. -14 (correct answer)
  2. -8
  3. -16
  4. -12

Explanation: For x = -3, we check the intervals: Is -3 < -2? Yes. So we use the first piece: f(x)=5x+1f(x) = 5x + 1f(x)=5x+1. Substituting x = -3: f(−3)=5(−3)+1=−15+1=−14f(-3) = 5(-3) + 1 = -15 + 1 = -14f(−3)=5(−3)+1=−15+1=−14. Choice D would result from using the second piece incorrectly.

Question 21

The graph of y=f(x)y = f(x)y=f(x) where f(x)=x2f(x) = x^2f(x)=x2 is transformed to y=f(x+3)−2y = f(x+3) - 2y=f(x+3)−2. How is the vertex of the parabola shifted?

  1. left 3 units and down 2 units. (correct answer)
  2. right 3 units and down 2 units.
  3. left 3 units and up 2 units.
  4. right 3 units and up 2 units.

Explanation: This is a graph transformations question testing vertex form of a parabola. Choice A (left 3 units, down 2 units) is correct — for y = f(x + 3) − 2: the "+3" inside the argument shifts the graph opposite to its sign (left 3 units), and the "−2" outside shifts down 2 units. So the vertex moves from (0, 0) to (−3, −2). Choice B (right 3, down 2) gets the vertical direction right but misreads the horizontal: seeing "+3" and thinking "right 3." Choice C (left 3, up 2) gets the horizontal direction right but misreads the vertical sign: seeing "−2" and thinking "up." Choice D (right 3, up 2) gets both directions wrong. Pro tip: Horizontal shifts are counterintuitive — f(x + h) shifts LEFT (not right) by h units. A helpful way to remember: the graph shifts to make the argument equal zero. For f(x + 3), set x + 3 = 0 → x = −3, so the vertex moves to x = −3 (left). Vertical shifts behave naturally: subtracting outside moves down, adding moves up.

Question 22

What is the length of arc AB if the circle's radius is 8 and the central angle is 90°?

  1. 4π4\pi4π (correct answer)
  2. 2π2\pi2π
  3. 8π8\pi8π
  4. 16π16\pi16π

Explanation: We need to find the arc length with radius 8 and central angle 90°. The arc length formula is s = (θ/360°) × 2πr. Substituting: s = (90°/360°) × 2π(8) = (1/4) × 16π = 4π. Choice B incorrectly uses half the angle, while choice C doubles the correct answer.

Question 23

Which of the following equals cos⁡(π3)\cos\left(\frac{\pi}{3}\right)cos(3π​)?​

  1. 32\frac{\sqrt{3}}{2}23​​
  2. 12\frac{1}{2}21​ (correct answer)
  3. 22\frac{\sqrt{2}}{2}22​​
  4. 111

Explanation: The angle π/3 radians equals 60°. From the unit circle, cos(60°) = 1/2. This is a standard value to memorize. Choice A gives √3/2, which is actually sin(60°), not cos(60°).

Question 24

A quadratic equation for a design constraint is x2−3x−10=0x^2-3x-10=0x2−3x−10=0. What are the solutions to x2−3x−10=0x^2-3x-10=0x2−3x−10=0?

  1. x=5x=5x=5 only
  2. x=2x=2x=2 and x=−5x=-5x=−5
  3. x=5x=5x=5 and x=−2x=-2x=−2 (correct answer)
  4. x=−5x=-5x=−5 and x=−2x=-2x=−2

Explanation: To solve x² - 3x - 10 = 0, factor by finding two numbers that multiply to -10 and add to -3. Since (2)(-5) = -10 and 2 + (-5) = -3, we get (x - 5)(x + 2) = 0. Setting each factor equal to zero gives x = 5 and x = -2. Choice B has the wrong signs for both solutions.

Question 25

A circle has circumference 30π30\pi30π. What is the diameter of the circle?

  1. 151515
  2. 303030 (correct answer)
  3. 606060
  4. 30π\dfrac{30}{\pi}π30​

Explanation: This question seeks the diameter of a circle with circumference 30π. The circumference formula is C = πd, so d = C/π = 30π/π = 30. Alternatively, C = 2πr implies r = 15, d = 30. Choice A halves it incorrectly to 15, the radius. Choice D divides by π again unnecessarily. Remember circumference directly gives diameter by dividing by π.