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ACT Math

ACT Math Practice Test: Practice Test 7

Practice Test 7 for ACT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

A quantity decreases from 909090 to 727272. What is the percent decrease from the original value?

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Question 1

A quantity decreases from 909090 to 727272. What is the percent decrease from the original value?

  1. 18%18\%18%
  2. 20%20\%20% (correct answer)
  3. 25%25\%25%
  4. 80%80\%80%

Explanation: We need to find the percent decrease from 90 to 72. Percent change is (new value - original value)/original value × 100. Here: (72 - 90)/90 × 100 = -18/90 × 100 = -0.2 × 100 = -20%. The negative indicates a decrease, so the percent decrease is 20%. Choice D (80%) incorrectly calculates what percent 72 is of 90.

Question 2

In the right triangle, if the opposite side is 8 and the hypotenuse is 17, what is sin⁡(θ)\sin(\theta)sin(θ)?

  1. 815\frac{8}{15}158​
  2. 1517\frac{15}{17}1715​
  3. 178\frac{17}{8}817​
  4. 817\frac{8}{17}178​ (correct answer)

Explanation: For angle θ, the opposite side is 8 and the hypotenuse is 17. Using SOH-CAH-TOA, sin(θ) = opposite/hypotenuse. Therefore, sin(θ) = 8/17. Choice B (15/17) would be adjacent/hypotenuse, which is cosine, not sine.

Question 3

A line shows a constant rate of change in a game score. The line passes through points (−3,−1)(-3,-1)(−3,−1) and (1,7)(1,7)(1,7). What is the slope of the line through points (−3,−1)(-3,-1)(−3,−1) and (1,7)(1,7)(1,7)?

  1. −12-\dfrac{1}{2}−21​
  2. −2-2−2
  3. 12\dfrac{1}{2}21​
  4. 222 (correct answer)

Explanation: We need to find the slope of the line through points (-3,-1) and (1,7). Using the slope formula m = (y₂-y₁)/(x₂-x₁), we calculate m = (7-(-1))/(1-(-3)) = (7+1)/(4) = 8/4 = 2. The slope represents the rate of change in the game score, which is 2 points per unit time. Choice B shows the negative of the correct answer, while choices C and D show fractional values that would result from calculation errors.

Question 4

A student wants to estimate the average price of a textbook at her college bookstore. She records the prices of the first 30 textbooks she sees on the “New Arrivals” shelf. That shelf mostly contains expensive new editions. Which characteristic makes this sample biased?

  1. The bookstore is on campus, so the results will generalize to all bookstores
  2. The sample size is 30, which is always too small to estimate an average
  3. Textbook prices are measured in dollars, which introduces measurement error
  4. The sample comes from a single shelf that may not represent all textbooks sold (correct answer)

Explanation: Sampling from only the 'New Arrivals' shelf creates bias because this shelf likely contains expensive new editions that don't represent the full range of textbook prices. Location-based sampling bias occurs when the sampling location systematically includes items with different characteristics than the overall population. The new arrivals shelf would overrepresent recently published, higher-priced textbooks compared to the complete bookstore inventory. Choice B incorrectly suggests sample size is always the issue, but 30 can be adequate if sampled representatively.

Question 5

A map uses the scale 1 inch:4 miles1\text{ inch} : 4\text{ miles}1 inch:4 miles. Two towns are 3.53.53.5 inches apart on the map. How many miles apart are the towns in real life?

  1. 7
  2. 10
  3. 14 (correct answer)
  4. 28

Explanation: This problem involves using a map scale to find the actual distance between two towns. The scale 1 inch : 4 miles means that each inch on the map represents 4 miles in real life. Set up the proportion: 1 inch / 4 miles = 3.5 inches / x miles. Cross-multiply: 1 × x = 4 × 3.5, so x = 14 miles. A common error would be to confuse the ratio direction and divide 3.5 by 4 instead of multiplying.

Question 6

A student tracks the temperature (in °C) of a cooling liquid, which changes linearly with time. The line passes through points (0,18)(0,18)(0,18) and (4,6)(4,6)(4,6). What is the y-intercept of the line through these points?​​

  1. 666
  2. 181818 (correct answer)
  3. −3-3−3
  4. 121212

Explanation: This question seeks the y-intercept of the line passing through points (0,18) and (4,6), which models the initial temperature of the cooling liquid. The y-intercept is the y-value when x = 0, and one point is already (0,18), directly giving the intercept. To confirm, calculate the slope m = (6 - 18)/(4 - 0) = -12/4 = -3, then the equation is y = -3x + 18, verifying the intercept is 18. This process underscores how the y-intercept represents the starting value in linear models. The correct answer is 18, which is choice B. A key distractor is choice A, 6, which is the y-value at x=4, confusing it with the intercept. Another is choice C, -3, mistaking the slope for the intercept.

Question 7

Two triangles have the following marked information: ∠A≅∠D\angle A \cong \angle D∠A≅∠D, ∠B≅∠E\angle B \cong \angle E∠B≅∠E, and a non-included side AC=13AC=13AC=13 equals DF=13DF=13DF=13. Which congruence criterion applies?

  1. ASA
  2. SSS
  3. AAS (correct answer)
  4. SAS

Explanation: The triangles are congruent by AAS (Angle-Angle-Side) criterion. We have angle A congruent to angle D, angle B congruent to angle E, and a non-included side AC = DF = 13. AAS requires two angles and a non-included side to be congruent, which matches our given information exactly. Note that AC is opposite to angle B, making it non-included relative to the two given angles.

Question 8

Which of the following represents f(x+3)f(x+3)f(x+3) if f(x)=x2−3xf(x) = x^2 - 3xf(x)=x2−3x?

  1. x2+3xx^2 + 3xx2+3x
  2. (x+3)2−9x(x + 3)^2 - 9x(x+3)2−9x
  3. (x+3)2−3(x+3)(x + 3)^2 - 3(x + 3)(x+3)2−3(x+3) (correct answer)
  4. x2+6x+9−3x−9x^2 + 6x + 9 - 3x - 9x2+6x+9−3x−9

Explanation: We need to find f(x + 3) when f(x) = x² - 3x. To find f(x + 3), we replace every x in the original function with (x + 3): f(x + 3) = (x + 3)² - 3(x + 3). This expression shows the direct substitution before any expansion. Choice C correctly represents this substitution form. Note that f(x + 3) means substituting (x + 3) for every x, not adding 3 to the result.

Question 9

A model uses the product (x+3i)(2−4i)(x+3i)(2-4i)(x+3i)(2−4i), where xxx is a real number. What is the real part of (x+3i)(2−4i)(x+3i)(2-4i)(x+3i)(2−4i) after using FOIL and applying i2=−1i^2=-1i2=−1?

  1. 2x−12i2x-12i2x−12i
  2. 2x−122x-122x−12
  3. −4x+6-4x+6−4x+6
  4. 2x+122x+122x+12 (correct answer)

Explanation: This problem asks for the real part of the product (x + 3i)(2 - 4i), expanded using FOIL. First, x2 = 2x; x(-4i) = -4xi; 3i2 = 6i; 3i(-4i) = -12i². Since i² = -1, -12i² = 12. The expression is 2x + 12 + (-4x + 6)i, so the real part is 2x + 12. Choice B might result from forgetting i² = -1 and leaving -12 as is.

Question 10

A local bakery sells muffins in boxes of 6 and donuts in boxes of 12. If a customer buys the same number of muffins as donuts, what is the smallest total number of baked goods (muffins plus donuts) the customer could have purchased?

  1. 12
  2. 18
  3. 24 (correct answer)
  4. 36

Explanation: This least common multiple (LCM) problem requires finding the smallest number that is divisible by both 6 and 12. Since the customer must buy equal numbers of muffins and donuts, you need a common multiple: LCM(6,12)=12LCM(6, 12) = 12LCM(6,12)=12. This means 2 boxes of muffins (12 muffins) and 1 box of donuts (12 donuts), for a total of 24 baked goods. Strategy: When a problem involves "same number" of items packaged differently, think LCM immediately.

Question 11

A phone plan’s total cost is modeled by y=45+0.08xy = 45 + 0.08xy=45+0.08x, where xxx is the number of text messages sent in a month and yyy is the total monthly cost in dollars.

Which variable represents the number of text messages?

  1. yyy
  2. 0.080.080.08
  3. xxx (correct answer)
  4. 454545

Explanation: This is a linear model y = 45 + 0.08x where y represents total monthly cost and x represents number of text messages. In this equation, x is explicitly defined as the independent variable representing the number of text messages sent. The coefficient 0.08 means each text message costs $0.08\$0.08$0.08, and 45 represents the base monthly fee. The variable y represents the dependent variable (total cost), while 45 and 0.08 are constants in the model. Choice D represents the base fee constant, not a variable.

Question 12

Simplify: 2x−(4x−6)2x - (4x - 6)2x−(4x−6)

  1. 2x−62x - 62x−6
  2. 2x+62x + 62x+6
  3. −2x−6-2x - 6−2x−6
  4. −2x+6-2x + 6−2x+6 (correct answer)

Explanation: Distribute the negative sign to both terms in the parentheses: 2x - (4x - 6) = 2x - 4x + 6. Combine like terms by grouping x terms and constants: (2x - 4x) + 6 = -2x + 6. The simplified form is -2x + 6.

Question 13

Two lines intersect at point OOO. One of the angles is labeled 120∘120^\circ120∘. The angle adjacent to it on the same straight line is labeled xxx.

   \ 120° /
    \    /
-----O-----
    / x  \
   /      \

What is the measure of angle xxx?

  1. 30∘30^\circ30∘
  2. 120∘120^\circ120∘
  3. 90∘90^\circ90∘
  4. 60∘60^\circ60∘ (correct answer)

Explanation: The angles 120∘120^\circ120∘ and xxx are adjacent angles that form a linear pair on a straight line. Adjacent angles on a straight line are supplementary, meaning they add up to 180∘180^\circ180∘. Therefore, 120∘+x=180∘120^\circ + x = 180^\circ120∘+x=180∘, which gives us x=180∘−120∘=60∘x = 180^\circ - 120^\circ = 60^\circx=180∘−120∘=60∘. Choice B (120∘120^\circ120∘) incorrectly assumes the angles are vertical angles (equal) rather than supplementary adjacent angles.

Question 14

Given i=−1i = \sqrt{-1}i=−1​, what is the simplified form of (4+5i)−(1−2i)(4 + 5i) - (1 - 2i)(4+5i)−(1−2i)?

  1. 3+3i3 + 3i3+3i
  2. 3+7i3 + 7i3+7i (correct answer)
  3. 5+3i5 + 3i5+3i
  4. 5+7i5 + 7i5+7i

Explanation: This is a complex numbers question testing subtraction with distribution. Choice B (3 + 7i) is correct — distribute the negative: (4 + 5i) − (1 − 2i) = 4 + 5i − 1 + 2i. Key step: −(−2i) = +2i. Combine real parts: 4 − 1 = 3. Combine imaginary parts: 5i + 2i = 7i. Result: 3 + 7i. Choice A (3 + 3i) correctly subtracts the real parts but fails to distribute the negative on the imaginary term: 5i − 2i = 3i instead of 5i + 2i = 7i. Choice C (5 + 3i) adds the real parts instead of subtracting: 4 + 1 = 5, and also gets the imaginary term wrong. Choice D (5 + 7i) adds real parts (correctly gets +7i from the imaginary) — two separate errors that partially cancel. Pro tip: When subtracting a complex number, rewrite it as addition of the negative first: (4 + 5i) + (−1 + 2i). This prevents sign errors by making every operation an addition. The most common mistake is treating −(−2i) as −2i instead of +2i.

Question 15

A complex number is −12+5i-12+5i−12+5i. What is its complex conjugate, used to form a real product with the original number?

  1. 12−5i12-5i12−5i
  2. −12−5i-12-5i−12−5i (correct answer)
  3. −12+5i-12+5i−12+5i
  4. 12+5i12+5i12+5i

Explanation: To find the complex conjugate, we change the sign of the imaginary part while keeping the real part the same. For the complex number −12+5i-12 + 5i−12+5i, the real part is −12-12−12 and the imaginary part is +5i+5i+5i. The complex conjugate is −12−5i-12 - 5i−12−5i. When multiplied together, a complex number and its conjugate produce a real result.

Question 16

Based on the piecewise function

4x-1, & x<5\\ (x-5)^2, & 5\le x<8\\ 20-x, & x\ge 8 \end{cases}$$ what is the value when $x=6$?
  1. 23
  2. 1 (correct answer)
  3. 14
  4. 0

Explanation: For x = 6, check which interval contains this value: 6<56 < 56<5? No. 5≤6<85 \le 6 < 85≤6<8? Yes. Since x = 6 satisfies the condition 5≤x<85 \le x < 85≤x<8, we use the second piece f(x)=(x−5)2f(x) = (x - 5)^2f(x)=(x−5)2. Substituting x = 6: f(6)=(6−5)2=12=1f(6) = (6 - 5)^2 = 1^2 = 1f(6)=(6−5)2=12=1. Choice A might result from incorrectly using the first piece where 4x−14x - 14x−1 would give 4(6)−1=234(6) - 1 = 234(6)−1=23.

Question 17

Two matrices represent consecutive transformations on a vector. Which of the following is the product ABABAB if $$A=\begin{bmatrix}0 & 2\ -1 & 3\end{bmatrix},\quad B=\begin{bmatrix}5 & -2\ 1 & 4\end{bmatrix}?$

  1. [2−4−512]\begin{bmatrix}2 & -4\\ -5 & 12\end{bmatrix}[2−5​−412​]
  2. [0−4−112]\begin{bmatrix}0 & -4\\ -1 & 12\end{bmatrix}[0−1​−412​]
  3. [28−214]\begin{bmatrix}2 & 8\\ -2 & 14\end{bmatrix}[2−2​814​] (correct answer)
  4. [28−814]\begin{bmatrix}2 & 8\\ -8 & 14\end{bmatrix}[2−8​814​]

Explanation: This problem requires computing the matrix product AB for consecutive transformations. To multiply matrices, compute dot products: for (1,1), row 1 of A dot column 1 of B is (0)(5)+(2)(1)=2(0)(5) + (2)(1) = 2(0)(5)+(2)(1)=2; for (1,2), (0)(−2)+(2)(4)=8(0)(-2) + (2)(4) = 8(0)(−2)+(2)(4)=8; for (2,1), (−1)(5)+(3)(1)=−2(-1)(5) + (3)(1) = -2(−1)(5)+(3)(1)=−2; for (2,2), (−1)(−2)+(3)(4)=14(-1)(-2) + (3)(4) = 14(−1)(−2)+(3)(4)=14. The product AB is [28−214]\begin{bmatrix} 2 & 8 \\ -2 & 14 \end{bmatrix}[2−2​814​]. Choice D miscalculates the (2,1) entry as -8, possibly from subtracting instead of adding in the dot product.

Question 18

If 7x=219\frac{7}{x} = \frac{21}{9}x7​=921​, what is the value of xxx?

  1. 3 (correct answer)
  2. 4
  3. 9
  4. 2

Explanation: In the proportion 7/x=21/97/x = 21/97/x=21/9, cross-multiply to get 7×9=21x7 \times 9 = 21x7×9=21x, so 63=21x63 = 21x63=21x. Dividing both sides by 21: x=3x = 3x=3. Students might make arithmetic errors or incorrectly set up the cross-multiplication.

Question 19

Based on the piecewise function f(x)={2x−3if x<−1x2+2if −1≤x<25x−1if x≥2f(x) = \begin{cases} 2x - 3 & \text{if } x < -1 \\ x^2 + 2 & \text{if } -1 \leq x < 2 \\ 5x - 1 & \text{if } x \geq 2 \end{cases}f(x)=⎩⎨⎧​2x−3x2+25x−1​if x<−1if −1≤x<2if x≥2​, what is f(2)?

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

Explanation: For x = 2, check intervals: 2<−12 < -12<−1? No. −1≤2<2-1 \leq 2 < 2−1≤2<2? No. 2≥22 \geq 22≥2? Yes. So use the third piece f(x)=5x−1f(x) = 5x - 1f(x)=5x−1. Substitute x = 2: f(2)=5(2)−1=10−1=9f(2) = 5(2) - 1 = 10 - 1 = 9f(2)=5(2)−1=10−1=9. Note that x = 2 falls in the third piece due to the ≥\geq≥ condition.

Question 20

If 4x+2y=184x + 2y = 184x+2y=18 and 2x−y=12x - y = 12x−y=1, what is the value of yyy?

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

Explanation: Use the substitution method from 2x - y = 1, so y = 2x - 1, into 4x + 2y = 18. This gives 4x + 2(2x - 1) = 18, 4x + 4x - 2 = 18, 8x = 20, x = 2.5, y = 2(2.5) - 1 = 5 - 1 = 4. The value of y is 4. Choice B of 3 might come from solving 8x = 18 incorrectly.

Question 21

What is log⁡4(1)\log_{4}(1)log4​(1)?

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

Explanation: This involves the fundamental logarithm property for the number 1. For any logarithm with a positive base (not equal to 1), loga(1)=0log_a(1) = 0loga​(1)=0. This is because any positive number raised to the power of 0 equals 1. Therefore, log⁡4(1)=0\log_4(1) = 0log4​(1)=0 because 40=14^0 = 140=1. This property holds for all valid logarithm bases.

Question 22

A student combines two polynomial expressions while modeling total cost: (2x2−3x+1)+(x2+5x−4)(2x^2-3x+1)+(x^2+5x-4)(2x2−3x+1)+(x2+5x−4). Which polynomial is the result in standard form?

  1. 3x2+2x−33x^2+2x-33x2+2x−3 (correct answer)
  2. 3x2−8x+53x^2-8x+53x2−8x+5
  3. x4+2x−3x^4+2x-3x4+2x−3
  4. 3x2+8x−33x^2+8x-33x2+8x−3

Explanation: To add polynomials, combine like terms: (2x² - 3x + 1) + (x² + 5x - 4) = 2x² + x² - 3x + 5x + 1 - 4 = 3x² + 2x - 3. Choice B incorrectly gets -8x instead of +2x when combining -3x + 5x.

Question 23

If 2x+3y=132x + 3y = 132x+3y=13 and 2x−3y=12x - 3y = 12x−3y=1, what is the value of yyy?

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

Explanation: Use the elimination method by adding 2x+3y=132x + 3y = 132x+3y=13 and 2x−3y=12x - 3y = 12x−3y=1. This eliminates y, giving 4x=144x = 144x=14, so x=14/4=3.5x = 14/4 = 3.5x=14/4=3.5. Substitute into 2x−3y=12x - 3y = 12x−3y=1: 2(3.5)−3y=12(3.5) - 3y = 12(3.5)−3y=1, 7−3y=17 - 3y = 17−3y=1, −3y=−6-3y = -6−3y=−6, y=2y = 2y=2. Alternatively, subtract the equations to get 6y=126y = 126y=12, y=2y = 2y=2 directly. Choice C of 3 might come from misadding to 4x=124x = 124x=12.

Question 24

The cost of attending a concert is modeled by the equation C=25n+5C = 25n + 5C=25n+5, where CCC is the total cost in dollars and nnn is the number of tickets. What is the cost for 3 tickets?

  1. $80 (correct answer)
  2. $75
  3. $30
  4. $5

Explanation: This concert cost model C = 25n + 5 shows total cost C for n tickets. The slope 25 means each ticket costs 25,andy−intercept5representsaflatservicefee.For3tickets:C=25(3)+5=75+5=25, and y-intercept 5 represents a flat service fee. For 3 tickets: C = 25(3) + 5 = 75 + 5 = 25,andy−intercept5representsaflatservicefee.For3tickets:C=25(3)+5=75+5=80. Choice B incorrectly uses the per-ticket cost without the service fee.

Question 25

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

  1. 1080°
  2. 1260°
  3. 1620°
  4. 1440° (correct answer)

Explanation: This question asks for the sum of interior angles of a decagon (10 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 10: 180(10-2) = 180(8) = 1440°. Choice A would result from incorrectly using 180(6) for an 8-sided polygon.