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ACT Math

ACT Math Practice Test: Practice Test 15

Practice Test 15 for ACT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

Which value of xxx satisfies the equation 3x+4=x+183x + 4 = x + 183x+4=x+18?

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Question 1

Which value of xxx satisfies the equation 3x+4=x+183x + 4 = x + 183x+4=x+18?

  1. 111111
  2. 777 (correct answer)
  3. −7-7−7
  4. 141414

Explanation: This equation has variables on both sides. Starting with 3x + 4 = x + 18, subtract x from both sides: 2x + 4 = 18. Subtract 4 from both sides: 2x = 14. Divide by 2: x = 7.

Question 2

A complex number is given by −9−12i-9 - 12i−9−12i. To find its distance from the origin, compute its absolute value. What is the absolute value of (−9−12i)(-9 - 12i)(−9−12i)? (Simplify the radical.)

  1. 225\sqrt{225}225​
  2. 212121
  3. 151515 (correct answer)
  4. 153\sqrt{153}153​

Explanation: The operation is finding the absolute value of -9 - 12i. Compute it as the square root of ((-9) squared plus (-12) squared), which is sqrt(81 + 144) = sqrt(225). Simplifying the radical gives 15. This is the distance from the origin. Note that choice A is the unsimplified form, but the question asks to simplify the radical.

Question 3

In the right triangle shown, the right angle is at BBB. The side lengths are AB=3AB=3AB=3, BC=4BC=4BC=4, and AC=5AC=5AC=5. Angle θ\thetaθ is at AAA (between ABABAB and ACACAC). What is sin⁡(θ)\sin(\theta)sin(θ)?

  1. 35\frac{3}{5}53​
  2. 45\frac{4}{5}54​ (correct answer)
  3. 54\frac{5}{4}45​
  4. 53\frac{5}{3}35​

Explanation: For angle θθθ at vertex A, we need to identify the sides relative to this angle. The opposite side to angle θθθ is BC=4BC = 4BC=4, the adjacent side is AB=3AB = 3AB=3, and the hypotenuse is AC=5AC = 5AC=5. Using SOH-CAH-TOA, sin⁡(θ)=oppositehypotenuse=BCAC=45\sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{4}{5}sin(θ)=hypotenuseopposite​=ACBC​=54​. Choice A gives 35\frac{3}{5}53​, which would be using the adjacent side instead of the opposite side.

Question 4

If  6(x+2)−3=21 \,6(x + 2) - 3 = 21\,6(x+2)−3=21, what is xxx?

  1. 111
  2. 222 (correct answer)
  3. 555
  4. −1-1−1

Explanation: First distribute the 6: 6x + 12 - 3 = 21. Simplify: 6x + 9 = 21. Subtract 9 from both sides: 6x = 12. Divide by 6: x = 2. The value of x is 2.

Question 5

A ball is thrown upward from a height of 5 meters. Its height after xxx seconds is modeled by the quadratic equation y=−4x2+12x+5y = -4x^2 + 12x + 5y=−4x2+12x+5, where yyy is height in meters.

According to the model, what is the height of the ball after 222 seconds?

  1. 212121 meters
  2. 131313 meters (correct answer)
  3. 555 meters
  4. −3-3−3 meters

Explanation: This is a quadratic model y=−4x2+12x+5y = -4x^2 + 12x + 5y=−4x2+12x+5 where y represents height in meters and x represents time in seconds. The coefficient 5 represents the initial height (5 meters when x = 0). To find the height after 2 seconds, substitute x = 2: y=−4(2)2+12(2)+5=−4(4)+24+5=−16+24+5=13y = -4(2)^2 + 12(2) + 5 = -4(4) + 24 + 5 = -16 + 24 + 5 = 13y=−4(2)2+12(2)+5=−4(4)+24+5=−16+24+5=13 meters. The model predicts the ball will be 13 meters high after 2 seconds. Choice A would result from calculation errors, while choice C represents the initial height rather than the height at t = 2.

Question 6

How many permutations of 5 letters taken 3 at a time?

  1. 60 (correct answer)
  2. 10
  3. 20
  4. 120

Explanation: Order matters in permutations by definition, so we use the permutation formula. We need P(5,3) = 5!/(5-3)! = 5!/2! = 5 × 4 × 3 = 60. The calculation shows we're selecting and arranging 3 letters from 5 available positions. Choice D incorrectly used 5! = 120 instead of the correct permutation formula.

Question 7

A shipment is packed into identical boxes. If the total number of items is given by 6x=546x = 546x=54, what is the value of xxx?

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

Explanation: This is a simple linear equation 6x = 54 representing total items in boxes, with x as items per box. Divide both sides by 6: x = 54 / 6. This simplifies to x = 9. Therefore, the value of x is 9, matching choice C. Choice B (8) might come from misdividing 48 by 6 or subtracting before dividing.

Question 8

Which formula gives the surface area of a right circular cylinder with radius rrr and height hhh (including both circular bases)?

  1. πr2h\pi r^2hπr2h
  2. 2πr2+2πrh2\pi r^2+2\pi rh2πr2+2πrh (correct answer)
  3. 43πr3\dfrac{4}{3}\pi r^334​πr3
  4. πr2+2πrh\pi r^2+2\pi rhπr2+2πrh

Explanation: The question asks for the formula giving the surface area of a right circular cylinder including both bases, with radius r and height h. The correct formula is SA = 2πr² + 2πrh, which accounts for the two circular bases (2πr²) and the lateral surface (2πrh). This combines the areas properly. Choice A is the volume formula, not surface area. Choice D might confuse by omitting one base or miscounting.

Question 9

In the diagram, △XYZ is given with angles θ=50o, and angle Y=70o\triangle XYZ \text{ is given with angles } \theta = 50^\text{o}, \text{ and } \text{angle } Y = 70^\text{o}△XYZ is given with angles θ=50o, and angle Y=70o. △PQR is given with angles P=50o and R=60o\triangle PQR \text{ is given with angles } P = 50^\text{o} \text{ and } R = 60^\text{o}△PQR is given with angles P=50o and R=60o. Are the triangles similar?

  1. Yes, by AA similarity. (correct answer)
  2. No, they are not similar.
  3. Yes, by SAS similarity.
  4. Yes, by ASA similarity.

Explanation: The triangles are similar by AA similarity criterion because they have two pairs of equal angles. Triangle XYZ has angles 50°, 70°, and 60° (since angles sum to 180°). Triangle PQR has angles 50°, 60°, and 70° (since angles sum to 180°). Since both triangles have the same three angle measures, they have two pairs of equal angles, confirming similarity by AA.

Question 10

To remove imaginary terms from a denominator later, you plan to multiply by a conjugate. What is the complex conjugate of 8−11i8 - 11i8−11i?

  1. −8+11i-8 + 11i−8+11i
  2. 8+11i8 + 11i8+11i (correct answer)
  3. −8−11i-8 - 11i−8−11i
  4. 8−11i8 - 11i8−11i

Explanation: The operation is finding the complex conjugate of 8 - 11i. The conjugate changes the sign of the imaginary part, so for a + bi it becomes a - bi, but here it's 8 + (-11)i, so conjugate is 8 + 11i. This is useful for rationalizing denominators. Choice D is the original number, a common distractor if forgetting to change the sign.

Question 11

In the diagram, a straight line forms a linear pair of adjacent angles. One angle measures 48∘48^\circ48∘, and the adjacent angle is labeled xxx. What is the measure of angle xxx?

  1. 132∘132^\circ132∘ (correct answer)
  2. 42∘42^\circ42∘
  3. 48∘48^\circ48∘
  4. 90∘90^\circ90∘

Explanation: The angles form a linear pair on a straight line. Angles in a linear pair are supplementary, meaning they add up to 180∘180^\circ180∘. To find x, subtract the given 48∘48^\circ48∘ from 180∘180^\circ180∘: x \= 180^\circ - 48^\circ \= 132^\circ. This calculation emphasizes the straight-line property. Choice C of 48∘48^\circ48∘ might confuse this with vertical angles, which are equal instead of supplementary.

Question 12

Two triangles are shown with side lengths. Triangle △GHI\triangle GHI△GHI has GH=7GH=7GH=7, HI=9HI=9HI=9, and GI=12GI=12GI=12. Triangle △JKL\triangle JKL△JKL has JK=14JK=14JK=14, KL=18KL=18KL=18, and JL=20JL=20JL=20.

Are the triangles similar? If so, why?

  1. Yes; SSSSSSSSS similarity because all three pairs of corresponding sides are proportional.
  2. Yes; SASSASSAS similarity because two sides are proportional and an included angle is equal.
  3. No; the side ratios are not all equal. (correct answer)
  4. No; triangles must have at least one equal angle to be similar.

Explanation: Triangles GHI and JKL are not similar because the corresponding sides are not proportional. Assuming correspondences G to J, H to K, and I to L based on side lengths, the ratios are JK/GH = 14/7 = 2, KL/HI = 18/9 = 2, but JL/GI = 20/12 = 5/3, which are inconsistent. Without a constant scale factor, similarity criteria like SSS or SAS are not met. This addresses the distractor that unequal ratios still allow similarity, confirming they do not.

Question 13

What is the median of the data set [9, 5, 12, 8, 7]?

  1. 8 (correct answer)
  2. 9
  3. 7
  4. 12

Explanation: The median is the middle value when data is sorted in order. First, sort the data: [5, 7, 8, 9, 12]. Since there are 5 values (odd count), the median is the middle (3rd) value = 8. The median is resistant to extreme values and represents the central tendency.

Question 14

A coffee shop sells muffins for $3\$3$3 each. Jordan buys 4 muffins and pays with a $20\$20$20 bill. How much change does Jordan receive?

  1. $8 (correct answer)
  2. $12
  3. $17
  4. $8 dollars

Explanation: We need to find how much change Jordan receives after buying 4 muffins at 3eachwitha3 each with a 3eachwitha20 bill. The total cost of the muffins is 4×3=124 \times 3 = 124×3=12. To find the change, we subtract the total cost from the amount paid: 20−12=820 - 12 = 820−12=8. Choice B ($12\$12$12) represents the total cost of the muffins, not the change received.

Question 15

In the diagram, lines mmm and nnn are parallel, and transversal ttt intersects them. The angle labeled 35∘35^\circ35∘ is an interior angle below line mmm and to the left of the transversal. Angle xxx is an interior angle above line nnn and to the right of the transversal (forming an alternate interior pair with the 35∘35^\circ35∘ angle). What is the measure of angle xxx?

  1. 145∘145^\circ145∘
  2. 55∘55^\circ55∘
  3. 35∘35^\circ35∘ (correct answer)
  4. 90∘90^\circ90∘

Explanation: The angles are alternate interior angles formed by a transversal intersecting parallel lines. Alternate interior angles are equal when the lines are parallel, as they lie on opposite sides of the transversal but inside the parallels. Therefore, angle x measures the same as the given 35° angle, so x = 35°. This equality is a key property of parallel lines. A distractor like 145° might arise from confusing them with supplementary angles.

Question 16

Simplify: 5a2−3a+2a2+7a5a^2-3a+2a^2+7a5a2−3a+2a2+7a.

  1. 3a2+4a3a^2+4a3a2+4a
  2. 7a2−10a7a^2-10a7a2−10a
  3. 7a4+4a7a^4+4a7a4+4a
  4. 7a2+4a7a^2+4a7a2+4a (correct answer)

Explanation: To simplify this expression, we need to combine like terms by grouping terms with the same variable and exponent. Group the a² terms: 5a² + 2a² = 7a², and group the a terms: -3a + 7a = 4a. The simplified form is 7a² + 4a. We cannot combine a² terms with a terms since they have different exponents. Choice B shows 7a² - 10a, which incorrectly combines -3a + 7a as -10a instead of +4a.

Question 17

The times (in minutes) it took a runner to complete a short course over 8 trials were: 9, 12, 11, 10, 13, 10, 14, 8. What is the median of {9,12,11,10,13,10,14,8}\{9, 12, 11, 10, 13, 10, 14, 8\}{9,12,11,10,13,10,14,8}?

  1. 10
  2. 11
  3. 10.5 (correct answer)
  4. 12

Explanation: The median is the middle value of a sorted data set; for an even number of values, it is the average of the two middle values. Sort {9, 12, 11, 10, 13, 10, 14, 8} to get 8, 9, 10, 10, 11, 12, 13, 14. With 8 values, the median is the average of the 4th and 5th values: (10 + 11) / 2 = 10.5. The median is 10.5. Choice A is the 3rd value, which might be mistaken for the median in an unsorted list.

Question 18

In the standard (x,y)(x, y)(x,y) coordinate plane, what is the slope of the line given by the equation 5x+2y=105x + 2y = 105x+2y=10?

  1. −5-5−5
  2. −52-\frac{5}{2}−25​ (correct answer)
  3. −25-\frac{2}{5}−52​
  4. 52\frac{5}{2}25​

Explanation: This is a slope-intercept form question testing equation rewriting. Choice B (−5/2) is correct — isolate y: 2y = −5x + 10, so y = (−5/2)x + 5. The slope is the coefficient of x: −5/2. Choice A (−5) reads the coefficient of x from the original equation before dividing by 2 — correctly identifying the −5 but forgetting that both sides must be divided by 2 when solving for y. Choice C (−2/5) inverts the slope fraction, giving the negative reciprocal of the correct answer (which would actually be the slope of a perpendicular line). Choice D (5/2) inverts the fraction AND drops the negative sign — two errors compounded. Pro tip: Never read the slope from an equation unless y is completely isolated. Rewrite 5x + 2y = 10 as y = (−5/2)x + 5 first, then the slope is right there as the coefficient of x.

Question 19

In a circle of radius 444, arc EF⌢\overset{\frown}{EF}EF⌢ is subtended by a central angle of 135∘135^\circ135∘. What is the length of arc EF⌢\overset{\frown}{EF}EF⌢?

  1. 8π3\dfrac{8\pi}{3}38π​
  2. 3π2\dfrac{3\pi}{2}23π​
  3. 6π6\pi6π
  4. 3π3\pi3π (correct answer)

Explanation: We need to find the arc length subtended by a 135° central angle in a circle of radius 4. The arc length formula is arc = (θ/360°) × 2πr. Substituting: arc = (135°/360°) × 2π(4) = (3/8) × 8π = 3π. Choice B gives half this result, while choice C incorrectly doubles the circumference.

Question 20

What is tan⁡(π4)\tan(\frac{\pi}{4})tan(4π​)?

  1. 000
  2. 111 (correct answer)
  3. 3\sqrt{3}3​
  4. 32\frac{\sqrt{3}}{2}23​​

Explanation: In the unit circle, π/4\pi/4π/4 radians equals 45∘45^\circ45∘. Using SOH-CAH-TOA, tangent represents opposite over adjacent. For the special angle π/4\pi/4π/4 (45∘45^\circ45∘), tan⁡(π/4)=1\tan(\pi/4) = 1tan(π/4)=1. Choice C (3\sqrt{3}3​) is actually tan⁡(π/3)\tan(\pi/3)tan(π/3) or tan⁡(60∘)\tan(60^\circ)tan(60∘).

Question 21

What is the measure of each interior angle in a regular decagon (10-gon)?

  1. 140∘140^\circ140∘
  2. 150∘150^\circ150∘
  3. 135∘135^\circ135∘
  4. 144∘144^\circ144∘ (correct answer)

Explanation: This problem asks for each interior angle in a regular decagon (10-gon). For a regular n-sided polygon, each interior angle is 180(n-2)/n degrees. Substituting n = 10: 180(10-2)/10 = 180(8)/10 = 1440/10 = 144°. Choice B represents the angle for a 12-sided polygon, while choice C represents an octagon's angle.

Question 22

The scale factor from model A to model B is 3. If the length of model A is 5 cm, what is the length of model B?

  1. 15 cm (correct answer)
  2. 10 cm
  3. 8 cm
  4. 12 cm

Explanation: A scale factor of 3 means model B is 3 times larger than model A. If model A has length 5 cm, then model B has length 5 × 3 = 15 cm. Students might confuse scale factor direction or think it means dividing instead of multiplying.

Question 23

In △ABC\triangle ABC△ABC, the length of side aaa is 5, the length of side bbb is 7, and the measure of ∠C\angle C∠C is 60°60°60°. What is the length of side ccc?

  1. 39\sqrt{39}39​ (correct answer)
  2. 74\sqrt{74}74​
  3. 888
  4. 109\sqrt{109}109​

Explanation: This is a law of cosines question. Choice A (√39) is correct — the Law of Cosines: c² = a² + b² − 2ab·cos(C) = 5² + 7² − 2(5)(7)·cos(60°) = 25 + 49 − 70(0.5) = 74 − 35 = 39. Therefore c = √39. Choice B (√74) results from omitting the cosine term entirely: c² = 5² + 7² = 74 — applying the Pythagorean theorem as if the triangle were a right triangle. Choice C (8) results from rounding or estimating √39 ≈ 6.24... possibly from a computational error that produces c² = 64. Choice D (√109) results from adding the cosine term instead of subtracting: c² = 25 + 49 + 35 = 109 — a sign error on the formula. Pro tip: The Law of Cosines formula always subtracts the cosine term: c² = a² + b² − 2ab·cos(C). For acute angles (less than 90°), cos(C) is positive, so the term 2ab·cos(C) reduces c² below the Pythagorean sum. For obtuse angles, cos(C) is negative, and c² is larger than a² + b².

Question 24

Solve the system: 3x+2y=123x + 2y = 123x+2y=12 and x−y=1x - y = 1x−y=1.

  1. (4, 0)
  2. (2, 3)
  3. (1, 5)
  4. (3, 2) (correct answer)

Explanation: Use substitution to solve this system. From x - y = 1, we get x = y + 1. Substituting into 3x + 2y = 12: 3(y + 1) + 2y = 12, which simplifies to 5y + 3 = 12, so y = 2. Then x = 2 + 1 = 3. The solution is (3, 2).

Question 25

The data set is {5,7,8,10,12}\{5, 7, 8, 10, 12\}{5,7,8,10,12}. If the value 100 is added to the data set, what happens to the median?

  1. It becomes 10.
  2. It becomes 8.
  3. It becomes 12.
  4. It becomes 9. (correct answer)

Explanation: The original median of {5, 7, 8, 10, 12} is 8 (the middle value). When 100 is added, the sorted set becomes {5, 7, 8, 10, 12, 100}, and with 6 values, the median is the average of the 3rd and 4th values. New median = (8 + 10) ÷ 2 = 9. The median changes from 8 to 9 when the extreme value 100 is added.