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ACT Math

ACT Math Practice Test: Practice Test 17

Practice Test 17 for ACT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

A store offers a 30% discount on a $120\$120$120 jacket. What is the sale price?

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Question 1

A store offers a 30% discount on a $120\$120$120 jacket. What is the sale price?

  1. $84 (correct answer)
  2. $90
  3. $100
  4. $80

Explanation: This question asks for the sale price after a 30% discount on a 120jacket.First,findthediscountamount:30120 jacket. First, find the discount amount: 30% of 120jacket.First,findthediscountamount:30120 = 0.30 × 120=120 = 120=36. Then subtract from the original price: 120−120 - 120−36 = 84.Alternatively,thesalepriceis7084. Alternatively, the sale price is 70% of the original: 0.70 × 84.Alternatively,thesalepriceis70120 = $84\$84$84. Choice C incorrectly kept the original price without applying the discount.

Question 2

In △XYZ\triangle XYZ△XYZ and △ABC\triangle ABC△ABC, △XYZ\triangle XYZ△XYZ has angles X=90oX = 90^\text{o}X=90o, Y=45oY = 45^\text{o}Y=45o, and △ABC\triangle ABC△ABC has angles A=90oA = 90^\text{o}A=90o, B=45oB = 45^\text{o}B=45o. Are the triangles similar?

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

Explanation: The triangles are similar by AA similarity criterion because they have two pairs of equal angles. Triangle XYZ has angles 90°, 45°, and 45° (since angles sum to 180°). Triangle ABC has angles 90°, 45°, and 45° (since angles sum to 180°). Having two pairs of equal angles (90° = 90° and 45° = 45°) confirms similarity by AA criterion.

Question 3

A fair six-sided die is rolled once. What is the probability of rolling an even number or a number greater than 4? (Assume outcomes are mutually exclusive only when appropriate.)

  1. 23\frac{2}{3}32​ (correct answer)
  2. 12\frac{1}{2}21​
  3. 56\frac{5}{6}65​
  4. 13\frac{1}{3}31​

Explanation: The sample space is {1, 2, 3, 4, 5, 6}. Even numbers are {2, 4, 6} and numbers greater than 4 are {5, 6}. Since 6 appears in both sets, favorable outcomes are {2, 4, 5, 6} = 4 outcomes. P(even or >4)=46=23P(\text{even or } >4) = \frac{4}{6} = \frac{2}{3}P(even or >4)=64​=32​. Choice D incorrectly treats these as mutually exclusive.

Question 4

Find the circumference of a circle with radius 14.

  1. 14π14\pi14π
  2. 28π28\pi28π (correct answer)
  3. 56π56\pi56π
  4. 42π42\pi42π

Explanation: We need to find the circumference of a circle with radius 14. Using the circumference formula C=2πrC = 2\pi rC=2πr: C=2π(14)=28πC = 2\pi(14) = 28\piC=2π(14)=28π. Choice A uses only the radius, choice C doubles the correct answer, and choice D uses 3 times the radius.

Question 5

What is f(0)f(0)f(0) for the function f(x)=3x−2f(x) = 3x - 2f(x)=3x−2?

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

Explanation: We need to find f(0) for the function f(x) = 3x - 2. To evaluate f(0), we substitute x = 0 into the function. f(0) = 3(0) - 2 = 0 - 2 = -2. The answer is -2, which corresponds to choice C. This represents the y-intercept of the linear function where the graph crosses the y-axis below the origin. Choice B incorrectly uses the coefficient of x instead of the constant term.

Question 6

A bar graph shows the number of minutes of screen time on four days.

How many days had screen time greater than 85 minutes?

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

Explanation: The question asks how many days had screen time greater than 85 minutes. Check each day's values: Day 1 (90), Day 2 (75), Day 3 (100), Day 4 (80). Day 1 (90) and Day 3 (100) are both greater than 85, so 2 days meet the criteria.

Question 7

A displacement is represented by v=⟨2, −7⟩\mathbf{v}=\langle 2,\,-7\ranglev=⟨2,−7⟩. What is −2v-2\mathbf{v}−2v?

  1. ⟨−2, 7⟩\langle -2,\,7\rangle⟨−2,7⟩
  2. ⟨4, −14⟩\langle 4,\,-14\rangle⟨4,−14⟩
  3. ⟨−4, −14⟩\langle -4,\,-14\rangle⟨−4,−14⟩
  4. ⟨−4, 14⟩\langle -4,\,14\rangle⟨−4,14⟩ (correct answer)

Explanation: This question requires computing -2 times the displacement vector v = ⟨2, -7⟩. Scalar multiplication by k of ⟨a, b⟩ yields ⟨k a, k b⟩. Calculate -2 × ⟨2, -7⟩ = ⟨-2 × 2, -2 × (-7)⟩ = ⟨-4, 14⟩. This reverses the direction and doubles the magnitude. Choice B is ⟨4, -14⟩, which forgets the sign change from the negative scalar.

Question 8

In △ABC\triangle ABC△ABC, the side lengths are AB=6AB=6AB=6, AC=8AC=8AC=8, and BC=10BC=10BC=10. In △DEF\triangle DEF△DEF, the side lengths are DE=9DE=9DE=9, DF=12DF=12DF=12, and EF=15EF=15EF=15. Which triangles are similar?

  1. △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF by SSS (correct answer)
  2. △ABC∼△DFE\triangle ABC \sim \triangle DFE△ABC∼△DFE by SSS
  3. The triangles are congruent by SSS
  4. The triangles are not similar because their perimeters are different

Explanation: To check if triangles are similar by SSS, we need all three pairs of corresponding sides to be proportional. For △ABC with sides 6, 8, 10 and △DEF with sides 9, 12, 15, we check the ratios: 6/9 = 2/3, 8/12 = 2/3, and 10/15 = 2/3. Since all three ratios equal 2/3, the triangles are similar by SSS. The correspondence is A↔D, B↔E, C↔F, making it △ABC ~ △DEF.

Question 9

Given the function f(x)=x2−3x+4f(x) = x^2 - 3x + 4f(x)=x2−3x+4, what is the value of f(−2)f(-2)f(−2)?

  1. 2
  2. 6
  3. 10
  4. 14 (correct answer)

Explanation: The correct answer is D (14). Substitute x = −2 into the function: f(−2) = (−2)² − 3(−2) + 4 = 4 + 6 + 4 = 14. The three key steps are: (1) square the input: (−2)² = +4, not −4; (2) multiply: −3(−2) = +6, not −6; (3) add: 4 + 6 + 4 = 14. A (2) comes from treating −3(−2) as −6: 4 − 6 + 4 = 2. B (6) comes from treating (−2)² as −4: −4 + 6 + 4 = 6. C (10) comes from a partial sign error. Pro tip: when substituting a negative value, write every step explicitly — sign errors on squared terms and products are the most common mistakes in function evaluation.

Question 10

Two fair six-sided dice are rolled. What is the probability that the sum is 7?

  1. 112\frac{1}{12}121​
  2. 16\frac{1}{6}61​ (correct answer)
  3. 736\frac{7}{36}367​
  4. 536\frac{5}{36}365​

Explanation: The sample space for two dice has 36 equally likely outcomes. The favorable outcomes for sum = 7 are {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}=6\{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\} = 6{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}=6 outcomes. P(sum=7)=636=16P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}P(sum=7)=366​=61​. This is the most likely sum when rolling two dice.

Question 11

A right triangle has hypotenuse length 171717 and one leg length 151515. What is the length of the other leg?

  1. 222
  2. 888 (correct answer)
  3. 514\sqrt{514}514​
  4. 64\sqrt{64}64​

Explanation: We need to find the unknown leg in a right triangle with hypotenuse 171717 and one leg 151515. Using the Pythagorean theorem: 152+leg2=17215^2 + \text{leg}^2 = 17^2152+leg2=172. Calculating: 225+leg2=289225 + \text{leg}^2 = 289225+leg2=289, so leg2=64\text{leg}^2 = 64leg2=64 and leg=8\text{leg} = 8leg=8. Choice D shows 64\sqrt{64}64​, which equals 8 but is not simplified.

Question 12

To determine the effectiveness of a new advertising campaign, a company surveys every 10th customer who makes a purchase online. Which sampling method is being used?

  1. Random sampling
  2. Convenience sampling
  3. Systematic sampling (correct answer)
  4. Stratified sampling

Explanation: This is systematic sampling because the company follows a regular pattern, surveying every 10th customer who makes an online purchase. Systematic sampling involves selecting subjects at fixed intervals from an ordered list or sequence, creating a predictable selection method. This approach provides good coverage of the customer base while being more efficient than random sampling. Random sampling (Choice A) would require each online customer to have equal probability of selection regardless of their purchase timing.

Question 13

A researcher surveyed 50 families and summarized her findings in the table. Of the families surveyed, what is the mean number of vehicles per family?

  1. 1.9 (correct answer)
  2. 2
  3. 2.1
  4. 2.5

Explanation: This is a mean from frequency table question testing the weighted average calculation. Choice A (1.9) is correct — multiply each vehicle count by its frequency, sum the products, then divide by total families. Total vehicles = (1 × 15) + (2 × 25) + (3 × 10) = 15 + 50 + 30 = 95. Mean = 95/50 = 1.9. Choice B (2.0) comes from computing (1 + 2 + 3)/3 = 2 — averaging the vehicle numbers without weighting by frequency. Choice C (2.1) results from an arithmetic error in one product, perhaps computing (2 × 25) = 52 instead of 50: 15 + 52 + 30 = 97, 97/50 = 1.94 ≈ 2.1... or another minor error. Choice D (2.5) averages only the vehicle counts (1 + 2 + 3 + 4)/4 type error, or computes (15 + 25 + 10)/something incorrectly. Pro tip: For frequency tables, NEVER average the category values directly. You must weight each value by how many times it appears. Think of it as expanding the table: 15 families with 1 vehicle = fifteen 1s; 25 families with 2 = twenty-five 2s; etc. Then sum and divide by total families (50).

Question 14

If f(x)=3x+2x−1f(x) = \dfrac{3x + 2}{x - 1}f(x)=x−13x+2​, what is the value of f−1(5)f^{-1}(5)f−1(5)?

  1. 12\dfrac{1}{2}21​
  2. 72\dfrac{7}{2}27​ (correct answer)
  3. 555
  4. 174\dfrac{17}{4}417​

Explanation: This is an inverse functions question testing a faster method than algebraically finding f⁻¹(x). Choice B (7/2) is correct — rather than deriving the full inverse function, directly solve f(x) = 5: (3x + 2)/(x − 1) = 5 → 3x + 2 = 5(x − 1) → 3x + 2 = 5x − 5 → 7 = 2x → x = 7/2. So f(7/2) = 5, which means f⁻¹(5) = 7/2. Choice A (1/2) comes from an arithmetic error in the solving step — perhaps writing 3x + 2 = 5x − 5 as 7 = 2x but then computing x = 7/14 = 1/2. Choice C (5) returns the input value — the student confuses f⁻¹(5) with the output of f, or interprets "f⁻¹(5) = 5" from f(5) = something. Choice D (17/4) evaluates f(5) instead of f⁻¹(5): f(5) = (15 + 2)/(5 − 1) = 17/4 — answering the wrong question. Pro tip: The fastest method for evaluating f⁻¹(a) is to solve f(x) = a directly. This avoids the algebraic work of finding the full inverse formula. Cross-multiplying (3x + 2)/(x − 1) = 5 gives 3x + 2 = 5x − 5. Collect x terms: 7 = 2x. Answer: x = 7/2.

Question 15

A complex impedance is given by −6+7i-6+7i−6+7i. What is the complex conjugate of −6+7i-6+7i−6+7i (flip the sign of the imaginary part only)?

  1. 6+7i6+7i6+7i
  2. −6−7i-6-7i−6−7i (correct answer)
  3. 6−7i6-7i6−7i
  4. −6+7i-6+7i−6+7i

Explanation: This problem asks for the complex conjugate of −6+7i-6 + 7i−6+7i, which is found by changing the sign of the imaginary part only. The real part is −6-6−6, and the imaginary part 7i7i7i becomes −7i-7i−7i. Thus, the conjugate is −6−7i-6 - 7i−6−7i. Choice D might result from incorrectly flipping the sign of the real part instead of the imaginary part.

Question 16

A map uses a scale of 111 inch to 444 miles. If two towns are 7.57.57.5 inches apart on the map, what is the actual distance between the towns in miles?

  1. 11
  2. 18
  3. 30 (correct answer)
  4. 32

Explanation: The map scale is 1 inch : 4 miles, meaning each inch on the map represents 4 miles in reality. To find the actual distance, multiply the map distance by the scale factor: 7.5 inches × 4 miles/inch = 30 miles. A common error is dividing instead of multiplying, which would give 1.875 miles.

Question 17

A deck of cards is shuffled and one card is drawn. What is the probability of drawing a King or a heart?

  1. 513\frac{5}{13}135​
  2. 213\frac{2}{13}132​
  3. 313\frac{3}{13}133​
  4. 413\frac{4}{13}134​ (correct answer)

Explanation: There are 4 Kings and 13 hearts in a standard deck, but the King of hearts appears in both categories. Using the inclusion-exclusion principle: P(Kingorheart)=P(King)+P(heart)−P(Kingandheart)=452+1352−152=1652=413P(King or heart) = P(King) + P(heart) - P(King and heart) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}P(Kingorheart)=P(King)+P(heart)−P(Kingandheart)=524​+5213​−521​=5216​=134​. Choice B (213\frac{2}{13}132​) might only count the Kings, while choice C (313\frac{3}{13}133​) incorrectly omits some cards.

Question 18

A student has 7 different pens and wants to place exactly 5 of them into a pencil case in a specific left-to-right order. How many different ordered selections are possible?

  1. 21
  2. 2520 (correct answer)
  3. 120
  4. 420

Explanation: Since the pens are placed in a specific left-to-right order, order matters, making this a permutation problem. We select and arrange 5 out of 7 using P(7,5) = 7! / (7-5)! = 7 × 6 × 5 × 4 × 3. Calculating: 7 × 6 = 42, 42 × 5 = 210, 210 × 4 = 840, 840 × 3 = 2,520. Alternatively, it's C(7,5) × 5! = 21 × 120 = 2,520. A key distractor is choice D (420), which is P(7,4), perhaps from choosing 4 instead of 5.

Question 19

How many sides does a regular polygon have if each exterior angle measures 45∘45^\circ45∘?

  1. 6
  2. 8 (correct answer)
  3. 10
  4. 12

Explanation: The question asks how many sides a regular polygon has if each exterior angle measures 45 degrees. The sum of exterior angles for any polygon is always 360 degrees, and for a regular polygon, each exterior angle is 360/n degrees, where n is the number of sides. Set 360/n=45, so n=360/45=8. Note that each interior angle is 180 minus the exterior angle, but here we use the exterior sum directly. Choice A, 6, might be from confusing with 60-degree exterior angles, as 360/60=6.

Question 20

Which equation represents a circle with center (-4, 5) and radius 6?

  1. (x−4)2+(y−5)2=6(x-4)^2 + (y-5)^2 = 6(x−4)2+(y−5)2=6
  2. (x−4)2+(y+5)2=36(x-4)^2 + (y+5)^2 = 36(x−4)2+(y+5)2=36
  3. (x+4)2+(y+5)2=6(x+4)^2 + (y+5)^2 = 6(x+4)2+(y+5)2=6
  4. (x+4)2+(y−5)2=36(x+4)^2 + (y-5)^2 = 36(x+4)2+(y−5)2=36 (correct answer)

Explanation: We need to write the equation of a circle with center (-4, 5) and radius 6. The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center. Substituting center (-4, 5) and r = 6: (x-(-4))² + (y-5)² = 6², which simplifies to (x+4)² + (y-5)² = 36. Choice B incorrectly places the center at (4, -5) instead of (-4, 5).

Question 21

Solve: 6−x<26 - x < 26−x<2

  1. x>4x > 4x>4 (correct answer)
  2. x<4x < 4x<4
  3. x>−4x > -4x>−4
  4. x<−4x < -4x<−4

Explanation: This is a linear inequality with a negative coefficient requiring sign reversal. Starting with 6 - x < 2, subtract 6 from both sides to get -x < -4. Multiply by -1 (negative, so flip the inequality sign) to get x > 4. The solution is x > 4, which matches choice A. Choice B forgot to flip the inequality sign when multiplying by a negative, leading to the incorrect x < 4. This type of sign error is very common when dealing with negative coefficients.

Question 22

If two dice are rolled, what is the probability of getting a sum of 7?

  1. 1/6 (correct answer)
  2. 1/12
  3. 5/36
  4. 1/36

Explanation: When rolling two dice, the sample space contains 36 equally likely outcomes. The event "sum equals 7" includes favorable outcomes {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}, giving us 6 favorable outcomes. Applying P(event) = favorable / total, we get P(sum = 7) = 6/36 = 1/6. Choice D (1/36) represents the probability of rolling a specific ordered pair like (1,6).

Question 23

A gym charges a one-time sign-up fee plus a monthly membership cost. In the equation y=25x+40y = 25x + 40y=25x+40, xxx is the number of months and yyy is the total cost (in dollars). What is the meaning of the yyy-intercept in this context?

  1. The monthly cost is $40 per month.
  2. The total cost after 40 months is $25.
  3. The one-time sign-up fee is $40. (correct answer)
  4. The total cost increases by $40 each month.

Explanation: This is a linear model y = 25x + 40 where y is total cost in dollars and x is number of months. The slope 25 means the monthly membership cost is 25permonth.They−intercept40representsthevaluewhenx=0,whichistheinitialcostbeforeanymonthlypayments−theone−timesign−upfeeof25 per month. The y-intercept 40 represents the value when x = 0, which is the initial cost before any monthly payments - the one-time sign-up fee of 25permonth.They−intercept40representsthevaluewhenx=0,whichistheinitialcostbeforeanymonthlypayments−theone−timesign−upfeeof40. Choice A incorrectly identifies the y-intercept as the monthly cost, which is actually the slope. Choice D confuses the y-intercept with the rate of change.

Question 24

A paint mixture uses a ratio of blue to yellow of 3:23:23:2. If the artist uses 181818 cups of blue paint, how many cups of yellow paint are needed to keep the same ratio?

  1. 10
  2. 12 (correct answer)
  3. 15
  4. 20

Explanation: The ratio of blue to yellow paint is 3:2, meaning for every 3 cups of blue paint, we need 2 cups of yellow paint. Set up the proportion: 3/2 = 18/y, where y is the cups of yellow paint needed. Cross-multiply: 3y = 36, so y = 12 cups of yellow paint. A common error would be to reverse the ratio and get 27 cups, but we must maintain the original 3:2 relationship.

Question 25

A price increases from $40\$40$40 to $50. What is the percent increase?

  1. 20%
  2. 25% (correct answer)
  3. 30%
  4. 15%

Explanation: This question asks for the percent increase from 40to40 to 40to50. Percent change is (new value - original value) ÷ original value × 100. Here: (50 - 40) ÷ 40 × 100 = 10 ÷ 40 × 100 = 0.25 × 100 = 25%. Choice A incorrectly calculated 20%, possibly using the wrong denominator or making an arithmetic error.