Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Opening subject page...

Loading your content

ACT Math

ACT Math Practice Test: Practice Test 6

Practice Test 6 for ACT Math: real questions and explanations from the Varsity Tutors practice-test pool.

0%

0 / 25 answered

Question 1 of 25

For positive numbers xxx and yyy, which expression is equivalent to log⁡(xy)\log(xy)log(xy) (base 10)?

Question Navigator

All questions

Question 1

For positive numbers xxx and yyy, which expression is equivalent to log⁡(xy)\log(xy)log(xy) (base 10)?

  1. log⁡(x)+log⁡(y)\log(x)+\log(y)log(x)+log(y) (correct answer)
  2. log⁡(x)−log⁡(y)\log(x)-\log(y)log(x)−log(y)
  3. log⁡(x)⋅log⁡(y)\log(x)\cdot\log(y)log(x)⋅log(y)
  4. log⁡ ⁣(xy)\log\!\left(\dfrac{x}{y}\right)log(yx​)

Explanation: We need the product rule for logarithms to simplify log(xy). The product rule states that log_a(xy) = log_a(x) + log_a(y) for positive x and y. Therefore, log(xy) = log(x) + log(y). Choice C incorrectly multiplies the logarithms, while choice B would apply to log(x/y).

Question 2

A sweater is discounted 20%20\%20% from its original price of \65$. What is the discounted price?

  1. \52$ (correct answer)
  2. \45$
  3. \78$
  4. \13$

Explanation: We need to find the discounted price of a 65sweaterwitha2065 sweater with a 20% discount. First find the discount amount: 20% of 65sweaterwitha2065 = 0.20 × 65=65 = 65=13. Then subtract from the original price: 65−65 - 65−13 = 52.ChoiceD(52. Choice D (52.ChoiceD(13) is just the discount amount, not the final price.

Question 3

A cylinder has a surface area of 150π square units and a radius of 5 units. What is the height of the cylinder?

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

Explanation: We need to find the height of a cylinder with surface area 150π square units and radius 5 units. Using SA = 2πr² + 2πrh where r = 5: 150π = 2π(5²) + 2π(5)h = 50π + 10πh. Solving: 150π = 50π + 10πh, so 100π = 10πh, therefore h = 10 units.

Question 4

What is the surface area of a cube with side length 7 units?

  1. 294 square units (correct answer)
  2. 147 square units
  3. 343 square units
  4. 49 square units

Explanation: This is finding the surface area of a cube with side length 7 units. The surface area formula for a cube is SA = 6s², where s is the side length, since a cube has six square faces. Substituting s = 7: SA = 6(7²) = 6(49) = 294 square units. Choice C uses s³ which gives volume, not surface area.

Question 5

For the function f(x)={x+2if x≤−33x−7if x>−3f(x) = \begin{cases} x + 2 & \text{if } x \leq -3 \\ 3x - 7 & \text{if } x > -3 \end{cases}f(x)={x+23x−7​if x≤−3if x>−3​, what is f(−3)f(-3)f(−3)?

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

Explanation: For x=−3x = -3x=−3, we check which interval contains it: Is −3≤−3-3 \leq -3−3≤−3? Yes. So we use the first piece: f(x)=x+2f(x) = x + 2f(x)=x+2. Substituting x=−3x = -3x=−3: f(−3)=(−3)+2=−1f(-3) = (-3) + 2 = -1f(−3)=(−3)+2=−1. The boundary point x=−3x = -3x=−3 is included in the first piece due to the ≤\leq≤ symbol.

Question 6

A set of 8 integers is: 1, 2, 2, 3, 4, 4, 5, 100. Which best describes the shape of the distribution?

  1. Symmetric
  2. Skewed right (correct answer)
  3. Skewed left
  4. Uniform

Explanation: The shape of a distribution can be skewed if there's a tail due to outliers. In {1,2,2,3,4,4,5,100}, values cluster low around 1-5, with 100 as a high outlier creating a tail to the right. This asymmetry indicates right skew. The distribution is skewed right. Choice C of skewed left would apply if the outlier was low, like a negative value pulling left.

Question 7

A student has taken 3 tests and received scores of 85, 92, and 78. What score must the student earn on the 4th test to have an overall mean score of exactly 85 for the 4 tests?

  1. 80
  2. 85 (correct answer)
  3. 88
  4. 90

Explanation: The correct answer is B (85). To find the required 4th test score, first calculate the target sum: 4 × 85 = 340. Then find the current sum of the first 3 scores: 85 + 92 + 78 = 255. The needed score is 340 − 255 = 85. A (80) is a trap for students who notice the current average is already 85 and guess a nearby number. C (88) results from an arithmetic error in computing the sum or target. D (90) overestimates how much the low score of 78 pulls the average down. Pro tip: always compute the target sum first, then subtract the known values — this method works for any weighted average problem on the ACT.

Question 8

Lines ppp and qqq are parallel, and line rrr is a transversal. If angle 222 is 110∘110^\circ110∘, what is the measure of the corresponding angle 444?

  1. 70°
  2. 110° (correct answer)
  3. 80°
  4. 100°

Explanation: Angles 222 and 444 are corresponding angles formed by parallel lines p and q cut by transversal r. When parallel lines are cut by a transversal, corresponding angles are equal. Since angle 222 is 110∘110^\circ110∘, angle 444 must also be 110∘110^\circ110∘. Choice A incorrectly uses 70∘70^\circ70∘, which would be the supplementary angle.

Question 9

A hot air balloon is descending at a rate of 5 meters per minute. If its initial altitude is 200 meters, what equation models the altitude yyy after xxx minutes?

  1. y=200−xy = 200 - xy=200−x
  2. y=5x+200y = 5x + 200y=5x+200
  3. y=200+5xy = 200 + 5xy=200+5x
  4. y=200−5xy = 200 - 5xy=200−5x (correct answer)

Explanation: This is a linear altitude model where y represents altitude and x represents time in minutes. The slope of -5 means the altitude decreases by 5 meters per minute (negative because it's descending). The y-intercept of 200 represents the initial altitude when x = 0 minutes. The equation y = 200 - 5x correctly models this decreasing relationship. Choice B incorrectly uses a positive slope, which would represent ascending rather than descending. Choice A uses the wrong rate of change.

Question 10

Given △GHI with sides GH=5,HI=12, and IG=13\triangle GHI \text{ with sides } GH = 5, HI = 12, \text{ and } IG = 13△GHI with sides GH=5,HI=12, and IG=13, and △JKL with sides JK=10,KL=24, and LJ=26\triangle JKL \text{ with sides } JK = 10, KL = 24, \text{ and } LJ = 26△JKL with sides JK=10,KL=24, and LJ=26. What is the scale factor from △GHI\triangle GHI△GHI to △JKL\triangle JKL△JKL?

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

Explanation: The triangles are similar by SSS since all corresponding sides are proportional. Triangle GHI has sides 5, 12, 13 and triangle JKL has sides 10, 24, 26. The ratios are 10/5 = 2, 24/12 = 2, and 26/13 = 2. Since all ratios equal 2, the scale factor from triangle GHI to triangle JKL is 2:1.

Question 11

For a 2×2 matrix [abcd]\begin{bmatrix}a & b\\ c & d\end{bmatrix}[ac​bd​], the determinant is ad−bcad-bcad−bc. What is the determinant of [25−14]\begin{bmatrix}2 & 5\\ -1 & 4\end{bmatrix}[2−1​54​]?

  1. 131313 (correct answer)
  2. 333
  3. −13-13−13
  4. −3-3−3

Explanation: This problem asks for the determinant of a 2×2 matrix using the formula ad - bc. For the matrix [[2, 5], [-1, 4]], we compute: determinant = (2)(4) - (5)(-1) = 8 - (-5) = 8 + 5 = 13. The result is 13.

Question 12

What is the slope of the line through points (-1, 4) and (2, -5)?

  1. 3
  2. -\frac{9}{3}
  3. \frac{9}{3}
  4. -3 (correct answer)

Explanation: To find the slope of the line through points (-1, 4) and (2, -5), we use the slope formula m = (y₂ - y₁)/(x₂ - x₁). Substituting the coordinates: m = (-5 - 4)/(2 - (-1)) = (-9)/(3) = -3. The slope is -3, indicating the line decreases by 3 units vertically for every 1 unit horizontally. Choice B shows -9/3, which equals -3 but isn't simplified, while choice C shows the fraction with opposite signs.

Question 13

A grocery store wants to know which of two shelf layouts leads to higher sales. For one month, Store 1 uses Layout A and Store 2 uses Layout B, then sales are compared. The stores are in different neighborhoods with different customer bases. Which characteristic makes this comparison potentially biased?

  1. The layouts are tested for a full month, so the sample is too large
  2. Different stores may have different customers, so layout is mixed with neighborhood effects (correct answer)
  3. Sales are measured in dollars, which is a quantitative outcome
  4. The study compares two layouts, which makes it an observational study

Explanation: Different customer bases between neighborhoods create confounding bias because layout effects cannot be separated from neighborhood demographic differences. The comparison is biased because Store 1 and Store 2 serve different populations who might have different shopping behaviors regardless of shelf layout. Any sales difference could be due to neighborhood characteristics rather than the layout intervention. Choice A incorrectly suggests sample size is the issue, but the problem is confounding variables, not sample size.

Question 14

Solve the system:

4x - y = 9\\ 2x + y = 3 \end{cases}$$ Which ordered pair satisfies both equations?
  1. (2,−1)(2,-1)(2,−1) (correct answer)
  2. (2,1)(2,1)(2,1)
  3. (1,1)(1,1)(1,1)
  4. (1,−1)(1,-1)(1,−1)

Explanation: Use elimination: add equations 4x−y=94x - y = 94x−y=9 and 2x+y=32x + y = 32x+y=3. This gives 6x=126x = 126x=12, so x=2x = 2x=2. Substitute x=2x = 2x=2 into 2x+y=32x + y = 32x+y=3: 2(2)+y=32(2) + y = 32(2)+y=3, so y=−1y = -1y=−1. The solution is (2,−1)(2, -1)(2,−1). Choice B has the wrong sign for y.

Question 15

Which equation represents a circle with center (3, -4) and radius 5?

  1. (x−3)2+(y+4)2=5(x - 3)^2 + (y + 4)^2 = 5(x−3)2+(y+4)2=5
  2. (x+3)2+(y−4)2=25(x + 3)^2 + (y - 4)^2 = 25(x+3)2+(y−4)2=25
  3. (x−3)2+(y+4)2=25(x - 3)^2 + (y + 4)^2 = 25(x−3)2+(y+4)2=25 (correct answer)
  4. (x−3)2+(y−4)2=5(x - 3)^2 + (y - 4)^2 = 5(x−3)2+(y−4)2=5

Explanation: We need the equation of a circle with center (3, -4) and radius 5. The standard form is (x - h)² + (y - k)² = r², where (h, k) is the center. Substituting: (x - 3)² + (y - (-4))² = 5², which gives (x - 3)² + (y + 4)² = 25. Choice B incorrectly changes the signs of the center coordinates.

Question 16

In a right triangle, the hypotenuse is 15 units and one leg is 9 units. What is the length of the other leg?

  1. 10 units
  2. 11 units
  3. 12 units (correct answer)
  4. 13 units

Explanation: We need to find the unknown leg of a right triangle with hypotenuse = 15 units and one leg = 9 units. Using the Pythagorean theorem: a² + b² = c². So 9² + b² = 15², which gives 81 + b² = 225, therefore b² = 144, and b = 12 units.

Question 17

If angle AAA and angle BBB are vertical angles and angle AAA is 50o50^\text{o}50o, what is the measure of angle BBB?

  1. 130°
  2. 50° (correct answer)
  3. 80°
  4. 100°

Explanation: Angles A and B are vertical angles formed when two lines intersect. Vertical angles are always equal in measure. Since angle A is 50°, angle B must also be 50°. Choice A incorrectly uses 130°, which would be the supplementary (adjacent) angle.

Question 18

Convert log⁡2(16)=d\log_{2}(16) = dlog2​(16)=d to exponential form.

  1. 216=d2^{16} = d216=d
  2. 16d=216^d = 216d=2
  3. d2=16d^2 = 16d2=16
  4. 2d=162^d = 162d=16 (correct answer)

Explanation: To convert from logarithmic to exponential form, we use the fundamental property that log_a(b) = c is equivalent to a^c = b. In the equation log₂(16) = d, the base is 2, the argument is 16, and the result is d. Converting to exponential form gives us 2^d = 16. Choice B incorrectly reverses the base and argument positions.

Question 19

A bacteria culture starts with 50 bacteria and doubles every hour. Let xxx be the number of hours and yyy be the number of bacteria. Which equation best models this relationship?

  1. y=50+2xy = 50 + 2xy=50+2x
  2. y=50x2y = 50x^2y=50x2
  3. y=2x+50y = 2^x + 50y=2x+50
  4. y=50⋅2xy = 50\cdot 2^xy=50⋅2x (correct answer)

Explanation: This is an exponential growth model where bacteria double every hour. Starting with 50 bacteria, after x hours there are 50 × 2^x bacteria, giving the model y = 50 · 2^x. The base 2 represents doubling, and 50 is the initial amount. Choice A (y = 50 + 2x) is linear growth adding 2 bacteria per hour. Choice B (y = 50x²) is quadratic. Choice C (y = 2^x + 50) would start with 1 bacterium that doubles, then adds 50.

Question 20

Which of the following expressions is equivalent to 4(a+2b)−2(a−3b)4(a + 2b) - 2(a - 3b)4(a+2b)−2(a−3b)?

  1. 2a+2b2a + 2b2a+2b
  2. 2a+5b2a + 5b2a+5b
  3. 6a+14b6a + 14b6a+14b
  4. 2a+14b2a + 14b2a+14b (correct answer)

Explanation: This is a distribution and simplification question testing the distributive property with negatives. Choice D (2a + 14b) is correct — distribute the 4: 4a + 8b. Distribute the −2 across (a − 3b): −2a + 6b. Note: −2 × (−3b) = +6b, not −6b. Combine like terms: (4a − 2a) + (8b + 6b) = 2a + 14b. Choice A (2a + 2b) results from treating the second distribution as −2(a − 3b) = −2a − 6b (wrong sign on 3b), giving 8b − 6b = 2b. Choice B (2a + 5b) is an arithmetic error in combining the b terms, possibly computing 8b − 3b. Choice C (6a + 14b) adds 4a + 2a = 6a instead of subtracting, getting the sign wrong on the a-coefficient of the second term. Pro tip: When distributing a negative number, BOTH terms inside the parentheses change sign. Write out −2(a − 3b) = −2a + 6b before combining anything.

Question 21

For the piecewise function

3x+2, & x<2\\ 8-x, & 2\le x<6\\ (x-6)^2, & x\ge 6 \end{cases}$$ what is $f(5)$?
  1. 17
  2. 3 (correct answer)
  3. 25
  4. 13

Explanation: For x=5x = 5x=5, check which interval contains this value: 5<25 < 25<2? No. 2≤5<62 \leq 5 < 62≤5<6? Yes. Since x=5x = 5x=5 satisfies the condition 2≤x<62 \leq x < 62≤x<6, we use the second piece f(x)=8−xf(x) = 8 - xf(x)=8−x. Substituting x=5x = 5x=5: f(5)=8−5=3f(5) = 8 - 5 = 3f(5)=8−5=3. Choice A might result from incorrectly using the first piece or misreading the intervals.

Question 22

Convert 0.45 to a percent.

  1. 4.5%
  2. 45% (correct answer)
  3. 0.45%
  4. 450%

Explanation: This question asks to convert the decimal 0.45 to a percent. To convert a decimal to a percent, multiply by 100. 0.45 × 100 = 45%. Choice A incorrectly calculated 4.5%, possibly dividing instead of multiplying, while choice C gave 0.45%, showing confusion about decimal placement.

Question 23

What is tan⁡(30∘)\tan(30^\circ)tan(30∘)?

  1. 13\frac{1}{\sqrt{3}}3​1​
  2. 3\sqrt{3}3​
  3. 33\frac{\sqrt{3}}{3}33​​ (correct answer)
  4. 1

Explanation: 30° is a common unit circle angle. Using SOH-CAH-TOA, tan(30∘)=oppositeadjacent=13=33tan(30^\circ) = \frac{opposite}{adjacent} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(30∘)=adjacentopposite​=3​1​=33​​. This is a standard unit circle value from the 30-60-90 triangle. Choice A gives 13\frac{1}{\sqrt{3}}3​1​ without rationalization, while D gives the rationalized form 33\frac{\sqrt{3}}{3}33​​.

Question 24

What is (8−5i)+(3+2i)(8 - 5i) + (3 + 2i)(8−5i)+(3+2i)?

  1. 11 - 3i (correct answer)
  2. 11 + 7i
  3. 5 - 3i
  4. 5 + 3i

Explanation: This problem requires adding two complex numbers. To add complex numbers, we combine like terms by adding the real parts together and the imaginary parts together. For (8 - 5i) + (3 + 2i), we get (8 + 3) + (-5i + 2i) = 11 + (-3i) = 11 - 3i. The result is in standard form a + bi.

Question 25

A circle has radius 666. What is the circumference of the circle?

  1. 24π24\pi24π
  2. 36π36\pi36π
  3. 6π6\pi6π
  4. 12π12\pi12π (correct answer)

Explanation: This question asks for the circumference of a circle with radius 6. The correct formula for circumference is C = 2πr. Substituting r = 6 gives C = 2π(6) = 12π. Choice D is correct, while choice B incorrectly uses the area formula πr², yielding 36π. Emphasize selecting the circumference formula and multiplying by 2 to avoid confusing it with area calculations.