ACT Math Practice Test: Mixed Review (20 Questions)
- Mixed Review: All Math Topics
- 20 questions
- 22 minutes
These 20 questions mix every ACT Math reporting area and get harder as you go, the way the real Math section does (45 questions in 50 minutes on the enhanced ACT, with a calculator allowed). Do not spend too long on one early question: the last ones carry the same weight but need more time.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
A bread recipe uses 3 cups of flour for every 2 cups of water. How many cups of flour are needed for 7 cups of water?
Correct answer: C
Flour is 32 times the water, so 7 × 32 = 10.5 cups. The choice 4 23 comes from multiplying by 23 instead, which flips the ratio.
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Question 2
What is the value of 2x2 − 3x when x = −4?
Correct answer: A
2(−4)2 − 3(−4) = 2(16) + 12 = 44. Getting 20 means subtracting 12 instead of adding it; 76 comes from squaring 2x instead of only x.
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Question 3
What is 0.6% of 450?
Correct answer: B
0.6% = 0.006, and 0.006 × 450 = 2.7. The answer 270 treats 0.6% as 0.6, and 27 treats it as 6%.
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Question 4
What value of x is the solution of the equation x3 + 4 = 10?
Correct answer: D
Subtract 4 to get x3 = 6, then multiply by 3: x = 18. The value 2 comes from dividing 6 by 3 instead of multiplying, and 42 from adding 4 instead of subtracting it.
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Question 5
What is the slope of the line through the points (−1, 4) and (3, −8) in the standard (x, y) coordinate plane?
Correct answer: A
Slope = −8 − 43 − (−1) = −124 = −3. The value −13 puts the change in x over the change in y.
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Question 6
A rectangle has an area of 72 square inches and a length of 9 inches. What is the perimeter of the rectangle, in inches?
Correct answer: B
The width is 72 ÷ 9 = 8 inches, so the perimeter is 2(9 + 8) = 34 inches. The value 17 is only one length plus one width, half the perimeter.
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Question 7
A bag contains 5 red, 7 blue and 8 green marbles. One marble is chosen at random. What is the probability that the marble is not blue?
Correct answer: C
There are 20 marbles and 5 + 8 = 13 of them are not blue, so the probability is 1320. The value 720 is the probability that the marble is blue, and 713 compares blue marbles with the other marbles, not with the total.
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Question 8
Which of the following expressions is equivalent to (2x + 5)(x − 3)?
Correct answer: D
FOIL gives 2x2 − 6x + 5x − 15 = 2x2 − x − 15. Writing +x reverses the sign of the middle term, and 2x2 − 15 leaves out the middle term completely.
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Question 9
Maya has 31 coins, all of them dimes and quarters. The coins are worth $5.35 in all. How many quarters does Maya have?
Correct answer: B
With q quarters and 31 − q dimes, 0.25q + 0.10(31 − q) = 5.35, so 0.15q = 2.25 and q = 15. Check: 15 quarters ($3.75) plus 16 dimes ($1.60) make $5.35; 16 is the number of dimes.
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Question 10
In triangle PQR, the measure of ∠P is x°, the measure of ∠Q is 2x°, and the exterior angle at vertex R measures 126°. What is the measure of the largest interior angle of the triangle?
Correct answer: C
An exterior angle equals the sum of the two remote interior angles, so 3x = 126 and x = 42. The angles are 42°, 84° and 180° − 126° = 54°, so the largest is 84°. The 126° angle is outside the triangle.
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Question 11
For all x ≠ 0, which of the following is equivalent to (2x3)44x5?
Correct answer: A
(2x3)4 = 16x12, and 16x12 ÷ 4x5 = 4x7. The choice 4x2 adds the exponents 3 and 4 instead of multiplying them, and 2x7 multiplies 2 by 4 instead of raising 2 to the 4th power.
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Question 12
The table shows some values of the linear function f.
x 1 3 5 7 f(x) 7 13 19 25 What is the value of f(10)?
Correct answer: D
Each increase of 2 in x adds 6, so the slope is 3 and f(x) = 3x + 4. Then f(10) = 34. Answering 30 ignores the constant 4, and 37 is f(11).
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Question 13
What are the solutions of the equation 2x2 + 3x − 20 = 0?
Correct answer: B
The equation factors as (2x − 5)(x + 4) = 0, so x = 52 or x = −4. Check: 2(6.25) + 7.5 − 20 = 0. The signs in −52 and 4 are reversed, and 5 forgets to divide by the 2 in 2x − 5.
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Question 14
In the standard (x, y) coordinate plane, point M(1, 2) is the midpoint of segment AB. Point A has coordinates (−3, 7). What are the coordinates of point B?
Correct answer: A
From A to M, x increases by 4 and y decreases by 5; repeating that step from M gives B(5, −3). The point (−1, 4.5) is the midpoint of A and M, and (4, −5) is only the change from A to M.
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Question 15
A circle is inscribed in a square whose sides are 10 inches long, so the circle touches all four sides. What is the area, in square inches, of the region inside the square but outside the circle?
Correct answer: D
The circle's diameter equals the side, 10 inches, so its radius is 5 and its area is 25π. The region's area is 100 − 25π. Using 10 as the radius gives 100 − 100π, which is negative; 40 − 10π subtracts the circumference from the perimeter instead of areas.
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Question 16
In an arithmetic sequence, the 4th term is 17 and the 9th term is 42. What is the 20th term?
Correct answer: C
Five steps raise the term by 25, so the common difference is 5 and the 1st term is 17 − 3(5) = 2. The 20th term is 2 + 19(5) = 97. The value 92 is the 19th term and 102 is the 21st.
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Question 17
The value of y varies inversely with the square of x. When x = 2, y = 12. What is the value of y when x = 4?
Correct answer: A
y = kx2 with k = 12 × 22 = 48, so y = 4816 = 3. The answer 6 treats y as varying inversely with x itself, and 48 is the constant k.
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Question 18
What is the period of the graph of y = 3 sin(4x) − 1?
Correct answer: B
The period of sin(bx) is 2πb, so here it is 2π4 = π2. The 3 changes the amplitude and the −1 shifts the graph down; neither affects the period. 8π multiplies 2π by 4 instead of dividing.
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Question 19
A lab culture starts with 500 bacteria, and the number of bacteria triples every 4 hours. After how many hours will the culture contain 40,500 bacteria?
Correct answer: D
40,500 ÷ 500 = 81 = 34, so the population triples 4 times, which takes 4 × 4 = 16 hours. The number 81 is the growth factor, not a number of hours.
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Question 20
A 4-digit code is made from the digits 0 through 9. The first digit cannot be 0, and no digit may be used more than once in a code. How many different codes are possible?
Correct answer: C
There are 9 choices for the first digit (1-9), then 9 for the second (0 is now allowed, but one digit is used), 8 for the third and 7 for the fourth: 9 × 9 × 8 × 7 = 4,536. The value 5,040 = 10 × 9 × 8 × 7 ignores the rule about 0, and 3,024 = 9 × 8 × 7 × 6 wrongly excludes 0 everywhere.
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