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ACT Math study materials
ACT Math study resources

ACT Math practice questions

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Question 1Number and Quantity

What is the value of (2/3) ÷ (4/9)?

  1. 3/8
  2. 27/8
  3. 8/27
  4. 3/2 Correct answer.

Explanation

Dividing by a fraction is the same as multiplying by its reciprocal, so (2/3) ÷ (4/9) = (2/3) × (9/4) = 18/12 = 3/2.

Question 2Number and Quantity

What is 15% of 240?

  1. 16
  2. 36 Correct answer.
  3. 24
  4. 40

Explanation

Convert the percent to a decimal and multiply: 0.15 × 240 = 36. Equivalently, 10% of 240 is 24 and 5% is 12, and 24 + 12 = 36.

Question 3Number and Quantity

Which expression shows 0.00042 correctly written in scientific notation?

  1. 0.42 × 10^-3
  2. 4.2 × 10^3
  3. 4.2 × 10^-4 Correct answer.
  4. 42 × 10^-5

Explanation

Scientific notation requires a coefficient between 1 and 10 multiplied by a power of 10. Moving the decimal point 4 places to the right gives 4.2, so the exponent is −4.

Question 4Number and Quantity

For all real numbers x, x^3 · x^5 is equal to which expression?

  1. 2x^8
  2. x^2
  3. x^8 Correct answer.
  4. x^15

Explanation

When multiplying powers with the same base, keep the base and add the exponents: x^3 · x^5 = x^(3+5) = x^8.

Question 5Number and Quantity

What is the value of |−7| − |3|?

  1. 4 Correct answer.
  2. −4
  3. 10
  4. −10

Explanation

Absolute value gives the distance from zero, so |−7| = 7 and |3| = 3. Therefore the expression equals 7 − 3 = 4.

Question 6Number and Quantity

If 3/5 = x/40, what is the value of x?

  1. 30
  2. 8
  3. 24 Correct answer.
  4. 120

Explanation

Cross-multiplying gives 3 · 40 = 5 · x, so 120 = 5x and x = 24. Equivalently, since 40 is 8 times 5, multiply the numerator by 8 to get 24.

Question 7Number and Quantity

At a shelter the ratio of cats to dogs is 3 to 5. If there are 24 cats, how many dogs are there?

  1. 40 Correct answer.
  2. 45
  3. 72
  4. 14

Explanation

The 3 parts representing cats equal 24, so each part is 24 ÷ 3 = 8. Dogs make up 5 parts, so there are 5 × 8 = 40 dogs.

Question 8Number and Quantity

Which number is equal to 2^(−3)?

  1. 1/8 Correct answer.
  2. −8
  3. 1/6
  4. −6

Explanation

A negative exponent means take the reciprocal of the positive power: 2^(−3) = 1/2^3 = 1/8.

Question 9Number and Quantity

The price of a jacket increases from $80 to $92. What is the percent increase?

  1. 87%
  2. 13%
  3. 12%
  4. 15% Correct answer.

Explanation

The amount of increase is $12, and percent change is measured relative to the original price: 12 ÷ 80 = 0.15, or 15%.

Question 10Number and Quantity

A price is decreased by 20%, and then the new price is increased by 20%. Compared with the original price, the final price is:

  1. 20% lower
  2. unchanged
  3. 4% higher
  4. 4% lower Correct answer.

Explanation

Apply the changes multiplicatively: 0.80 × 1.20 = 0.96, which is 96% of the original price. The final price is therefore 4% lower.

Question 11Number and Quantity

Which of these numbers is irrational?

  1. √2 Correct answer.
  2. √49
  3. 0.75
  4. 22/7

Explanation

An irrational number cannot be written as a ratio of two integers. The square root of 2 is not a perfect square and has a non-terminating, non-repeating decimal expansion.

Question 12Number and Quantity

What is √50 + √18 − √8 in simplest radical form?

  1. 6√2 Correct answer.
  2. 10√2
  3. 2√15
  4. 4√2

Explanation

Simplify each radical first: √50 = 5√2, √18 = 3√2, and √8 = 2√2. Combining like radicals gives (5 + 3 − 2)√2 = 6√2.

Question 13Number and Quantity

Which expression is equivalent to 3/√5?

  1. 3√5/5 Correct answer.
  2. 9/5
  3. √15/5
  4. 3√5

Explanation

To rationalize the denominator, multiply numerator and denominator by √5: (3√5)/(√5 · √5) = 3√5/5, since √5 · √5 = 5.

Question 14Number and Quantity

What is the value of i^23, where i^2 = −1?

  1. i
  2. −1
  3. −i Correct answer.
  4. 1

Explanation

Powers of i repeat in a cycle of four: i, −1, −i, 1. Since 23 = 4(5) + 3, i^23 = i^3 = −i.

Question 15Number and Quantity

What is the product (3 + 2i)(1 − 4i)?

  1. −5 − 10i
  2. 11 − 10i Correct answer.
  3. 3 − 8i
  4. 11 + 10i

Explanation

Distribute: 3(1) + 3(−4i) + 2i(1) + 2i(−4i) = 3 − 12i + 2i − 8i^2. Since i^2 = −1, this becomes 3 − 10i + 8 = 11 − 10i.

Question 16Number and Quantity

A 3.5-pound bag of mixed nuts costs $8.75. What is the price per pound?

  1. $2.75
  2. $2.50 Correct answer.
  3. $2.25
  4. $3.50

Explanation

Price per pound is total cost divided by total weight: 8.75 ÷ 3.5 = 2.50. So the nuts cost $2.50 per pound.

Question 17Number and Quantity

When a positive integer n is divided by 9, the remainder is 4. What is the remainder when 2n + 3 is divided by 9?

  1. 8
  2. 5
  3. 2 Correct answer.
  4. 7

Explanation

Write n = 9k + 4. Then 2n + 3 = 18k + 8 + 3 = 18k + 11 = 9(2k + 1) + 2, so the remainder is 2.

Question 18Number and Quantity

Two lights flash on a timer, one every 12 seconds and the other every 18 seconds. If they flash together right now, after how many seconds will they next flash together?

  1. 36 seconds Correct answer.
  2. 72 seconds
  3. 6 seconds
  4. 30 seconds

Explanation

The next simultaneous flash occurs at the least common multiple of 12 and 18. Since 12 = 2^2 · 3 and 18 = 2 · 3^2, the LCM is 2^2 · 3^2 = 36.

Question 19Number and Quantity

Which fraction is exactly equal to the repeating decimal 0.363636...?

  1. 11/30
  2. 36/100
  3. 4/11 Correct answer.
  4. 4/99

Explanation

Let x = 0.363636...; then 100x = 36.363636..., so 99x = 36 and x = 36/99 = 4/11.

Question 20Number and Quantity

Ana's first three test scores are 82, 90, and 86. What score does she need on the fourth test to have a mean score of exactly 87?

  1. 87
  2. 88
  3. 92
  4. 90 Correct answer.

Explanation

A mean of 87 over four tests requires a total of 4 × 87 = 348 points. Her current total is 82 + 90 + 86 = 258, so she needs 348 − 258 = 90.

Question 21Number and Quantity

Simplify (2x^3·y^(−2))^2 ÷ (4x^(−1)·y^3), assuming x and y are nonzero.

  1. x^5/y
  2. x^7/y^7 Correct answer.
  3. x^7·y^7
  4. x^5·y

Explanation

Squaring the first factor gives 4x^6·y^(−4). Dividing by 4x^(−1)·y^3 subtracts exponents: x^(6−(−1)) = x^7 and y^(−4−3) = y^(−7), so the result is x^7/y^7.

Question 22Number and Quantity

How many solutions does the equation |x − 3| = 2x − 1 have, and what are they?

  1. x = −2 only
  2. x = 4/3 only Correct answer.
  3. x = 4/3 and x = −2
  4. There is no solution

Explanation

The right side must be nonnegative, which rules out negative candidates. Solving x − 3 = 2x − 1 gives x = −2, but then 2x − 1 = −5, so it fails. Solving 3 − x = 2x − 1 gives x = 4/3, and checking: |4/3 − 3| = 5/3 = 2(4/3) − 1.

Question 23Number and Quantity

What is (1 + i)/(2 − i) written in the form a + bi?

  1. 3/5 + (1/5)i
  2. 1/5 + (3/5)i Correct answer.
  3. 3/5 − (1/5)i
  4. 1/3 + i

Explanation

Multiply numerator and denominator by the conjugate 2 + i. The numerator becomes (1 + i)(2 + i) = 1 + 3i, and the denominator becomes (2 − i)(2 + i) = 5, giving 1/5 + (3/5)i.

Question 24Number and Quantity

Which of these numbers is the greatest?

  1. 3√2 Correct answer.
  2. 4
  3. √13
  4. 2√3

Explanation

Squaring each positive number preserves order: (2√3)^2 = 12, (√13)^2 = 13, 4^2 = 16, and (3√2)^2 = 18. The largest square is 18, so 3√2 is the greatest.

Question 25Number and Quantity

Two positive integers have a greatest common factor of 6 and a least common multiple of 72. If one of the integers is 24, what is the other integer?

  1. 48
  2. 36
  3. 12
  4. 18 Correct answer.

Explanation

For any two positive integers, GCF × LCM equals the product of the integers. So the product is 6 × 72 = 432, and the other integer is 432 ÷ 24 = 18, which checks since GCF(24, 18) = 6 and LCM(24, 18) = 72.

Question 26Number and Quantity

What is 0.00042 written in scientific notation?

  1. 4.2 × 10^−4 Correct answer.
  2. 42 × 10^−3
  3. 4.2 × 10^−5
  4. 4.2 × 10^4

Explanation

Scientific notation writes a number as a × 10^n, where 1 ≤ |a| < 10. Moving the decimal point in 0.00042 four places to the right gives 4.2, so the exponent is −4.

Question 27Number and Quantity

What is (4 − 3i) + (2 + 5i)?

  1. 2 − 8i
  2. 6 − 8i
  3. 6 + 2i Correct answer.
  4. 2 + 2i

Explanation

Add the real parts and the imaginary parts separately: (4 + 2) + (−3i + 5i) = 6 + 2i.

Question 28Number and Quantity

What is the value of (2/3)^(−2)?

  1. 4/9
  2. −9/4
  3. 9/4 Correct answer.
  4. −4/9

Explanation

A negative exponent means take the reciprocal and then apply the positive exponent: (2/3)^(−2) = (3/2)^2 = 9/4.

Question 29Number and Quantity

Which of the following numbers is rational?

  1. π/2
  2. √25 Correct answer.
  3. 0.1010010001...
  4. √7

Explanation

A rational number can be written as a ratio of two integers. Since √25 = 5, it is rational.

Question 30Number and Quantity

What is 16^(3/4)?

  1. 12
  2. 32
  3. 64
  4. 8 Correct answer.

Explanation

A rational exponent a^(m/n) means the nth root of a raised to the mth power. The fourth root of 16 is 2, and 2^3 = 8.

Question 31Number and Quantity

Which expression is equivalent to x^(2/3) · x^(1/6) for x > 0?

  1. x^(1/2)
  2. x^(5/6) Correct answer.
  3. x^(1/3)
  4. x^(4/9)

Explanation

When multiplying powers with the same base, add the exponents. Write 2/3 as 4/6, then 4/6 + 1/6 = 5/6, so the product is x^(5/6).

Question 32Number and Quantity

What is (3 × 10^5)(4 × 10^−8) in scientific notation?

  1. 1.2 × 10^−3
  2. 1.2 × 10^−2 Correct answer.
  3. 12 × 10^−2
  4. 1.2 × 10^13

Explanation

Multiply the coefficients and add the exponents: 3 · 4 = 12 and 10^5 · 10^−8 = 10^−3. Then rewrite 12 × 10^−3 as 1.2 × 10^−2 in scientific notation.

Question 33Number and Quantity

What is the product (2 + 3i)(2 − 3i)?

  1. 13 − 12i
  2. 4 + 9i
  3. 4 − 9i
  4. 13 Correct answer.

Explanation

This is a difference of squares: (2 + 3i)(2 − 3i) = 2^2 − (3i)^2. Since (3i)^2 = −9, the result is 4 − (−9) = 13.

Question 34Number and Quantity

What is the magnitude of the vector ⟨−5, 12⟩?

  1. √7
  2. 13 Correct answer.
  3. 60
  4. 17

Explanation

The magnitude of ⟨a, b⟩ is √(a^2 + b^2). Here, √((−5)^2 + 12^2) = √(25 + 144) = √169 = 13.

Question 35Number and Quantity

What is the determinant of the matrix with first row 3, 4 and second row 2, 5?

  1. 15
  2. 23
  3. 7 Correct answer.
  4. −7

Explanation

For a 2 × 2 matrix with entries a, b in the first row and c, d in the second row, the determinant is ad − bc. Here, (3)(5) − (4)(2) = 15 − 8 = 7.

Question 36Number and Quantity

How many positive integer factors does 72 have?

  1. 18
  2. 10
  3. 11
  4. 12 Correct answer.

Explanation

The prime factorization is 72 = 2^3 · 3^2. The number of positive factors is (3 + 1)(2 + 1) = 4 · 3 = 12.

Question 37Number and Quantity

What is ∛54 + ∛16 in simplest radical form?

  1. 6∛2
  2. 2∛70
  3. 3∛2
  4. 5∛2 Correct answer.

Explanation

Factor out perfect cubes: ∛54 = ∛(27 · 2) = 3∛2, and ∛16 = ∛(8 · 2) = 2∛2. Adding like radicals gives 3∛2 + 2∛2 = 5∛2.

Question 38Number and Quantity

What is the remainder when 7^100 is divided by 5?

  1. 4
  2. 1 Correct answer.
  3. 3
  4. 2

Explanation

Since 7 ≡ 2 (mod 5), 7^100 ≡ 2^100 (mod 5). Powers of 2 modulo 5 cycle 2, 4, 3, 1 with period 4; because 100 is a multiple of 4, the remainder is 1.

Question 39Number and Quantity

If z = 3 − 4i, what is z^2?

  1. −7 − 24i Correct answer.
  2. 25
  3. 9 − 16i
  4. −7 + 24i

Explanation

Square the binomial: (3 − 4i)^2 = 9 − 24i + 16i^2. Since i^2 = −1, this becomes 9 − 24i − 16 = −7 − 24i.

Question 40Number and Quantity

A student measures a length as 48.6 cm, but the actual length is 45.0 cm. To the nearest tenth of a percent, what is the percent error?

  1. 7.4%
  2. 3.6%
  3. 10.8%
  4. 8.0% Correct answer.

Explanation

Percent error is |measured − actual| / actual × 100%. Here, |48.6 − 45.0| / 45.0 = 3.6 / 45.0 = 0.08, which is 8.0%.

Question 41Algebra

If 3x − 7 = 14, what is the value of x?

  1. 21
  2. 7/3
  3. −7
  4. 7 Correct answer.

Explanation

Add 7 to both sides to get 3x = 21, then divide both sides by 3 to isolate x. This gives x = 7, which checks because 3(7) − 7 = 14.

Question 42Algebra

Which expression is equivalent to (2x³)(5x⁴)?

  1. 7x¹²
  2. 7x⁷
  3. 10x¹²
  4. 10x⁷ Correct answer.

Explanation

Multiply the coefficients, 2 · 5 = 10, and add the exponents of the common base, x³ · x⁴ = x⁷. The product is therefore 10x⁷.

Question 43Algebra

What is the slope of the line that passes through the points (2, 3) and (6, 11)?

  1. 1/2
  2. 4
  3. −2
  4. 2 Correct answer.

Explanation

Slope is the change in y divided by the change in x: (11 − 3) / (6 − 2) = 8/4. Simplifying gives a slope of 2.

Question 44Algebra

If f(x) = 3x² − 2x + 1, what is f(2)?

  1. 11
  2. 5
  3. 7
  4. 9 Correct answer.

Explanation

Substitute x = 2 into every term: 3(2)² − 2(2) + 1 = 12 − 4 + 1. Combining the terms gives 9.

Question 45Algebra

Which values of x satisfy the inequality −2x > 8?

  1. x < −4 Correct answer.
  2. x > 4
  3. x > −4
  4. x < 4

Explanation

Divide both sides by −2, and because you are dividing by a negative number, reverse the direction of the inequality. The result is x < −4.

Question 46Algebra

Which product is equal to x² − 9?

  1. (x + 9)(x − 1)
  2. (x − 3)(x + 3) Correct answer.
  3. (x − 3)²
  4. (x − 9)(x + 1)

Explanation

x² − 9 is a difference of two squares, and a² − b² factors as (a − b)(a + b) with a = x and b = 3. Therefore x² − 9 = (x − 3)(x + 3).

Question 47Algebra

Which expression is equivalent to x⁵ / x² for x ≠ 0?

  1. x³ Correct answer.
  2. x¹⁰
  3. x^2.5
  4. x⁷

Explanation

When dividing powers that share the same base, subtract the exponents. Here 5 − 2 = 3, so the quotient is x³.

Question 48Algebra

What is the solution set of |x − 4| = 6?

  1. x = −10 or x = 2
  2. x = 2 or x = 10
  3. x = −2 or x = 10 Correct answer.
  4. x = 10 only

Explanation

An absolute value equation splits into two cases: x − 4 = 6 and x − 4 = −6. Solving these gives x = 10 and x = −2.

Question 49Algebra

What is the equation of the line with slope −3 that passes through the point (4, −1)?

  1. y = −3x − 11
  2. y = −3x + 11 Correct answer.
  3. y = 3x − 13
  4. y = −3x + 13

Explanation

Point-slope form gives y − (−1) = −3(x − 4). Distributing and solving for y produces y = −3x + 11, and substituting x = 4 confirms y = −1.

Question 50Algebra

What are the solutions of x² − 5x − 24 = 0?

  1. x = −8 or x = 3
  2. x = 8 or x = −3 Correct answer.
  3. x = 12 or x = −2
  4. x = 6 or x = −4

Explanation

The quadratic factors as (x − 8)(x + 3) because −8 and 3 multiply to −24 and add to −5. Setting each factor equal to zero gives x = 8 and x = −3.