ACT Math: Number and Quantity (Drill 1)

Question 1. Which of the following numbers is irrational?

• A) √9
• B) 0.333…
• C) √7 ✓
• D) −4/5

Explanation: An irrational number cannot be expressed as a ratio of two integers and has a non-repeating, non-terminating decimal expansion. √9 = 3 (rational), 0.333… = 1/3 (rational), and −4/5 is already a fraction (rational). √7 ≈ 2.6457… has a non-terminating, non-repeating decimal expansion, so it is irrational.

Question 2. On the real number line, how many integers are strictly between −7/2 and 13/4?

• A) 4
• B) 5 ✓
• C) 6
• D) 7

Explanation: Convert to decimals: −7/2 = −3.5 and 13/4 = 3.25. The integers strictly between −3.5 and 3.25 are −3, −2, −1, 0, 1, 2, and 3, a total of 5 integers. Choice A results from starting the count at −2 instead of −3. Choice C results from mistakenly including −4 by misreading −3.5 as a full unit below −3. Choice D results from treating the bounds as integers and including both −4 and 4.

Question 3. Which of the following is equivalent to (3.2 × 10⁴) × (2.5 × 10³)?

• A) 8.0 × 10⁶
• B) 8.0 × 10⁷ ✓
• C) 5.7 × 10⁷
• D) 3.2 × 10¹²

Explanation: Multiply the coefficients and add the exponents: 3.2 × 2.5 = 8.0, and 10⁴ × 10³ = 10⁷. The product is 8.0 × 10⁷. Choice A uses 10⁶, a common error from computing 4 + 3 incorrectly as 6. Choice C adds the coefficients (3.2 + 2.5 = 5.7) instead of multiplying them. Choice D multiplies the exponents (4 × 3 = 12) instead of adding them.

Question 4. For which of the following values of x is √3x − 9 NOT a real number?

• A) 3
• B) 4
• C) 7
• D) 2 ✓

Explanation: A square root produces a real number only when the radicand is greater than or equal to zero, so we need 3x − 9 ≥ 0, which means x ≥ 3. Testing each choice: x = 3 gives √0 = 0 (real); x = 4 gives √3 (real); x = 7 gives √12 (real); x = 2 gives √−3, which is not a real number. The answer is x = 2.

Question 5. The population of a city is approximately 4.8 × 10⁵. A neighboring city has a population approximately 6 times as large. Which of the following best represents the population of the neighboring city?

• A) 2.88 × 10⁵
• B) 2.88 × 10⁶ ✓
• C) 4.86 × 10⁵
• D) 2.88 × 10¹¹

Explanation: Multiply 4.8 × 10⁵ by 6: 6 × 4.8 = 28.8, giving 28.8 × 10⁵. Since 28.8 is not between 1 and 10, rewrite in proper scientific notation: 28.8 × 10⁵ = 2.88 × 10⁶. Choice A keeps the original exponent of 5 and fails to adjust after the coefficient exceeds 10. Choice C adds 6 to the coefficient (4.8 + 0.06 ≈ 4.86) instead of multiplying. Choice D results from adding the exponents (5 + 6 = 11) as if multiplying two numbers in scientific notation, rather than multiplying the coefficient by 6.