Pre-Algebra and Number Properties

Pre-Algebra and Number Properties form the foundation of the ACT Math section, accounting for roughly 20-25% of all questions. This area tests your understanding of basic arithmetic, number theory, and fundamental mathematical operations that underpin every other topic on the exam.

Order of Operations (PEMDAS)

Always follow the correct order: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). A common ACT trap is to present expressions where skipping a step leads to a wrong answer choice.

Example 1: Evaluate 5 × (3 + 2)² − 40 ÷ 8
Step 1: Parentheses: (3 + 2) = 5
Step 2: Exponents: 5² = 25
Step 3: Multiply/Divide left to right: 5 × 25 = 125 and 40 ÷ 8 = 5
Step 4: Subtract: 125 − 5 = 120

Example 2: Evaluate 12 − 3 × (8 − 2²) + 1
Step 1: Inner parentheses first — exponent inside: 2² = 4, then 8 − 4 = 4
Step 2: Multiply: 3 × 4 = 12
Step 3: Left to right: 12 − 12 + 1 = 1

Example 3: Evaluate −2³ + (−2)³
−2³ means −(2³) = −8. (−2)³ means (−2)(−2)(−2) = −8.
So −2³ + (−2)³ = −8 + (−8) = −16. Watch out: many students think −2³ = 8 because "negative times negative is positive" — but exponents apply only to what is directly attached.

Fractions, Decimals, and Percents

You must be comfortable converting between these three forms. Key relationships to memorize:

• 1/2 = 0.5 = 50%
• 1/3 ≈ 0.333 = 33.3%
• 1/4 = 0.25 = 25%
• 1/5 = 0.20 = 20%
• 1/8 = 0.125 = 12.5%
• 3/4 = 0.75 = 75%
• 2/3 ≈ 0.667 = 66.7%

To find a percent of a number, convert the percent to a decimal and multiply. For example, 35% of 80 = 0.35 × 80 = 28.

Example 1: A store sells a jacket originally priced at $60 for 15% off. What is the sale price?
Discount = 0.15 × 60 = $9. Sale price = 60 − 9 = $51.

Example 2: A population grows from 400 to 500. What is the percent increase?
Change = 500 − 400 = 100. Percent increase = (100 / 400) × 100 = 25%.
Shortcut: divide the new by the old: 500/400 = 1.25 → 25% increase.

Example 3: What is 3/8 as a percent?
3 ÷ 8 = 0.375 = 37.5%. Alternatively, 3/8 = (3 × 12.5)% = 37.5%.

Factors, Multiples, and Primes

A factor divides evenly into a number. A multiple is the product of a number and any integer. Prime numbers have exactly two factors: 1 and themselves. The first several primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Remember: 2 is the only even prime and 1 is NOT prime.

Greatest Common Factor (GCF) and Least Common Multiple (LCM)
GCF: the largest number that divides evenly into two or more numbers. LCM: the smallest number that is a multiple of two or more numbers. Use prime factorization to find both efficiently.

Example: Find the GCF and LCM of 18 and 48.
18 = 2 × 3² and 48 = 2⁴ × 3. GCF = 2¹ × 3¹ = 6. LCM = 2⁴ × 3² = 144.
Quick check: GCF × LCM = 6 × 144 = 864 = 18 × 48. This relationship always holds.

Properties of Numbers

• Commutative: a + b = b + a and a × b = b × a
• Associative: (a + b) + c = a + (b + c)
• Distributive: a(b + c) = ab + ac
• Identity: a + 0 = a and a × 1 = a

Absolute Value, Ratios, and Proportions

The absolute value |x| represents the distance from zero on the number line, so it is always non-negative. |−7| = 7, |3| = 3, and |0| = 0.

A ratio compares two quantities. A proportion states that two ratios are equal: a/b = c/d. Cross-multiply to solve: ad = bc.

Example 1: If 3 notebooks cost $7.50, how much do 8 notebooks cost?
Set up the proportion: 3/7.50 = 8/x. Cross-multiply: 3x = 60. So x = $20.00.

Example 2: A map uses a scale of 1 inch = 25 miles. Two cities are 3.5 inches apart on the map. What is the actual distance?
1/25 = 3.5/x → x = 25 × 3.5 = 87.5 miles.

Example 3: The ratio of boys to girls in a class is 3:5. If there are 40 students total, how many are boys?
Total parts = 3 + 5 = 8. Boys = (3/8) × 40 = 15.

Exponents, Roots, and Scientific Notation

Key exponent rules: a^m × aⁿ = a^(m+n), (a^m)ⁿ = a^(mn), a⁰ = 1 (for a ≠ 0), and a⁽⁻ⁿ⁾ = 1/aⁿ.
Square roots: √(ab) = √a × √b, and √(a/b) = √a / √b.

Scientific Notation: Numbers written as a × 10ⁿ where 1 ≤ a < 10. For example, 4,500,000 = 4.5 × 10⁶ and 0.0032 = 3.2 × 10⁽⁻³⁾.

Example: Simplify (3 × 10⁴)(2 × 10⁵).
Multiply the coefficients: 3 × 2 = 6. Add the exponents: 4 + 5 = 9. Answer: 6 × 10⁹.

Common Mistakes

Quick Reference — Formula Sheet

ACT Strategies for Pre-Algebra

• Memorize conversions: Know common fraction-decimal-percent equivalents to save 30+ seconds per question.
• Base-100 trick: When a problem involves percents, assume the total is 100 to simplify calculations.
• Cross-multiply to compare fractions: To compare a/b and c/d, check if ad > bc (then a/b > c/d).
• Last-digit shortcut: Only compute the last digit of each factor to quickly eliminate wrong answers.
• Backsolving: If a question looks time-consuming, plug answer choices back in to find the correct one.
• Estimation: Round numbers to get a quick ballpark — this eliminates obviously wrong answers on the ACT.
• Watch for sign traps: Negative × Negative = Positive. The ACT loves to include sign errors as distractors.
• Divisibility rules: By 2 (even), by 3 (digit sum divisible by 3), by 5 (ends in 0 or 5), by 9 (digit sum divisible by 9). These save time on factor questions.
• Successive percents: A 20% increase then a 20% decrease is NOT the original — it is 0.96 of the original. Multiply the multipliers: 1.20 × 0.80 = 0.96.

Practice Walkthrough

ACT-Style Problem: A shirt originally priced at $40 is marked 25% off. Sales tax of 8% is applied to the sale price. What is the total cost?

A. $32.40    B. $33.60    C. $34.56    D. $35.20

Step 1 — Find the sale price: 25% off means you pay 75%. Sale price = 0.75 × $40 = $30.00.
Step 2 — Apply tax: Tax = 0.08 × $30 = $2.40. Total = $30 + $2.40 = $32.40.
Step 3 — Eliminate wrong answers: Choice D ($35.20) is close to just applying 8% to $40 with a small discount — too high. Choice B ($33.60) would result from taxing before discounting. Choice C ($34.56) comes from applying tax to the original price then discounting. Only A ($32.40) correctly discounts first, then taxes.