Fractions, Decimals & Percentages

Strategy/Review

Fraction ↔ Decimal ↔ Percent Conversions

Memorize these common equivalents until they're automatic:

Knowing 1/8 = 0.125 lets you instantly compute 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875 by multiplying 0.125 × 3, × 5, × 7.

Fraction Arithmetic

Addition/Subtraction: Find a common denominator, convert, then add/subtract numerators. Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12

Multiplication: Multiply numerators, multiply denominators. Simplify before multiplying if possible (cross-cancel). Example: 3/8 × 4/9 = (3×4)/(8×9) = 12/72 = 1/6 Better: cancel first — the 3 and 9 share a factor of 3; the 4 and 8 share a factor of 4 → 1/2 × 1/3 = 1/6

Division: Multiply by the reciprocal. "Keep, Change, Flip." Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1⅞

Percent Change Formula

This is the single most important percent formula for the DAT:

% Change = [(New − Old) / Old] × 100

Key distinctions the DAT tests:

Percent increase: A $50 item increases to $60. % increase = [(60 − 50) / 50] × 100 = (10/50) × 100 = 20% increase

Percent decrease: A $60 item drops to $50. % decrease = [(50 − 60) / 60] × 100 = (−10/60) × 100 = −16.7% (a 16.7% decrease)

Percent OF vs. Percent MORE THAN:

• "X is 20% OF Y" → X = 0.20 × Y
• "X is 20% MORE THAN Y" → X = Y + 0.20Y = 1.20Y
• "X is 20% LESS THAN Y" → X = Y − 0.20Y = 0.80Y

This distinction is one of the most common DAT percent traps. Read "of" vs. "more than" vs. "less than" carefully in every problem.

Successive Percent Changes: Two 10% increases do NOT equal a 20% increase. Start with $100. After first 10% increase: $100 × 1.10 = $110 After second 10% increase: $110 × 1.10 = $121 Net increase: 21%, not 20%.

Ratios and Proportions

A ratio compares two quantities: a:b. A proportion states that two ratios are equal: a/b = c/d.

Cross-multiplication: a/b = c/d → ad = bc. This is your go-to for solving proportions.

Example: If 5 apples cost $3.75, how much do 8 apples cost? 5/3.75 = 8/x → 5x = 30 → x = $6.00

Three-term ratios: If a:b = 2:3 and b:c = 4:5, find a:b:c. Make b the same in both: a:b = 8:12, b:c = 12:15 → a:b:c = 8:12:15

Mental Math Strategies

10% trick: To find 10% of any number, move the decimal one place left. Then 5% is half of 10%. 15% is 10% + 5%. 20% is double 10%.

Example: Find 35% of 80. 10% = 8 30% = 24 5% = 4 35% = 28

Fraction benchmarks: When estimating, know that 1/3 ≈ 33%, 2/3 ≈ 67%, 1/6 ≈ 17%, etc. This lets you approximate without the calculator.

Doubling and halving for multiplication: 16 × 35 = (8 × 70) = 560 (halve one, double the other)

Common Traps

Trap 1 — Percent OF vs. Percent MORE THAN. "X is 150% of Y" means X = 1.5Y. "X is 150% more than Y" means X = Y + 1.5Y = 2.5Y. The word "more" adds 100 percentage points.

Trap 2 — Reversing the Percent Change Base. If a price increases 25% and then decreases 25%, you're NOT back to the original. The decrease is calculated from the new, higher base, so you end up lower. $100 → +25% = $125 → −25% = $93.75.

Trap 3 — Dividing Fractions by Multiplying Wrong Reciprocal. 3/4 ÷ 2/3 is NOT 3/4 × 3/2 (correct). It's not 3/4 × 2/3 (wrong — that's multiplication). "Keep, Change, Flip" only flips the SECOND fraction.

Trap 4 — Adding Fractions Without Common Denominator. 2/3 + 1/4 ≠ 3/7. You CANNOT add denominators. Must convert to common denominator first.

Trap 5 — Rounding Too Early. When converting fractions to decimals, the DAT sometimes requires precision. 2/3 = 0.666... Don't round to 0.67 mid-calculation if subsequent steps amplify the error.