Core answer: To add fractions, find a common denominator first: 1/4 + 1/6 = 3/12 + 2/12 = 5/12. To multiply, go straight across: 2/3 × 4/5 = 8/15. To divide, flip the second and multiply: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Always finish by simplifying with the GCD.
The four operations
| Operation | Rule | Example |
|---|---|---|
| Add / subtract | common denominator, then add numerators | 1/4 + 1/6 = 5/12 |
| Multiply | top × top, bottom × bottom | 3/7 × 2/5 = 6/35 |
| Divide | multiply by the reciprocal | 3/7 ÷ 2/5 = 15/14 |
| Simplify | divide both by GCD | 12/18 → ÷6 → 2/3 |
Worked examples
Example 1 — Recipe scaling. A recipe for 4 needs 3/4 cup of flour; cooking for 6 means 3/4 × 6/4 = 18/16 = 9/8 = 1 1/8 cups.
Example 2 — Splitting a bill with a tip. ¥240 bill, 3 people, one pays half: 240 × 1/2 = 120; the rest split 1/2 equally: 240 × 1/2 ÷ 2 = 60 each.
Example 3 — Comparing discounts. Is 1/3 off better than 30% off? 1/3 = 33.33% — yes, slightly. Convert to decimals (0.333 vs 0.30) to compare instantly.
Fractions, decimals, percentages — one triangle
- Fraction → decimal: divide (7/8 = 0.875)
- Decimal → percentage: ×100 (0.875 = 87.5%)
- Percentage → fraction: over 100 then simplify (62.5% = 62.5/100 = 5/8)
The 8ths are worth memorizing: 1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5% — they appear in finance (basis-point moves), cooking, and hardware (inch fractions).
Common mistakes and myths
- Adding denominators — 1/2 + 1/3 ≠ 2/5; only numerators add after commonizing: 5/6.
- Cross-cancelling wrongly — cancel one top against one bottom only: (4/9) × (3/8) = (1/3) × (1/2) = 1/6 by cancelling 4-8 and 3-9.
- Forgetting to flip when dividing — dividing by 1/2 *doubles* (÷ 1/2 = × 2); "half of" and "divided by half" are opposites.
- Mixed-number multiplication trap — 2 1/2 × 2 1/2 ≠ 4 1/4; convert first: 5/2 × 5/2 = 25/4 = 6 1/4.
- Believing bigger denominator = bigger fraction — 1/8 < 1/6; the more slices, the smaller each slice.