Ratio notation and simplification
A ratio compares quantities by division. Where fractions ask "what fraction is part of the whole?", ratios ask "how do these parts compare to each other?".
Notation — colons and parts
A ratio is written with colons: 3 : 5 ("3 to 5"). Multi-part ratios extend: 2 : 3 : 5. Each number is a share or part of a larger whole.
If a recipe uses 2 parts flour to 1 part butter, that's 2 : 1. If we double the recipe, it's 4 : 2 — but it's the same ratio, just unsimplified.
Simplifying — divide all parts by HCF
Reduce a ratio to simplest form by dividing every part by their highest common factor (HCF).
Worked example: simplify 12 : 18.
- HCF(12, 18) = 6.
- 12 ÷ 6 : 18 ÷ 6 = 2 : 3.
Worked example: simplify 24 : 36 : 60.
- HCF = 12. → 2 : 3 : 5.
Ratios with units
Convert to the same unit first.
Worked example: simplify 50 cm : 1 m.
- 1 m = 100 cm.
- 50 : 100 = 1 : 2.
Ratios with decimals or fractions
Multiply both parts by a power of 10 (decimals) or by the LCM of denominators (fractions) to clear them.
Worked example: simplify 1.2 : 0.5.
- × 10: 12 : 5. HCF = 1, already simplest.
- Answer: 12 : 5.
Worked example: simplify ⅔ : ¾.
- LCM(3, 4) = 12. × 12: 8 : 9.
- Answer: 8 : 9.
Ratios in form 1 : n or n : 1
Sometimes you must rewrite a ratio so one side is 1. Divide both sides by the chosen part.
Worked example: write 5 : 8 in the form 1 : n.
- Divide by 5: 1 : 8/5 = 1 : 1.6.
This form is especially common in map scales (e.g. 1 : 50 000) and gear ratios.
⚠Common mistakes
- Forgetting same units — 30 cm : 1 m is 3 : 10, not 30 : 1.
- Subtracting instead of dividing when simplifying.
- Reversing the ratio order — read carefully: "boys to girls = 4 : 5" means there are MORE GIRLS.
- Treating ratio like a fraction — 3 : 5 means 3 parts to 5 parts, total 8 parts (NOT 3/5 of the total).
- Stopping too early — always check whether the simplified ratio can be reduced further.
➜Try this— Quick check
Simplify 45 : 60 : 75. HCF = 15 → 3 : 4 : 5.
AI-generated · claude-opus-4-7 · v3-deep-ratio