Ratio notation and simplification

Ratios are everywhere in OCR J560 — recipes, scaling, sharing money, mixtures. Simplifying a ratio is mechanically the same as simplifying a fraction.

Reading ratios

A ratio compares two or more quantities of the same kind.

• "Boys to girls is 3 : 4" — for every 3 boys, 4 girls.
• The ratio 3 : 4 has parts 3 + 4 = 7. Boys are 3/7 of class; girls 4/7.

Simplifying ratios

To simplify, divide every part by their HCF.

• 12 : 18 → divide by 6 → 2 : 3.
• 24 : 36 : 48 → divide by 12 → 2 : 3 : 4.

If parts contain decimals or fractions, multiply through to clear them first.

• 0.5 : 0.75 → multiply by 4 → 2 : 3.
• 1/2 : 3/4 → multiply by 4 → 2 : 3.

Mixed units

Convert to the same unit before simplifying.

• 50 cm : 1 m → 50 cm : 100 cm → 1 : 2.
• 15 minutes : 1 hour → 15 : 60 → 1 : 4.

1 : n form

Express a ratio so the first part is 1.

• 12 : 18 → divide both by 12 → 1 : 1.5.
• Useful for currency conversion, scale drawings.

Linking ratio and fraction

If A : B = a : b, then:

• A/total = a/(a+b)
• B/total = b/(a+b)
• A/B = a/b (fraction).

Sharing in a ratio

To share quantity Q in ratio a : b : c…:

1. Add the parts: total parts = a + b + c.
2. Compute one part = Q / total parts.
3. Multiply each share's parts by one-part value.

Example: share £180 in ratio 2 : 3 : 4.

• Total parts = 9. One part = 180/9 = £20.
• Shares: 2 × 20 = £40, 3 × 20 = £60, 4 × 20 = £80.
• Check: £40 + £60 + £80 = £180 ✓.

OCR mark scheme conventions

• B1 for correct simplification (divide by HCF).
• M1 + A1 for sharing-in-ratio: M1 for "one part" calculation, A1 for the correct shares.
• Mixed-units: B1 for converting to common units before simplifying.