Ratio, Proportion, and Rates

Ratio

A ratio compares quantities: 12 : 8 simplifies to 3 : 2 (divide by the HCF).

Sharing in a ratio

Share 60 in the ratio 3 : 2.

Total parts = 5, one part = 60 ÷ 5 = 12.

Shares: 3 × 12 = 36 and 2 × 12 = 24.

Direct proportion

If 4 pens cost $6, then 1 pen costs $1.50 and 10 pens cost $15 (unitary method).

Inverse proportion

If 6 workers take 10 days, 1 worker takes 60 days, so 4 workers take 15 days. More workers, less time.

Compound measures

Measure	Formula	Units
speed	distance ÷ time	km/h, m/s
density	mass ÷ volume	g/cm³
pressure	force ÷ area	N/m²

Example. A car travels 150 km in 2 h 30 min. Speed = 150 ÷ 2.5 = 60 km/h.

⚠️ Convert minutes to hours as a decimal: 30 min = 0.5 h, 20 min = 1/3 h (not 0.2).

Unit conversions

• 1 m/s = 3.6 km/h.
• 1 m² = 10 000 cm²; 1 m³ = 1 000 000 cm³.

Map scales

On a 1 : 50 000 map, 4 cm represents 4 × 50 000 = 200 000 cm = 2 km.

Quick summary

• Simplify ratios by the HCF; share by finding one part.
• Direct: both increase; inverse: one increases as the other decreases.
• Speed, density, pressure: learn the formula triangles and convert units.