Ratios & Proportions

What is a Ratio?

A ratio compares two or more quantities. It can be written as a:b or a/b.

Example: In a class of 30 students, 18 are girls and 12 are boys. The ratio of girls to boys is 18:12 = 3:2

Simplifying Ratios

Divide all parts of the ratio by their GCF.

Example: 15:25:10 → GCF = 5 → 3:5:2

Dividing a Quantity in a Given Ratio

1. Add the parts of the ratio to get the total number of shares
2. Divide the total quantity by the total shares to find one share
3. Multiply each part by one share

Example: Share $120 in the ratio 3:2:1
Total shares = 3+2+1 = 6   |   One share = $120÷6 = $20
→ $60 : $40 : $20

Equivalent Ratios

Like equivalent fractions — multiply or divide all parts by the same number.

2:5 = 4:10 = 6:15 = 10:25

Direct Proportion

Two quantities are in direct proportion if they increase/decrease at the same rate.

Example: If 5 pens cost $3, how much do 8 pens cost?
Cost per pen = $3÷5 = $0.60 → 8 × $0.60 = $4.80

Scale and Maps

A scale of 1:50,000 means 1 cm on the map = 50,000 cm (or 500 m) in real life.

Example: A map distance of 4 cm represents 4 × 50,000 = 200,000 cm = 2 km

Key Tips

• Always simplify ratios to their lowest terms before answering
• Check the order of a ratio matches the order in the question
• Unitary method (find '1 of something' first) works for most proportion problems