What is a proportion?
A proportion is an equation that says two ratios are equal:
3/4 = 9/12
Solving a proportion means finding an unknown term.
Method 1: Scale factor
Look for the scale factor that turns one ratio into the other.
Example: 3/5 = x/40
- 5 × 8 = 40, so multiply the numerator by 8 too.
- x = 3 × 8 = 24
Fast when the numbers fit nicely.
Method 2: Cross-multiplication
In a/b = c/d, the cross products are equal: a × d = b × c.
Example: 7/x = 21/30
- 7 × 30 = 21 × x
- 210 = 21x
- x = 10
Example: x/12 = 5/8
- 8x = 60
- x = 7.5
Cross-multiplication works every time, even with awkward numbers.
Method 3: Unit rate
Find the value for 1, then scale. This is the unit rate method.
Example: 5 notebooks cost $8.75. How much do 8 cost?
- 1 notebook: 8.75 ÷ 5 = $1.75
- 8 notebooks: 1.75 × 8 = $14
Setting up a proportion correctly
Keep the same units in the same position.
Problem: A car uses 6 L of gas to go 80 km. How much gas for 200 km?
litres/km = litres/km → 6/80 = x/200
- 80x = 1,200
- x = 15 L
Writing 6/80 = 200/x would mix up the units and give the wrong answer.
Checking
Substitute back: 6/80 = 0.075 and 15/200 = 0.075 ✓.
Worked examples
Example 1: x/9 = 8/12
12x = 72 → x = 6.
Example 2: A map scale is 2 cm : 25 km. Two towns are 9 cm apart. Real distance?
2/25 = 9/x → 2x = 225 → x = 112.5 km.
Example 3: 4 workers paint a fence in 6 hours. Is "8 workers take 12 hours" a proportion problem?
No. More workers take less time. That’s inverse variation, not a proportion.
Common mistakes
- Mismatched units when setting up.
- Multiplying straight across (numerator × numerator) instead of diagonally.
- Using a proportion when the relationship isn’t proportional.
Try it yourself
- 4/7 = x/35
- 15/x = 5/9
- x/2.5 = 6/10
- 3 kg of apples cost $7.20. How much for 5 kg?
- A photo 4 cm wide by 6 cm tall is enlarged to 10 cm wide. How tall?
Answers: 1) 20 2) 27 3) 1.5 4) $12.00 5) 15 cm
Quick recap
- A proportion says two ratios are equal.
- Solve with a scale factor, cross-multiplication or the unit rate method.
- Keep units in matching positions and check your answer.