Proportion

Proportion

Proportion is the equality of two ratios. Two quantities are in proportion if their ratio remains constant as both change. The statement \( \frac{a}{b} = \frac{c}{d} \) (or \( a : b = c : d \)) says that the four quantities are in proportion. This is the formal definition, and it underlies every proportional reasoning problem.

The Unitary Method

The most reliable approach to proportion problems: find the value of one unit, then scale up or down.

  1. Divide to find the value of one unit (one item, one hour, one metre, etc.).
  2. Multiply by the required number of units.
\[ \text{If 8 metres of fabric costs £28, then 1 metre costs } £28 \div 8 = £3.50 \implies 5 \text{ m costs } 5 \times £3.50 = £17.50 \]

The Proportion Equation

Proportion problems can also be solved by setting up an equation. If \( y \) and \( x \) are in direct proportion:

\[ \frac{y_1}{x_1} = \frac{y_2}{x_2} \qquad \text{or equivalently} \qquad y_1 x_2 = y_2 x_1 \quad \text{(cross-multiplication)} \]

Best Buy Problems

Compare the unit price (or rate) for each option. The option with the lower unit price is better value. Always compare using the same units.

Worked Examples

5 identical pens cost £3.75. How much do 8 pens cost?
\[ 1 \text{ pen} = £3.75 \div 5 = £0.75 \qquad 8 \text{ pens} = 8 \times £0.75 = £6.00 \]
A car uses 12 litres of petrol for 180 km. How far can it travel on a full 50-litre tank?
\[ 1 \text{ litre} \to \frac{180}{12} = 15 \text{ km} \qquad 50 \text{ litres} \to 50 \times 15 = 750 \text{ km} \]
If 3 kg of apples cost £4.20, how much do 7 kg cost? Solve using a proportion equation.
\[ \frac{4.20}{3} = \frac{x}{7} \implies x = \frac{4.20 \times 7}{3} = \frac{29.40}{3} = £9.80 \]
A recipe uses 250 ml of milk for 12 biscuits. How much milk is needed for 30 biscuits?
\[ \frac{250}{12} = \frac{m}{30} \implies m = \frac{250 \times 30}{12} = \frac{7500}{12} = 625 \text{ ml} \]
Which is better value: 3 for £2.55 or 5 for £4.00?
\[ \text{Option A: } £2.55 \div 3 = £0.85 \text{ each} \qquad \text{Option B: } £4.00 \div 5 = £0.80 \text{ each} \]

Option B is better value at 80p each.

 Key Takeaways

  • Proportion = equality of ratios: \( a : b = c : d \implies \frac{a}{b} = \frac{c}{d} \).
  • Unitary method: divide to find one unit, then multiply to find the required amount.
  • Proportion equation: \( \frac{y_1}{x_1} = \frac{y_2}{x_2} \) — cross-multiply to solve for the unknown.
  • Best buy: compute unit price for each option; lower unit price = better value.
  • Always use consistent units when setting up a proportion — mixing km with miles, or litres with pints, invalidates the ratio.