Proportion and the unitary method

If $3$ apples cost $\mathbb{?}\,30$, then $6$ apples cost $\mathbb{?}\,60$ , every child knows that. The maths underneath is the proportion

$\dfrac{3}{30} = \dfrac{6}{60}.$

Two ratios that are equal form a proportion. Recognising and using proportions is one of the most powerful problem-solving tools in arithmetic.

Concept

A proportion is a statement that two ratios are equal: $\dfrac{a}{b} = \dfrac{c}{d}$, sometimes written $a : b :: c : d$, read "$a$ is to $b$ as $c$ is to $d$".

Cross-multiplication. If $\dfrac{a}{b} = \dfrac{c}{d}$, then $ad = bc$.

This lets us find any one of the four numbers when the other three are known.

Example: $\dfrac{4}{5} = \dfrac{x}{20}$. Cross-multiply: $4 \cdot 20 = 5 \cdot x$, so $x = 16$.

The unitary method. A safer and more flexible approach is to find the unit value first.

Recipe: "If $3$ apples cost $\mathbb{?}\,30$, how much do $7$ apples cost?"

• Step 1: cost of $1$ apple $= 30 / 3 = \mathbb{?}\,10$.
• Step 2: cost of $7$ apples $= 7 \times 10 = \mathbb{?}\,70$.

It is exactly the same arithmetic as $\dfrac{3}{30} = \dfrac{7}{x}$ but the reasoning is more transparent.

When does proportion work? Only when two quantities scale together , doubling one doubles the other, tripling one triples the other. We call this direct proportion.

• $\checkmark$ Cost of identical items vs number of items.
• $\checkmark$ Distance at constant speed vs time.
• $\checkmark$ Wages vs hours worked.
• $\times$ Number of workers vs time to finish a job , that is inverse proportion. (More workers, less time.)
• $\times$ Side of a square vs its area , area grows as the square of the side, not directly.

Always pause to ask whether your quantities really scale linearly before applying a proportion.

Notation. $y \propto x$ means $y = kx$ for a positive constant $k$ called the constant of proportionality. If $y$ depends on $x$ this way, then the ratio $\dfrac{y}{x}$ is always the same value, $k$.

Worked examples

Example 1. If $9$ pens cost $\mathbb{?}\,135$, find the cost of $5$ pens.

• Cost of $1$ pen $= 135 / 9 = \mathbb{?}\,15$.
• Cost of $5$ pens $= 5 \cdot 15 = \mathbb{?}\,75$.

Example 2. A car travels $240$ km on $20$ L of petrol. How far on $35$ L?

• Distance per litre $= 240 / 20 = 12$ km/L.
• $35$ L gives $35 \cdot 12 = 420$ km.

Example 3. Solve $\dfrac{6}{x} = \dfrac{8}{12}$.

• Cross-multiply: $6 \cdot 12 = 8 x$.
• $x = 72 / 8 = 9$.

Example 4. On a map, $4$ cm represents $30$ km. What does $14$ cm represent?

• $1$ cm $\to 30/4 = 7.5$ km.
• $14$ cm $\to 14 \times 7.5 = 105$ km.

Try it yourself

1. If $5$ kg of rice cost $\mathbb{?}\,250$, find the cost of $8$ kg.
2. A train travels $360$ km in $4$ hours. How far in $7$ hours at the same speed?
3. Solve $\dfrac{7}{x} = \dfrac{14}{20}$.
4. A worker earns $\mathbb{?}\,720$ for $6$ hours. How much for $11$ hours?
5. The ratio of girls to boys is $3 : 5$. If there are $24$ girls, how many boys?
6. If $15$ books cost $\mathbb{?}\,375$, how many books can be bought with $\mathbb{?}\,500$?
7. Is "more workers, less time" a direct or inverse proportion?
8. The cost of $11$ kg of sugar is $\mathbb{?}\,506$. What is the cost of $4$ kg?

Activity

Tape recipe. Get a tape measure and measure your own height in cm. Then measure the height of one floor of a building. The ratio "your height : building height" is now a proportion. Use it to estimate how tall a $10$-storey building would be without measuring it.