Proportion

A proportion is an equality of two ratios, like $\displaystyle {3:4}={6:8}$.

• A ratio is also a fraction, so the same proportion can be written $\displaystyle {\frac{3}{4}}={\frac{6}{8}}$.
• $\displaystyle 3:4$ and $\displaystyle 5:7$ are not equal ratios, so $\displaystyle {3:4}={5:7}$ is not a proportion.

In a proportion, the numerator of each fraction times the denominator of the other – the two products are equal:

• Why: multiply both sides of $\displaystyle {\frac{a}{b}}={\frac{c}{d}}$ by $\displaystyle b\cdot d$. On the left $\displaystyle b$ cancels, on the right $\displaystyle d$ – and $\displaystyle {a\cdot d}={c\cdot b}$ is left.
• In the example: in $\displaystyle {\frac{3}{4}}={\frac{6}{8}}$ the products are $\displaystyle {3\cdot8}={24}$ and $\displaystyle {4\cdot6}={24}$.
• The multiplication goes diagonally, in an X – so it is called cross-multiplication.

It works the other way too: if the cross products are equal, the ratios are equal.

• $\displaystyle 4:6$ and $\displaystyle 10:15$: $\displaystyle {4\cdot15}={60}$ and $\displaystyle {6\cdot10}={60}$ – the products are equal, so $\displaystyle {4:6}={10:15}$.
• $\displaystyle 2:3$ and $\displaystyle 5:8$: $\displaystyle {2\cdot8}={16}$ and $\displaystyle {3\cdot5}={15}$ – the products differ, so the ratios are not equal.

Check by cross-multiplication: are the ratios equal?

• $\displaystyle 3:5$ and $\displaystyle 9:15$
• $\displaystyle 4:7$ and $\displaystyle 8:15$
• $\displaystyle 1.5:2$ and $\displaystyle 6:8$

When one of the numbers in a proportion is missing, write $\displaystyle x$ in its place and solve an equation.
Let us find $\displaystyle x$ in $\displaystyle {\frac{x}{12}}={\frac{3}{4}}$:

• By cross-multiplication: $\displaystyle {4\cdot x}={12\cdot3}$, that is $\displaystyle {4x}={36}$.
• Divide both sides by 4: $\displaystyle {x}={9}$.

Solve.

A pancake recipe uses 4 eggs for 6 servings.
How many eggs are needed for 15 servings?

The ratio of eggs to servings stays the same. Let $\displaystyle x$ be the number of eggs for 15 servings, and write a proportion – eggs in the numerator, servings in the denominator, in both fractions:

By cross-multiplication $\displaystyle {6x}={60}$, so $\displaystyle {x}={10}$: 10 eggs are needed.

3 kg of apples hold 18 apples.
About how many apples are there in 5 kg of the same apples?
Write a proportion and solve it.

In a climbing club the ratio of instructors to members is $\displaystyle 2:15$.
The club has 45 members.
How many instructors does it have?
Write a proportion and solve it.