Ratios and Proportions

What is a ratio?

A ratio is a way to compare two quantities — to say how much of one thing there is compared with another. If a fruit bowl has 4 apples and 6 oranges, the ratio of apples to oranges is 4 to 6, which we write as 4 : 6.

Ratios are everywhere: mixing paint, following a recipe, reading a map scale, sharing money fairly, or comparing speeds. Once you understand them, a lot of real-world maths suddenly clicks into place.

You can write a ratio three ways, and they all mean the same thing:

• with a colon: 4 : 6
• in words: 4 to 6
• as a fraction: 4/6

The fraction form is a big hint — it tells you that the tools you already use for fractions, like simplifying, will work on ratios too. A quick look at Introduction to Fractions will pay off here.

Order matters

A ratio carries information in its order. "The ratio of apples to oranges is 4 : 6" is not the same as "oranges to apples is 4 : 6." Always read carefully and keep the quantities in the order the problem describes them.

Simplifying ratios

Just like fractions, ratios are usually written in their simplest form. To simplify, divide both parts by the same number — their greatest common factor (GCF).

Example: simplify 4 : 6.

Both 4 and 6 divide by 2:

$$ 4 : 6 \;\;\to\;\; (4 \div 2) : (6 \div 2) \;\;=\;\; 2 : 3 $$

So for every 2 apples there are 3 oranges. The numbers are smaller, but the relationship is identical.

Example: simplify 15 : 25. The GCF is 5, so 15 : 25 = 3 : 5.

Equivalent ratios

Equivalent ratios describe the same relationship using different numbers. You make them by multiplying or dividing both parts by the same value.

Start	× or ÷	Equivalent ratio
2 : 3	× 2	4 : 6
2 : 3	× 5	10 : 15
12 : 18	÷ 6	2 : 3
1 : 4	× 10	10 : 40

This is the secret behind scaling recipes up or down. If a sauce uses sugar and vinegar in the ratio 2 : 3, then 4 : 6 and 10 : 15 taste exactly the same.

What is a proportion?

A proportion is a statement that two ratios are equal. For example:

$$ \frac{2}{3} = \frac{4}{6} $$

This is true because 4 : 6 is just 2 : 3 scaled up. Proportions are powerful because they let us find a missing number.

Solving proportions with cross multiplication

When one value in a proportion is unknown, we can solve for it using cross multiplication. The rule comes from a simple fact: if two fractions are equal, their cross products are equal too.

$$ \frac{a}{b} = \frac{c}{d} \quad\Rightarrow\quad a \times d = b \times c $$

Worked example 1. Solve 2/5 = x/20.

1. Cross multiply: 2 × 20 = 5 × x
2. Simplify: 40 = 5x
3. Divide both sides by 5: x = 8

Check: 2/5 = 8/20, and 8/20 simplifies back to 2/5. ✓ This balancing-both-sides idea is exactly the one used in Solving Linear Equations.

Worked example 2 — a recipe. A pancake recipe for 4 people uses 6 eggs. How many eggs are needed for 6 people?

Set up a proportion (eggs over people, in the same order on both sides):

$$ \frac{6}{4} = \frac{x}{6} $$

Cross multiply: 6 × 6 = 4 × x → 36 = 4x → x = 9 eggs.

Sharing in a given ratio

A very common exam question shares a total amount in a given ratio. The trick is to think in parts.

Worked example 3. Share £40 between two people in the ratio 3 : 5.

1. Add the parts: 3 + 5 = 8 parts in total.
2. Find one part: £40 ÷ 8 = £5 per part.
3. Multiply: first person 3 × £5 = £15; second person 5 × £5 = £25.

Check: £15 + £25 = £40. ✓ The amounts add back to the total, which tells you the split is correct.

Worked example 4 — a class

In a class the ratio of girls to boys is 3 : 2, and there are 30 students in total.

• Total parts = 3 + 2 = 5
• One part = 30 ÷ 5 = 6 students
• Girls = 3 × 6 = 18, Boys = 2 × 6 = 12

Practice activity

Work these out on paper. Simplify where you can, and check each answer.

1. Simplify the ratio 18 : 24.
2. Write two ratios equivalent to 5 : 2.
3. Solve the proportion 3/7 = 12/x.
4. Share 35 sweets between Ana and Ben in the ratio 4 : 3.
5. A map scale is 1 : 50 000. A road is 4 cm on the map. How long is it in real life, in km?

Answers: 1) 3 : 4 2) e.g. 10 : 4 and 15 : 6 3) x = 28 4) Ana 20, Ben 15 5) 4 × 50 000 = 200 000 cm = 2 km.

Why this matters

Ratios and proportions are the maths of comparison and scaling. Cooks scale recipes, builders read scale drawings, scientists mix solutions, and shoppers compare prices per gram using exactly these ideas. The core moves are simple: simplify like a fraction, build equivalent ratios by multiplying both parts, and use cross multiplication to find anything that is missing. Master those, and proportional thinking will serve you in science, money, and design for years to come.