Writing a Ratio in the Form 1:n (GCSE Maths)

What does writing a ratio in the form 1:n mean?

The form 1:n is called the unitary form of a ratio — "unitary" because the first term is 1. It tells you directly: "for every 1 of the first quantity, there are n of the second."

For example, 1:3.5 means for every 1 part of the first quantity there are 3.5 parts of the second. The value of n does not have to be a whole number — decimals and fractions are perfectly valid.

How do you write a ratio in the form 1:n?

Method — two steps:

1. Write the ratio as a:b.
2. Divide both parts by a (the first term), giving 1 : b/a.

Worked example: Write the ratio 4:14 in the form 1:n.

1. Divide both parts by 4.
2. 4 ÷ 4 : 14 ÷ 4 = 1 : 3.5.

Worked example: Write the ratio 6:9 in the form 1:n.

1. Divide both parts by 6.
2. 6 ÷ 6 : 9 ÷ 6 = 1 : 1.5.

Worked example: Write the ratio 5:3 in the form 1:n.

1. Divide both parts by 5.
2. 5 ÷ 5 : 3 ÷ 5 = 1 : 0.6.

Note that when the second part is smaller than the first, n is less than 1 — this is still a valid answer.

How do you use the 1:n form to compare two ratios?

To compare two ratios, convert both to the 1:n form and then compare the value of n.

Worked example: A recipe A uses flour to sugar in the ratio 5:2. Recipe B uses flour to sugar in the ratio 3:1. Which recipe uses more sugar relative to flour?

Recipe A:

• 5:2 → divide by 5 → 1 : 0.4

Recipe B:

• 3:1 → divide by 3 → 1 : 0.333…

Since 0.4 > 0.333…, Recipe A uses more sugar relative to flour.

What if the ratio has three or more parts?

You can apply the same method. Divide all parts by the first term.

Example: Write 4:6:10 in the form 1:n:m.

Divide each part by 4: 4 ÷ 4 : 6 ÷ 4 : 10 ÷ 4 = 1 : 1.5 : 2.5.

What if the ratio involves measurements in different units?

Always convert to the same unit before writing the ratio.

Example: Write the ratio 50 cm to 2 m in the form 1:n.

1. Convert 2 m to cm: 2 m = 200 cm.
2. Ratio is 50:200.
3. Divide both by 50: 1:4.

This is the same skill as simplifying a ratio — the unit-conversion step must come first.

What mistakes should you avoid?

Mistake 1 — Dividing by the wrong term. The form 1:n requires dividing by the FIRST term, not the second. Dividing by the second term gives n:1, a different form.

Mistake 2 — Rounding prematurely. Keep the full decimal (or fraction) as long as possible. Only round to the number of decimal places specified in the question at the very end.

Mistake 3 — Assuming n must be a whole number. The n in 1:n is nearly always a decimal in these questions. For example, 3:7 in the form 1:n gives 1 : 2.333…, which is 1 : 7/3, or 1 : 2⅓.

Frequently asked questions

Does it matter which term becomes the 1?

Yes — the form 1:n places the 1 in the FIRST position. If a question asks for the form n:1 instead (the second term equals 1), you divide by the second term. Always read the question carefully to see which part should be 1.

Is the 1:n form the same as simplifying a ratio?

Not exactly. Simplifying a ratio means dividing both parts by their highest common factor to get the smallest whole-number ratio (for example, 8:12 simplifies to 2:3). Writing in 1:n form specifically makes the first term equal to 1, which may produce a decimal. The two techniques are related but serve different purposes.

When would I be asked to write a ratio in the form 1:n in an exam?

GCSE questions most commonly ask for the 1:n form when comparing two different rates or proportions — for example, comparing population densities, ingredient ratios across recipes, or unit prices. It is also used in map scale problems, where a scale of 1:25 000 means every 1 cm on the map represents 25 000 cm in real life.

Can n be a fraction instead of a decimal?

Yes. For example, 4:3 in the form 1:n gives 1 : ¾. Whether you write ¾ or 0.75 depends on what the question asks for. Fractions are exact; decimals may need rounding, so fractions are often preferable unless the question specifies decimal places.

---

For more ratio and proportion practice with Professor Pi step-by-step, visit aitutors.me.