Ratio Toolkit (Edexcel IGCSE Maths): Revision notes

Section 1

What is a ratio and how do I simplify one?

A ratio compares two or more quantities of the same kind, written as $a:b$ (or $a:b:c$ for three quantities). Ratios must use consistent units before comparing — convert everything to the same unit first.

To simplify a ratio, divide every part by their highest common factor (HCF), just like simplifying a fraction.

$8:12 = 2:3$ (divide both by 4)
$15:25:40 = 3:5:8$ (divide all by 5)

If the ratio contains decimals or fractions, multiply through first to clear them, then simplify.

$0.4:1.2 \rightarrow 4:12 \rightarrow 1:3$
$\frac{1}{2}:\frac{3}{4} \rightarrow 2:3$ (multiply both by 4)

Key termsratiosimplifyhighest common factor (HCF)

Common mistake

Forgetting to convert units before simplifying — e.g. $50\text{cm}:2\text{m}$ must become $50:200 = 1:4$ , not $50:2$ .

Section 2

How do I write a ratio in the form $1:n$ or $n:1$ ?

To write a ratio in the form $1:n$ , divide both sides by the first number so the left-hand side becomes 1. For $n:1$ , divide both sides by the second number instead.

Example: Write $4:9$ in the form $1:n$ . $4:9 = \frac{4}{4}:\frac{9}{4} = 1:2.25$

Example: Write $4:9$ in the form $n:1$ . $4:9 = \frac{4}{9}:\frac{9}{9} = 0.444...:1$

This form is especially useful for comparing rates, scale drawings, and exchange rates, since it lets you read off directly "for every 1 unit of X, there are $n$ units of Y".

Key terms1:n formunitary ratio

Exam tip

Whichever side you want to become 1, divide both parts by that same number.

Section 3

How do I share an amount in a given ratio?

To share a quantity in a given ratio:

Add the parts of the ratio together to find the total number of parts.
Divide the total amount by the total number of parts to find the value of one part.
Multiply one part by each ratio number to find each share.

Example: Share £60 in the ratio $2:3:5$ .

Step	Working
Total parts	$2+3+5=10$
Value of 1 part	$60 \div 10 = 6$
Shares	$2\times6=12$ , $3\times6=18$ , $5\times6=30$

Check: $12+18+30=60$ ✓ — always verify the shares add back to the original total.

Key termsshare in a ratioparts

Example

Divide 45 sweets between Amir and Zara in the ratio $4:5$ . Total parts $=9$ , one part $=45\div9=5$ , so Amir gets $20$ and Zara gets $25$ .

Section 4

How do I work backwards when I only know one share or the difference?

Sometimes you are given one share (or the difference between shares) instead of the total.

Given one share: find the value of one part using that share, then scale up.

Example: In the ratio $3:7$ , the smaller share is $21$ . One part $=21\div3=7$ . The larger share $=7\times7=49$ . Total $=3\times7+7\times7=21+49=70$ .

Given a difference: find how many parts the difference represents, divide to find one part, then scale up.

Example: Two amounts are in ratio $3:8$ and differ by $30$ . The difference is $8-3=5$ parts, so one part $=30\div5=6$ . The amounts are $18$ and $48$ .

Key termsdifference in ratio

Common mistake

Do not divide the difference by the total number of parts — divide it by the difference between the parts instead.

Section 5

How do ratios connect to fractions and scale?

A ratio $a:b$ can be written as fractions of the whole: the first quantity is $\frac{a}{a+b}$ of the total, and the second is $\frac{b}{a+b}$ .

Example: In the ratio $2:3$ , the first share is $\frac{2}{5}$ of the total and the second is $\frac{3}{5}$ .

Ratios also describe scale (e.g. map scales, recipes, and similar shapes). A map scale of $1:25000$ means $1\text{cm}$ on the map represents $25000\text{cm}$ ( $250\text{m}$ ) in real life. Recipe ratios stay constant when scaling up or down — if a recipe needs flour and sugar in ratio $5:2$ , doubling the recipe keeps the ratio $5:2$ , only the total changes.

Key termsscaleequivalent ratio

Think of it like this

Think of ratio parts like ingredients in a recipe — scaling the recipe up or down changes the amounts but never the balance between ingredients.

Must Know

Simplify a ratio by dividing all parts by their HCF; clear decimals/fractions first.
Always convert to the same units before comparing or simplifying a ratio.
To write $a:b$ as $1:n$ , divide both parts by $a$ ; for $n:1$ , divide both parts by $b$ .
To share a total: add the parts, divide the total by the number of parts, then multiply by each part.
If given one share or a difference, find the value of one part from that information first, then scale up to find the rest.
A ratio $a:b$ means the first quantity is $\frac{a}{a+b}$ of the whole and the second is $\frac{b}{a+b}$ .

That's the notes covered.

Carry on to the next subtopic.