What you'll learn
This topic covers ratio — simplifying ratios, dividing a quantity in a given ratio, and solving ratio problems. In this guide you will learn how to write and simplify ratios, how to share amounts, how to use ratios with one known part, and how ratio links to fractions and proportion. These skills appear in recipes, scale, money sharing and many real contexts.
Key terms and definitions
Ratio — a comparison of two or more quantities (e.g. 3 : 2).
Simplify — divide all parts of a ratio by a common factor.
Unitary ratio — a ratio written as 1 : n.
Proportion — a statement that two ratios are equal.
Parts — the shares a ratio divides a quantity into.
Core concepts
Writing and simplifying ratios
A ratio compares quantities in order, such as 3 : 2. Simplify by dividing all parts by their highest common factor — 12 : 8 simplifies to 3 : 2. Units must match before simplifying (convert first, e.g. 50 cm : 1 m = 50 : 100 = 1 : 2).
Dividing a quantity in a ratio
To share an amount in a ratio, add the parts to find the total number of parts, divide the amount by that total to find one part, then multiply for each share. For example, £40 in 3 : 5 has 8 parts, so one part is £5, giving £15 and £25.
Using one known part
Sometimes you know one part's value rather than the total. Find the value of one part from the known share, then scale up the others. If the ratio is 2 : 3 and the "2" part is 10, one part is 5, so the "3" part is 15.
Ratio and fractions
Each share is a fraction of the whole: in 3 : 2 the first quantity is 3/5 of the total and the second is 2/5. This link helps solve problems and check answers.
Unitary form and proportion
Writing a ratio as 1 : n (unitary form) makes comparisons and scaling easy, and is used in maps and best-buy problems. Proportion sets two ratios equal and lets you solve for an unknown by scaling.
Worked examples
Example 1: Simplifying
Simplify 18 : 24.
Divide by 6: 3 : 4.
Example 2: Sharing
Share £60 in the ratio 2 : 3.
5 parts, one part £12, so £24 and £36.
Example 3: One known part
A ratio is 4 : 7. The "4" part is 12. Find the "7" part.
One part = 12 ÷ 4 = 3, so 7 × 3 = 21.
Common mistakes and how to avoid them
Not matching units. Convert to the same unit before simplifying.
Dividing by the wrong total. Add the parts to find the total number of parts.
Treating one part as the whole. Find the value of a single part first.
Mixing up the order. Keep the quantities in the ratio's order.
Forgetting to simplify. Reduce the ratio to its simplest form.
Exam technique for Ratio
Match units and simplify by the highest common factor.
Add the parts to share a quantity, then find one part.
Use a known part to find one part, then scale the rest.
Link to fractions (each part over the total) to check.
Use 1 : n form for comparisons and best-buy questions.
Quick revision summary
A ratio compares quantities in order; simplify by dividing all parts by their highest common factor (12 : 8 = 3 : 2), making sure units match first. To share an amount, add the parts, divide the amount by the total parts to find one part, then multiply for each share (£40 in 3 : 5 → 8 parts, one part £5, giving £15 and £25). When you know one part's value, find a single part and scale the others up. Each share is also a fraction of the whole (3 : 2 means 3/5 and 2/5), which helps solve and check problems. Unitary form (1 : n) is handy for comparisons, scale and best-buy questions, and proportion equates two ratios to find an unknown. The common errors are mismatched units, dividing by the wrong total, treating one part as the whole, and losing the order. Match units, simplify, add the parts, find one part, and link ratios to fractions.