Simplifying ratios: how to simplify a ratio step by step (divide every part by the highest common factor), ratios with different units (convert first), with decimals and fractions (multiply up to whole numbers first), three-part ratios, writing a ratio in the form 1 : n or n : 1, ratios and fractions, twelve worked examples, the mistakes, and a printable sheet with practice

"Simplifying ratios" is searched about 450 times a month in the UK with a traffic potential of about the same again, by KS3 and GCSE students (ratio is the first of the six GCSE topic areas by weight on Foundation papers), and the same skill opens the ratio units in Australian Year 7, US Grade 6 and CBSE Class 6. The rule is one line; the marks are in units, decimals, fractions, three parts and the 1 : n form. This page gives the method, each special case, twelve worked examples, the link to fractions, the mistakes and a printable sheet with practice.

Download the simplifying ratios sheet with practice (PDF) (A4, the method and every special case with worked examples on page 1, the 1 : n form, ratios and fractions and the practice set with answers on page 2, free).

What a ratio is

A ratio compares two or more quantities of the same kind by division. "2 parts squash to 5 parts water" is written 2 : 5 and read "two to five". The order matters: 2 : 5 is not 5 : 2. A ratio has no units once it is written, because both parts are in the same unit and the unit cancels.

Two ratios are equivalent when one is a multiple of the other: 2 : 5, 4 : 10, 6 : 15 and 20 : 50 all describe the same mixture. Simplifying means finding the equivalent ratio with the smallest whole numbers.

The method

Divide every part by the same number until nothing bigger than 1 divides them all.

12 : 18

• Both divide by 6 (the highest common factor): 12 ÷ 6 : 18 ÷ 6 = 2 : 3.
• Or in stages: ÷ 2 gives 6 : 9, then ÷ 3 gives 2 : 3. Same answer, more steps.

Check: 2 and 3 share no factor but 1, so the ratio is in its simplest form (lowest terms).

Finding the HCF: list the factors, or use the times tables backwards (what is the biggest number in both the 12 and the 18 times tables?). For big numbers, divide by 2, 3, 5 in turn.

Ratios with different units

Convert to the same unit first, then drop the units.

• 1 m : 40 cm → 100 cm : 40 cm → 100 : 40 → 5 : 2.
• 2 hours : 45 minutes → 120 : 45 → 8 : 3.
• £2 : 50p → 200p : 50p → 4 : 1.
• 3 kg : 750 g → 3,000 : 750 → 4 : 1.

Converting to the smaller unit keeps the numbers whole. Writing 1 : 40 for "1 m : 40 cm" is the classic wrong answer.

Ratios with decimals

Multiply every part by 10, 100 or whatever makes them whole numbers, then simplify.

• 1.5 : 2 → × 2 → 3 : 4. (Or × 10 → 15 : 20 → 3 : 4.)
• 0.25 : 1 → × 4 → 1 : 4.
• 2.4 : 3.6 → × 10 → 24 : 36 → 2 : 3.
• 0.3 : 0.45 → × 100 → 30 : 45 → 2 : 3.

Ratios with fractions

Multiply every part by the lowest common denominator to clear the fractions, then simplify.

• ½ : ⅓ → × 6 → 3 : 2.
• ¾ : 1½ → × 4 → 3 : 6 → 1 : 2.
• ⅖ : ⅗ → × 5 → 2 : 3.
• 1⅓ : 2 → × 3 → 4 : 6 → 2 : 3.

Three-part ratios

Divide all three by their HCF.

• 12 : 18 : 30 → ÷ 6 → 2 : 3 : 5.
• 10 : 15 : 25 → ÷ 5 → 2 : 3 : 5.
• 8 : 12 : 20 → ÷ 4 → 2 : 3 : 5.
• 6 : 9 : 15 → ÷ 3 → 2 : 3 : 5.

A number that divides only two of the parts is not a common factor: in 4 : 6 : 9, the 2 divides 4 and 6 but not 9, so the ratio is already simplest.

The form 1 : n (and n : 1)

Sometimes a question asks for the ratio with one part equal to 1, which may leave a decimal in the other.

• 4 : 10 in the form 1 : n → ÷ 4 → 1 : 2.5.
• 5 : 2 in the form 1 : n → ÷ 5 → 1 : 0.4. In the form n : 1 → ÷ 2 → 2.5 : 1.
• A map scale of 2 cm : 5 km → 2 : 500,000 → 1 : 250,000.

The 1 : n form is for comparing (which squash is stronger, 1 : 4 or 1 : 3.5?) and for scales; it is not the "simplest form", which uses whole numbers. Read the question.

Ratios and fractions

A ratio splits a whole into parts; each part is a fraction of the whole with the total of the parts as the denominator. In 2 : 3 there are 5 parts: the first share is ⅖ of the whole and the second ⅗. (Not ⅔: the common trap is to make a fraction from the two ratio numbers.)

Sharing £40 in the ratio 2 : 3: 5 parts, each £8, so £16 and £24. The simplest form of the ratio does not change the shares, which is why simplifying first is safe.

A ratio of 2 : 3 also means the first quantity is ⅔ of the second, and the second is 1½ times the first; those are the comparisons GCSE "a is to b" questions ask for.

Twelve worked examples

The mistakes that cost marks

• Dividing only one part.
• Swapping the order (45 : 30 is 3 : 2, not 2 : 3).
• Leaving different units in (1 m : 40 cm is not 1 : 40).
• Stopping too early (6 : 9 is not simplest; 2 : 3 is).
• Giving 1 : 2.5 when "simplest form" was asked (2 : 5), or 2 : 5 when "the form 1 : n" was asked (1 : 2.5).
• Making a fraction from the two ratio numbers (2 : 3 is ⅖ and ⅗, not ⅔).
• Writing the ratio as a fraction with a line in the middle; use the colon.

Where it sits in each curriculum

Practice (answers in the PDF)

1. Simplify 24 : 36, 18 : 45 and 56 : 21.
2. Simplify 2 m : 75 cm, 3 hours : 20 minutes and £1.50 : 60p.
3. Simplify 1.6 : 2.4 and 0.75 : 2.
4. Simplify ⅗ : ⅖ and 2½ : 1¼.
5. Simplify 20 : 30 : 50 and 9 : 12 : 21.
6. Write 8 : 20 in the form 1 : n, and 20 : 8 in the form n : 1. Then share £84 in the ratio 8 : 20.

Download the sheet and the practice set (PDF)

Where to get help

A 1-on-1 online maths class in UK after-school hours with a tutor who holds a master's in mathematics and teaches KS2, KS3 and GCSE, the Australian Years 7 and 8, US Grade 6 and 7 and CBSE Class 6 to 8 every week: ratio taught as "divide everything by the same number", with the units-and-decimals cases practised until they are routine, then shares, fractions and the 1 : n form. The GCSE maths page explains how the classes run; the free 30-minute first class is a real lesson on this week's homework. Related pages: graphing proportional relationships and fractions to decimals to percentages.

Questions parents ask

See how we teach this, 1-on-1 online →