OCR GCSE Mathematics: Ratio: simplifying, dividing, ratios and fractions (J560)

This study guide is written for OCR GCSE (9-1) Mathematics (J560), aligned to the specification Version 2.1 (August 2026), first assessment 2017. It covers section 5.01, Calculations with ratio, which is part of section 5, Ratio, Proportion and Rates Of Change. Everything here is Foundation and Higher content: there are no Higher tier only statements in this unit. Splitting a quantity into three or more parts (5.01b) is listed for Foundation tier learners in the specification’s second column, so Higher tier learners need it too. The specification says all content may be assessed on either the calculator or non-calculator papers, so practise every method below by hand.

Next, use the revision notes to memorise the methods and the practice questions to test them. Related pages: fractions, decimals and percentages and number operations and integers. Course hub: OCR GCSE Mathematics. Printable checklist. To find your weak spots first, try the free 10-minute diagnostics.

What this unit covers

Ref	What you must be able to do	Tier
5.01a	Write the ratio of quantities as a : b and simplify it; write it in the form 1 : n	Both
5.01b	Split a quantity into two parts in a given ratio; express a division as a ratio; find one quantity from another given their ratio	Both
5.01b	Split a quantity into three or more parts in a given ratio	Both (Foundation column 2)
5.01c	Interpret a ratio of two parts as a fraction of a whole	Both
5.01d	Solve ratio and proportion problems, such as adapting a recipe; understand the link between ratio and linear functions	Both

5.01a Equivalent ratios

A ratio compares the sizes of quantities. The ratio 3 : 5 says that for every 3 of the first thing there are 5 of the second. You can multiply or divide every part by the same non-zero number and the ratio stays equivalent, in the same way that 1/2 and 3/6 are equivalent fractions.

To simplify a ratio:

1. Write every quantity in the same unit.
2. Divide all the parts by their highest common factor.
3. If the parts are decimals or fractions, multiply first to make them whole numbers.

Worked example 1: mixed units

Simplify 45 cm : 1.8 m.

1.8 m = 180 cm
45 : 180
HCF of 45 and 180 is 45
45 ÷ 45 : 180 ÷ 45 = 1 : 4

The answer is 1 : 4. Notice that there are no units in a ratio once the quantities are in the same unit.

Worked example 2: mixed units, three digits

Simplify 2.4 kg : 900 g.

2.4 kg = 2400 g
2400 : 900
÷ 300 → 8 : 3

The answer is 8 : 3.

Worked example 3: fractions

Simplify 2/3 : 5/6. Multiply both parts by 6, the common denominator.

(2/3) × 6 = 4        (5/6) × 6 = 5
4 : 5

The answer is 4 : 5.

The form 1 : n

To write a : b as 1 : n, divide both parts by a. Then n = b ÷ a. The value of n may be a decimal or a fraction.

Write 18 : 45 in the form 1 : n.

18 ÷ 18 = 1
45 ÷ 18 = 2.5
18 : 45 = 1 : 2.5

The answer is 1 : 2.5. This form is useful because n tells you how many of the second quantity there are for each single unit of the first. Check against the earlier example: 8 : 3 in the form 1 : n is 1 : 0.375, because 3 ÷ 8 = 0.375.

Finding a missing term

Find the missing number in 7 : 4 = 35 : ?. The first part has been multiplied by 5 (7 × 5 = 35), so multiply the second part by 5 as well: 4 × 5 = 20.

5.01b Division in a given ratio

To share a quantity in a ratio, use the “parts” method.

1. Add the parts of the ratio to find the total number of parts.
2. Divide the quantity by the total number of parts to find the value of one part.
3. Multiply the value of one part by each number in the ratio.
4. Check that the shares add up to the original quantity.

Worked example 4: three parts

Share £840 in the ratio 2 : 3 : 7.

Total parts: 2 + 3 + 7 = 12
One part: 840 ÷ 12 = 70
Shares: 2 × 70 = 140,  3 × 70 = 210,  7 × 70 = 490
Check: 140 + 210 + 490 = 840

The shares are £140, £210 and £490. The method is the same for two parts and for three or more parts, and it works for amounts of money with pence too: £2.50 in the ratio 2 : 3 has 5 parts of 50p, giving £1.00 and £1.50.

Worked example 5: two parts

Divide 63 litres of water between a tank and a barrel in the ratio 4 : 5.

Total parts: 4 + 5 = 9
One part: 63 ÷ 9 = 7
Tank: 4 × 7 = 28 litres,  barrel: 5 × 7 = 35 litres

Expressing a division as a ratio

Sometimes the quantities have already been shared and you must write the ratio. Three friends raise £45, £75 and £120 for charity. Write the ratio of their amounts.

45 : 75 : 120
HCF is 15 → 3 : 5 : 8

The ratio is 3 : 5 : 8.

Finding one quantity from another

If you are given one quantity and the ratio, scale the ratio until the known part matches.

A shelter has cats and dogs in the ratio 5 : 8. There are 35 cats. How many dogs are there?

5 parts = 35 cats, so 1 part = 7
Dogs: 8 × 7 = 56

There are 56 dogs.

If you are given the difference between two shares, the difference is a number of parts. Two shares are in the ratio 8 : 3 and differ by 55. The difference is 8 - 3 = 5 parts, so 5 parts = 55 and one part = 11. The shares are 8 × 11 = 88 and 3 × 11 = 33.

5.01c Ratios and fractions

A ratio of two parts can be turned into fractions of the whole. If the ratio is a : b, the whole has a + b parts, so the first part is a/(a + b) of the whole and the second part is b/(a + b). The specification’s example is £9 split in the ratio 2 : 1, giving 2/3 of £9 and 1/3 of £9.

Worked example 6

Two groups are in the ratio 5 : 3. What fraction of the total is each group? Find each share of £96.

Total parts: 5 + 3 = 8
First group: 5/8      Second group: 3/8
5/8 of 96 = 96 ÷ 8 × 5 = 12 × 5 = 60
3/8 of 96 = 12 × 3 = 36

The fractions are 5/8 and 3/8, and the shares are £60 and £36.

For three parts the same idea applies. In the ratio 2 : 3 : 5 there are 10 parts, so the fractions are 2/10 = 1/5, 3/10 and 5/10 = 1/2.

Going the other way

A fraction can be turned back into a ratio. If red paint is 3/7 of a mixture, the rest, 4/7, is not red. The ratio of red to other paint is 3 : 4. Be careful: the ratio compares part to part (3 : 4), while the fraction compares part to whole (3/7).

If A is 2/3 of B, then A : B = 2 : 3.

5.01d Ratio and proportion problems

Adapting a recipe

A soup recipe for 6 people uses 450 g of lentils, 1.2 litres of stock and 3 onions. Adapt it for 4 people and for 10 people.

Method: find the amount for 1 person, then multiply.

Per person: 450 ÷ 6 = 75 g,  1.2 ÷ 6 = 0.2 litres,  3 ÷ 6 = 0.5 onion

For 4 people:  300 g,  0.8 litres,  2 onions
For 10 people: 750 g,  2 litres,    5 onions

You can also use a scale factor: 4 ÷ 6 = 2/3, and 450 × 2/3 = 300.

Changing a ratio

An orchestra has strings and wind players in the ratio 7 : 3. There are 42 string players. Six more wind players join. Find the new ratio of strings to wind.

7 parts = 42, so 1 part = 6
Wind players: 3 × 6 = 18
After 6 join: 18 + 6 = 24
New ratio: 42 : 24
HCF is 6 → 7 : 4

The new ratio is 7 : 4. Always work with real numbers, find the new quantities, and only then form the new ratio.

Ratio and linear functions

If two quantities are always in the ratio 1 : 3, then y = 3x, where x is the first quantity and y is the second. Cement and sand mixed in the ratio 1 : 3 give these values:

Cement x (kg)	2	5	8
Sand y (kg)	6	15	24

Plotted, the points lie on a straight line that starts at the origin (0, 0). Its gradient is 3, which is the n in 1 : n. In general, for quantities in the ratio a : b, the second quantity is y = (b/a) x. For paint in the ratio 4 : 7, y = 7x/4. When x = 12, y = 7 × 12 ÷ 4 = 21. When y = 35, x = 35 × 4 ÷ 7 = 20.

A linear function does not always describe a ratio. A taxi fare y = 2x + 3 (in pounds, for x km) has a fixed charge. For 2 km the fare is 7 and for 4 km it is 11. The ratios 7 : 2 and 11 : 4 are not equal, so the fare is not in a constant ratio to the distance. Only a line through the origin, y = kx, gives a constant ratio.

Common errors

• Simplifying before the units match. 45 cm : 1.8 m is not 45 : 1.8.
• Dividing the quantity by one of the ratio numbers, not by the total number of parts.
• Mixing up part-to-part and part-to-whole. The ratio 2 : 3 gives fractions 2/5 and 3/5, not 2/3 and 3/3.
• Adding the new amount to the ratio numbers in a changing-ratio problem, instead of to the real quantities.
• Giving a final answer such as 3 : 5 : 8 without checking that the shares add up to the total.
• Treating every linear function as a ratio. Only y = kx has a constant ratio.

Next steps

Work through the practice questions, then return to the revision notes for the quick-recall tables.

Official syllabus

OCR, GCSE (9-1) Mathematics (J560), specification Version 2.1 (August 2026), first assessment 2017, section 5.01 Calculations with ratio.