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

These revision notes condense section 5.01, Calculations with ratio, from the OCR GCSE (9-1) Mathematics (J560) specification, Version 2.1 (August 2026), first assessment 2017. The content is Foundation and Higher: no statement in this unit is Higher tier only. The study guide has the full explanations and longer worked solutions. Then try the practice questions.

Links: OCR GCSE Mathematics hub, printable checklist, free 10-minute diagnostics, and the related page on fractions, decimals and percentages.

Key ideas at a glance

Ref	Key idea
5.01a	Equivalent ratios: multiply or divide every part by the same number. Simplify with the highest common factor. Form 1 : n means divide by the first part.
5.01b	Share in a ratio: add the parts, find one part, multiply. Three or more parts use the same method.
5.01b	Express a division as a ratio; find one quantity from another from the ratio.
5.01c	Ratio a : b gives fractions a/(a + b) and b/(a + b) of the whole.
5.01d	Ratio and proportion problems, including recipes; a constant ratio is a linear function y = kx.

Definitions and facts

• Ratio: compares quantities, written a : b. Order matters. 2 : 5 is not 5 : 2.
• Equivalent ratios: 3 : 4, 6 : 8 and 1.5 : 2 all describe the same comparison.
• Simplest form: whole numbers with no common factor other than 1.
• Form 1 : n: first part is 1. n can be a decimal or fraction.
• Part-to-part compares one part with another (3 : 4). Part-to-whole compares a part with the total (3/7).
• A constant ratio y : x = b : a means y = (b/a) x, a straight line through the origin.

Method in steps

Simplifying a ratio

1. Convert to the same unit.
2. Clear decimals or fractions by multiplying.
3. Divide by the highest common factor.

Reminder: 2.5 km : 500 m is 2500 : 500, which simplifies to 5 : 1. And 72 : 108 simplifies to 2 : 3. For three parts, 35 : 28 : 21 simplifies to 5 : 4 : 3 by dividing by 7.

Fractions: 3/4 : 5/8. Multiply by 8 to get 6 : 5, so the answer is 6 : 5.

Writing as 1 : n

Divide both parts by the first part. For 25 : 40, 40 ÷ 25 = 1.6, so the ratio is 1 : 1.6. Check by multiplying back: 25 × 1.6 = 40.

Sharing in a ratio

1. Total parts = sum of the ratio numbers
2. One part   = quantity ÷ total parts
3. Each share = ratio number × one part
4. Check      = shares add up to the quantity

Reminder: share £510 in the ratio 1 : 2 : 3. Total parts 6, one part 85, shares £85, £170, £255. Check: 85 + 170 + 255 = 510.

Using a difference

If the shares differ by d and the ratio is a : b, then (a - b) parts = d. Ratio 5 : 2 and difference 36: 3 parts = 36, one part = 12, shares 60 and 24.

Using one known share

Ratio 5 : 3, smaller share 27. Smaller share is 3 parts, so one part = 9. Larger share = 5 × 9 = 45.

Ratio to fractions

For a : b, the fractions of the whole are a/(a + b) and b/(a + b). For 4 : 3 the whole has 7 parts. Of 245, the first share is 4/7 × 245 = 140 and the second is 3/7 × 245 = 105. Check: 140 + 105 = 245.

Scaling a recipe

Find the amount for one portion, then multiply. Rice for 4 people is 280 g. One portion is 280 ÷ 4 = 70 g, so for 7 people you need 70 × 7 = 490 g.

Changing ratios

Find the real quantities, change them, then write the new ratio. Do not add to the ratio numbers.

Ratio and linear functions

Ratio of x : y	Equation	Example
1 : n	y = nx	1 : 4 gives y = 4x
a : b	y = (b/a) x	3 : 8 gives y = 8x/3; x = 9 gives y = 24

A line y = mx + c with c not zero does not describe a constant ratio.

Worked reminders

Known share, unknown share

Jo and Kit have pocket money in the ratio 3 : 7. Jo has £18. One part is 18 ÷ 3 = £6, so Kit has 7 × 6 = £42.

Total given, ratio given

A drink is made from concentrate and water in the ratio 1 : 4, and 650 ml is made in total. There are 5 parts, so one part is 650 ÷ 5 = 130 ml. Concentrate is 130 ml and water is 4 × 130 = 520 ml.

Changing ratio

A library has fiction and non-fiction books in the ratio 5 : 2, with 35 fiction books. One part is 7, so there are 2 × 7 = 14 non-fiction books. After 14 more non-fiction books arrive there are 28. The new ratio is 35 : 28, which simplifies by 7 to 5 : 4.

Must-know distinctions

Idea	Compare	Why it matters
Part-to-part and part-to-whole	3 : 4 and 3/7	The denominator of the fraction is the sum of the parts
Ratio and amount	2 : 3 and “2 and 3”	A ratio gives relative size, not actual size
Order of a ratio	cats : dogs and dogs : cats	Reversing the order reverses the answer
Ratio and difference	difference in parts	A difference of 36 means (a - b) parts, not 36 split in total

Checks to build into every answer

• Sum check: after sharing, add the shares. They must equal the original quantity.
• Ratio check: divide your two answers back down. If you shared in the ratio 3 : 7, the shares should simplify to 3 : 7.
• Size check: the larger ratio number must get the larger share. If it does not, you have swapped the order.
• Sense check: a recipe for more people needs more of every ingredient, and a recipe for fewer people needs less.
• Unit check: write the unit on a share or an amount, but never on a ratio.

Choosing a method

The question gives you	Start by finding
The total and the ratio	The total number of parts, then one part
One share and the ratio	The value of one part from that share
The difference between shares	The number of parts the difference represents
A fraction of a whole	The ratio of the part to the rest, using the denominator minus the numerator
A recipe for one group size	The amount for one portion

Quick self-test

Answer these without looking, then check below.

1. Simplify 90 : 60.
2. Simplify 1.2 m : 30 cm.
3. Write 14 : 35 in the form 1 : n.
4. Share 150 in the ratio 2 : 3.
5. Share 84 in the ratio 1 : 2 : 4.
6. Find the missing number: 3 : 5 = 21 : ?
7. What fraction of the total is the larger part when two parts are in the ratio 4 : 7?
8. Two quantities are in the ratio 5 : 3 and the smaller is 27. Find the larger.
9. Two shares are in the ratio 3 : 2 and differ by 14. Find both shares.
10. A pancake mix for 4 people uses 300 ml of milk. How much milk is needed for 10 people?
11. Quantities x and y are in the ratio 2 : 5. Write y in terms of x, and find y when x = 8.

Answers

1. 90 : 60 divided by 30 gives 3 : 2.
2. 1.2 m = 120 cm, so 120 : 30 = 4 : 1.
3. 14 : 35 = 2 : 5, so 35 ÷ 14 = 2.5 and the ratio is 1 : 2.5.
4. 5 parts, one part 30, shares 60 and 90.
5. 7 parts, one part 12, shares 12, 24 and 48.
6. 21 ÷ 3 = 7, so 5 × 7 = 35.
7. The whole has 11 parts, so the larger part is 7/11.
8. 27 ÷ 3 = 9, so 5 × 9 = 45.
9. 1 part = 14, so shares are 3 × 14 = 42 and 2 × 14 = 28.
10. 300 ÷ 4 = 75 ml per person, 75 × 10 = 750 ml.
11. y = 5x/2, and when x = 8, y = 20.

Where marks are usually lost

• Leaving units in different forms, such as cm and m, before simplifying.
• Stopping at a ratio that can still be simplified.
• Dividing by one number from the ratio rather than by the total number of parts.
• Using 2/3 for the fraction in a 2 : 3 split, when the fraction is 2/5.
• Giving the shares in the wrong order, so the larger share goes to the smaller part.
• Finding a difference in parts but forgetting to multiply by the value of one part.
• Adding an extra quantity to a ratio number rather than to the real amount.
• Skipping the check that the shares add up to the total.
• Not showing the value of one part, which leaves nothing to credit if an arithmetic slip occurs later.
• Writing 1 : n with n the wrong way up. Divide the second part by the first.

Official syllabus

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