OCR GCSE Mathematics: Fractions and decimal fractions (J560) – Revision Notes

These notes condense sections 2.01 Fractions and 2.02 Decimal fractions of OCR GCSE (9-1) Mathematics, J560, in Version 2.1 of the specification (August 2026). The study guide has the full explanations and longer worked examples. Test yourself with the practice questions. Everything is for both tiers except recurring decimals to fractions, which is Higher tier only. The course page is the OCR GCSE Mathematics hub and the printable checklist lets you tick off each reference. To find weak spots first, use the free diagnostics.

Specification map

Spec ref	Skill	Tier
2.01a	Equivalent fractions, mixed numbers	Both
2.01b	Four operations with fractions, mixed numbers, negatives, improper fractions	Both
2.01c	Fraction of a quantity, fractions greater than 1, one quantity as a fraction of another	Both
2.02a	Fraction and terminating decimal, place value, division to convert	Both
2.02a	Recurring decimal and exact fraction	Higher tier only
2.02b	Add, subtract, multiply decimals including negatives, by hand	Both
2.02c	Divide a decimal by a whole number, then by a decimal, by hand	Both

Paper 2 (Foundation) and Paper 5 (Higher) do not allow a calculator, so practise every method on paper.

Equivalent fractions and mixed numbers (2.01a)

• Multiply or divide top and bottom by the same number to make an equivalent fraction.
• Simplify by dividing by the highest common factor.
• Mixed to improper: (whole × bottom + top)/bottom. 5 2/3 = 17/3.
• Improper to mixed: divide top by bottom; the remainder is the new top.

Method: simplify in steps. Find the highest common factor. If you cannot see it, divide by any common factor and repeat until the top and bottom share nothing. Example: 45/75 = 9/15 = 3/5.

Calculations with fractions (2.01b)

Operation	Method	Reminder
Add or subtract	Common denominator, then combine the tops only	Use the lowest common multiple
Multiply	Tops × tops, bottoms × bottoms	Cancel before multiplying
Divide	Keep, change, flip	Flip the second fraction only
Mixed numbers	Convert to improper first	Never work on whole parts separately
Negatives	Decide the sign, then use the magnitudes	Same signs positive, different signs negative

Method: add or subtract in four steps.

1. Convert any mixed numbers to improper fractions.
2. Find the lowest common multiple of the denominators.
3. Rewrite each fraction over it and combine the tops.
4. Simplify, and convert back to a mixed number if asked.

Worked reminder. 1/6 + 7/10: common denominator 30, so 5/30 + 21/30 = 26/30 = 13/15.

Worked reminder. 1 2/5 × 5/7 = 7/5 × 5/7 = 1. The 5s and 7s cancel diagonally.

Worked reminder. −4/5 ÷ 2/15 = −4/5 × 15/2 = −60/10 = −6. Cancelling first: −(2 × 3)/(1 × 1) = −6.

Worked reminder. 2 3/5 − 1 3/4: convert to 13/5 − 7/4, use denominator 20, so 52/20 − 35/20 = 17/20. The answer is 17/20.

Method: multiply or divide with mixed numbers.

1. Convert every mixed number to an improper fraction.
2. For a division, flip the second fraction and change the sign to multiply.
3. Cancel common factors between any top and any bottom.
4. Multiply across, then convert to a mixed number if the question asks.

Fractions of a quantity (2.01c)

• Fraction of a quantity: divide by the bottom, multiply by the top.
• Fraction greater than 1: same method; the answer exceeds the quantity.
• One quantity as a fraction of another: first over second, same units, simplified.

Worked reminder. 5/6 of £4.20: £4.20 ÷ 6 = £0.70, and £0.70 × 5 = £3.50.

Worked reminder. 11/5 of 45 m: 45 ÷ 5 = 9, and 9 × 11 = 99 m. A fraction above 1 always gives more than the original quantity.

Worked reminder. 18 minutes as a fraction of 1 hour 30 minutes: 18/90 = 1/5.

Decimals and fractions (2.02a)

Place value. Reading right of the point: tenths, hundredths, thousandths. In 6.208, the 2 is worth 2 tenths and the 8 is worth 8 thousandths.

Direction	Method	Example
Terminating decimal to fraction	Digits over 10, 100 or 1000, then simplify	0.65 = 65/100 = 13/20
Simple fraction to decimal	Scale to a power of 10, or divide top by bottom	9/40 = 0.225
Division that repeats	Divide until a remainder repeats	1/6 = 0.1666…

A fraction in simplest form gives a terminating decimal only if its denominator has no prime factors except 2 and 5. Otherwise the decimal recurs. This is a useful check, not something you must quote.

Recurring decimal to fraction (Higher tier only)

1. Let x equal the decimal.
2. Multiply by a power of 10 so the repeating parts line up.
3. Subtract to cancel the repeating part.
4. Solve for x and simplify.

Decimal	Working	Fraction
0.545454… (54 repeating)	100x − x = 54, so 99x = 54	6/11
0.1666… (6 repeating)	100x = 16.666…, 10x = 1.666…, 90x = 15	1/6

For a pure repeating block, write the block over 9s: one digit over 9, two digits over 99, three over 999. Then simplify. To go back, divide: 6/11 = 0.545454…

Worked reminder: division with a remainder pattern. 7/30 = 0.2333… because 7 ÷ 30 gives 0.2, then a remainder of 1 that produces 3 forever. Write it as 0.2 followed by a recurring 3.

Decimal arithmetic (2.02b and 2.02c)

Task	Method
Add or subtract	Line up the decimal points, pad with zeros
Multiply	Multiply as whole numbers; decimal places in the answer equal the total in both factors
Divide by a whole number	Divide as normal, keep the point in line
Divide by a decimal	Scale both numbers by one power of 10 so the divisor becomes whole
Negatives	Fix the sign first, then do the arithmetic on the magnitudes

Worked reminder. 6.3 − 2.75 = 6.30 − 2.75 = 3.55.

Worked reminder. 1.2 × (−0.07): 12 × 7 = 84 with 3 decimal places is 0.084; different signs give −0.084.

Worked reminder. 0.195 ÷ 0.15: multiply both by 100 to get 19.5 ÷ 15 = 1.3.

Worked reminder. −0.36 ÷ 0.9: multiply both by 10 to get −3.6 ÷ 9 = −0.4. The sign comes from the numbers: different signs give a negative.

Check by estimating. Before you finish any decimal calculation, round both numbers to one significant figure and see whether your answer is about the right size. 6.3 − 2.75 is near 6 − 3 = 3, so 3.55 is sensible.

Must-know distinctions

• Common denominator vs multiply across. Addition needs a common denominator. Multiplication does not.
• Fraction of vs fraction as. “3/5 of 40” gives a quantity. “12 out of 40 as a fraction” gives a fraction.
• Terminating vs recurring. 0.25 stops. 0.2525… repeats, so it equals 25/99, not 25/100.
• Dividing by a decimal vs dividing by a whole number. Dividing by 0.5 doubles, not halves.
• Improper vs mixed. Use improper fractions to calculate and mixed numbers only to present, unless the question says otherwise.

Quick self-test

1. Simplify 45/75.
2. Write 5 2/3 as an improper fraction.
3. Work out 1/6 + 7/10.
4. Work out −4/5 ÷ 2/15.
5. Find 5/6 of £4.20.
6. Write 0.65 as a fraction in its simplest form.
7. Write 9/40 as a decimal without a calculator.
8. Work out 6.3 − 2.75.
9. Work out 1.2 × (−0.07).
10. Work out 0.195 ÷ 0.15.
11. (Higher tier only) Write 0.545454… as a fraction in its simplest form.

Answers

1. 3/5
2. 17/3
3. 13/15
4. −6
5. £3.50
6. 13/20
7. 0.225 (9/40 = 225/1000 after multiplying top and bottom by 25)
8. 3.55
9. −0.084
10. 1.3
11. 6/11

Where marks are usually lost

• Adding the numerators and the denominators, for example 1/2 + 1/3 = 2/5.
• Leaving an answer unsimplified, or as an improper fraction when the question asks for a mixed number.
• Cancelling across a sum instead of a product, which is not allowed.
• Forgetting to convert mixed numbers before multiplying or dividing.
• Losing a minus sign in a negative fraction calculation.
• Putting the quantities the wrong way round, or not converting units, when writing one quantity as a fraction of another.
• Misplacing the decimal point in a product, or lining digits up instead of the points in a sum.
• Multiplying the divisor by a power of 10 but not the other number.
• Writing a recurring decimal like 0.7272… as 72/100 or 72/90 instead of 72/99.
• Giving a bare answer on a calculation that asks you to show working. Show each step so a slip can still earn credit.

Official syllabus

OCR, GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026), for first assessment in 2017, sections 2.01 Fractions and 2.02 Decimal fractions. Published by OCR (Oxford Cambridge and RSA).