Cambridge Primary Checkpoint · Mathematics 0096 · Stage 6 · Number

Adding and Subtracting Fractions

Estimate, add and subtract fractions with different denominators using common denominators and equivalent fractions — with fraction-bar diagrams, worked examples and a Checkpoint-style quiz.

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

In the curriculum

Estimate, add and subtract fractions with different denominators (6Nf.05)

You already know how to add fractions with the same denominator: 2/7 + 3/7 = 5/7. In Stage 6 you add and subtract fractions with different denominators, such as 1/2 + 1/3, and you estimate the answer first so you can check it.

Key words

Word / phrase	Meaning / use
denominator	المقام — tells you the size of the parts (halves, thirds, sixths…)
common denominator	مقام مشترك — a denominator that both fractions can be changed to
multiple	مضاعف — a number in the times table: multiples of 4 are 4, 8, 12, 16 …
lowest common multiple (LCM)	المضاعف المشترك الأصغر — the smallest number in both times tables (LCM of 4 and 6 is 12)
equivalent fraction	كسر مكافئ — same value, different numbers: 1/2 = 3/6
benchmark fractions	كسور مرجعية — easy fractions to compare with: 0, 1/2 and 1
estimate	تقدير — a quick, close answer to check your working
simplify	يبسّط — divide numerator and denominator by a common factor

1. Why the denominators must be the same

You can only add or subtract parts that are the same size. A half and a third are different-sized pieces, so 1/2 + 1/3 is not 2/5. First cut both into pieces of the same size — sixths — then count them.

Halves and thirds can both be cut into sixths.

Never add the denominators. 1/3 + 1/4 is not 2/7: 2/7 is less than 1/3, and adding 1/4 must make the answer bigger than 1/3.

2. The method

Estimate the answer using 0, 1/2 and 1 (see section 3).
Find a common denominator: a number that both denominators divide into. The lowest common multiple (LCM) keeps the numbers small.
Change each fraction to an equivalent fraction with that denominator (multiply top and bottom by the same number).
Add or subtract the numerators only. Keep the denominator the same.
Simplify the answer, and change an improper fraction to a mixed number if needed. Compare with your estimate.

Choosing the common denominator — three cases:

Case	Example	Common denominator
One denominator is a multiple of the other	2/3 + 1/6	6 (only change 2/3 = 4/6)
The denominators have no common factor	3/5 + 1/2	5 × 2 = 10
The denominators share a factor	3/4 + 1/6	LCM = 12 (4 × 6 = 24 also works, but you must simplify more)

Calculation	Working	Answer
1/2 + 1/3	3/6 + 2/6	5/6
2/3 + 1/6	4/6 + 1/6	5/6
3/4 + 1/6	9/12 + 2/12	11/12
3/5 + 1/2	6/10 + 5/10 = 11/10	1 1/10
5/6 − 1/4	10/12 − 3/12	7/12
7/8 − 1/2	7/8 − 4/8	3/8
4/5 − 2/3	12/15 − 10/15	2/15
5/6 − 1/3	5/6 − 2/6 = 3/6	1/2 (simplified)

Subtracting: 3/4 − 1/6 using twelfths.

3. Estimating with 0, 1/2 and 1

Before calculating, decide whether each fraction is close to 0, 1/2 or 1. Compare the numerator with the denominator:

Numerator very small compared with the denominator → close to 0 (1/9, 1/10, 2/15).
Numerator about half of the denominator → close to 1/2 (3/8, 4/9, 5/11, 6/13).
Numerator nearly equal to the denominator → close to 1 (4/5, 7/8, 11/12).

Benchmark fractions on a number line from 0 to 1.

Calculation	Estimate	Exact answer
3/8 + 4/5	1/2 + 1 = 1 1/2	15/40 + 32/40 = 47/40 = 1 7/40
7/8 + 5/6	1 + 1 = 2	21/24 + 20/24 = 41/24 = 1 17/24
11/12 − 1/10	1 − 0 = 1	55/60 − 6/60 = 49/60
4/9 + 3/7	1/2 + 1/2 = 1 (but both are a little less than 1/2, so the answer is a little less than 1)	28/63 + 27/63 = 55/63

Using an estimate to check. If you work out 3/8 + 4/5 and get 7/13, your estimate (about 1 1/2) tells you at once that something is wrong — 7/13 is about 1/2.

4. Word problems and mixed numbers

"What fraction is left?" The whole is 1 = the same number on top and bottom. Omar eats 1/3 of a cake and his sister eats 1/4. Together: 4/12 + 3/12 = 7/12. Left: 12/12 − 7/12 = 5/12.

Mixed numbers. Add or subtract the whole numbers and the fractions separately, or change to improper fractions:

1 1/3 + 1/2 = 1 + 2/6 + 3/6 = 1 5/6
2 1/2 − 3/4: change to improper fractions: 5/2 − 3/4 = 10/4 − 3/4 = 7/4 = 1 3/4

القاعدة الذهبية

ممنوع تجمع أو تطرح المقامات! وحّد المقامات الأول، وبعدين اجمع أو اطرح البسط بس والمقام يفضل زي ما هو.
مثال: ١/٢ + ١/٣ = ٣/٦ + ٢/٦ = ٥/٦ (مش ٢/٥).
قبل ما تحل، قدّر: الكسر ده قريب من صفر ولا نص ولا واحد؟ وبعد ما تحل، قارن إجابتك بالتقدير.

Worked examples

Worked example 1 — Work out 2/5 + 1/3. [1 mark]

Estimate: 2/5 is a bit less than 1/2, 1/3 is between 0 and 1/2 → answer a bit less than 1.
5 and 3 have no common factor, so the common denominator is 15.
2/5 = 6/15 (× 3) and 1/3 = 5/15 (× 5).
6/15 + 5/15 = 11/15. It cannot be simplified, and it is a bit less than 1 ✓.

Worked example 2 — Work out 5/6 − 3/8. [1 mark]

Multiples of 6: 6, 12, 18, 24. Multiples of 8: 8, 16, 24. LCM = 24. 5/6 = 20/24 and 3/8 = 9/24. 20/24 − 9/24 = 11/24. Common mistake: writing 2/2 (subtracting top and bottom separately).

Worked example 3 — Work out 3/4 + 2/3. Give your answer as a mixed number. [1 mark]

Common denominator 12: 9/12 + 8/12 = 17/12. 17 ÷ 12 = 1 remainder 5, so the answer is 1 5/12. Estimate: 1 + 1/2 = 1 1/2 — close ✓.

Worked example 4 — Without working out the exact answer, ring the best estimate of 4/7 + 8/9: 1/2 1 1 1/2 2 [1 mark]

4/7 is close to 1/2 (4 is about half of 7) and 8/9 is close to 1. Estimate 1/2 + 1 = 1 1/2. (Exact: 36/63 + 56/63 = 92/63 = 1 29/63, which is about 1.46 ✓.)

Worked example 5 — Yara reads 1/4 of her book on Monday and 3/8 of it on Tuesday. What fraction of the book does she still have to read? [2 marks]

Fraction read: 1/4 + 3/8 = 2/8 + 3/8 = 5/8 (1 method mark).
Whole book = 8/8. Still to read: 8/8 − 5/8 = 3/8.

Common mistake: giving 5/8 as the final answer. Read the question again — it asks what is left.

Worked example 6 — Leila says: "1/2 + 1/5 = 2/7." Explain how you know she is wrong, and write the correct answer. [2 marks]

Answer: 2/7 is less than 1/2 (half of 7 is 3.5, and 2 is less than 3.5), but adding 1/5 to 1/2 must give more than 1/2. She added the numerators and the denominators. Correct: 5/10 + 2/10 = 7/10.

Common mistakes

Adding or subtracting the denominators: 1/3 + 1/4 ≠ 2/7.
Changing the denominator but not the numerator: 2/3 is 4/6, not 2/6.
Adding the denominators after making them equal: 3/6 + 2/6 = 5/6, not 5/12.
Not simplifying: 3/6 should be written as 1/2 when the question asks for simplest form.
In "what fraction is left?" questions, stopping before subtracting from 1.
Not checking with an estimate — an answer smaller than one of the fractions you added must be wrong.

How Checkpoint asks this

"Work out 3/4 − 1/6" with an answer box. On Paper 1 (no calculator) you must show the equivalent fractions.
"Give your answer in its simplest form" or "as a mixed number" — follow the instruction for the mark.
Missing-number problems: 1/2 + ☐/10 = 9/10.
Estimation: "Ring the best estimate" or "Is the answer more or less than 1? Explain."
Two-step word problems (eaten, spent, read … what is left?). Write the first step clearly for a method mark.
"Explain the mistake" questions — use an estimate ("the answer must be more than 1/2") or show the correct method.

Practice

Remember

Estimate first: is each fraction close to 0, 1/2 or 1?
Make the denominators the same (use the lowest common multiple), using equivalent fractions.
Add or subtract the numerators only — the denominator stays the same.
Simplify, and change improper fractions to mixed numbers when asked.
"What is left?" → subtract from 1 whole (12/12, 8/8 …).

All pages in this course

Start here

Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

Practice tests