To add or subtract fractions, you must first make the denominators the same. Once both fractions share a common denominator, add or subtract the numerators and simplify the result. This single rule applies whether you are working with proper fractions, improper fractions, or mixed numbers.
Why do you need a common denominator?
Fractions represent parts of a whole, but the size of each part depends on the denominator. You cannot add thirds and quarters directly any more than you can add centimetres and inches without converting first. Once the denominators match, every part is the same size and you can simply count them.
The lowest common denominator (LCD) is the smallest number that both denominators divide into exactly. Using the LCD keeps numbers small and avoids extra simplification at the end.
How to add fractions with the same denominator
When the denominators are already equal, add the numerators and keep the denominator unchanged.
Rule: a/c + b/c = (a + b)/c
Worked example 1
Calculate 3/7 + 2/7.
The denominators are both 7, so:
3/7 + 2/7 = (3 + 2)/7 = 5/7
Answer: 5/7
How to add fractions with different denominators
Step 1 — Find the lowest common denominator
List multiples of each denominator until you find the smallest number in both lists.
Step 2 — Convert each fraction to an equivalent fraction with that denominator
Multiply numerator and denominator by the same number.
Step 3 — Add the numerators
Keep the new denominator.
Step 4 — Simplify if possible
Divide numerator and denominator by their highest common factor (HCF).
Worked example 2
Calculate 1/3 + 1/4.
Step 1 — LCD of 3 and 4: Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. LCD = 12.
Step 2 — Convert:
1/3 = 4/12(multiply top and bottom by 4)1/4 = 3/12(multiply top and bottom by 3)
Step 3 — Add:
4/12 + 3/12 = 7/12
Step 4 — Simplify: 7 and 12 share no common factors other than 1, so 7/12 is already in its simplest form.
Answer: 7/12
Worked example 3
Calculate 5/6 + 2/9.
LCD of 6 and 9: Multiples of 6: 6, 12, 18. Multiples of 9: 9, 18. LCD = 18.
Convert:
5/6 = 15/18(multiply by 3)2/9 = 4/18(multiply by 2)
Add: 15/18 + 4/18 = 19/18
Simplify: 19 and 18 share no common factor, so the answer is the improper fraction 19/18, or equivalently the mixed number 1 1/18.
Answer: 19/18 (or 1 1/18)
How to subtract fractions
The process is identical to addition, except in Step 3 you subtract the numerators instead.
Rule: a/c − b/c = (a − b)/c
Worked example 4
Calculate 5/8 − 1/8.
Same denominator, so:
5/8 − 1/8 = (5 − 1)/8 = 4/8
Simplify: HCF of 4 and 8 is 4, so 4/8 = 1/2.
Answer: 1/2
Worked example 5
Calculate 3/4 − 2/5.
LCD of 4 and 5: 4 and 5 share no common factors, so LCD = 4 × 5 = 20.
Convert:
3/4 = 15/20(multiply by 5)2/5 = 8/20(multiply by 4)
Subtract: 15/20 − 8/20 = 7/20
HCF of 7 and 20 is 1, so no further simplification needed.
Answer: 7/20
How to add and subtract mixed numbers
A mixed number has a whole part and a fraction part (e.g. 2 3/5). There are two strategies:
Strategy A — Convert to improper fractions first, then proceed as above.
Strategy B — Add the whole-number parts and fraction parts separately.
Worked example 6 (Strategy A)
Calculate 1 2/3 + 2 1/4.
Convert to improper fractions:
1 2/3 = (3 × 1 + 2)/3 = 5/32 1/4 = (4 × 2 + 1)/4 = 9/4
LCD of 3 and 4 = 12.
Convert:
5/3 = 20/129/4 = 27/12
Add: 20/12 + 27/12 = 47/12
Convert back to mixed number: 47 ÷ 12 = 3 remainder 11, so 47/12 = 3 11/12.
Answer: 3 11/12
Worked example 7 (Strategy B — same problem)
Whole parts: 1 + 2 = 3.
Fraction parts: 2/3 + 1/4. LCD = 12. 8/12 + 3/12 = 11/12.
Combine: 3 + 11/12 = 3 11/12. ✓
Both strategies give the same answer.
Summary table
| Situation | Method |
|---|---|
| Same denominator | Add/subtract numerators; keep denominator |
| Different denominators | Find LCD; convert; then add/subtract numerators |
| Mixed numbers | Convert to improper fractions, OR handle whole/fraction parts separately |
| Result is improper fraction | Convert to mixed number if required |
| Always simplify | Divide numerator and denominator by their HCF |
Common mistakes to avoid
Mistake 1 — Adding the denominators too.
1/3 + 1/4 ≠ 2/7. The denominator does NOT get added. Only the numerators are added once the denominators match.
Mistake 2 — Forgetting to convert both fractions. When finding the LCD, both fractions must be converted, not just one.
Mistake 3 — Not simplifying the final answer.
4/8 should be simplified to 1/2. KS3 mark schemes expect answers in their simplest form.
Mistake 4 — Errors with mixed numbers.
When using Strategy A, double-check your conversion: 2 3/5 = (5 × 2 + 3)/5 = 13/5, not 10/5 (a common slip).
How fractions fit the KS3 national curriculum
The Department for Education's KS3 mathematics programme of study requires pupils to "use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers." Adding and subtracting fractions is therefore assessed across Year 7, Year 8, and Year 9, and underpins later GCSE topics such as algebraic fractions. BBC Bitesize's KS3 maths resources treat fraction arithmetic as a core number skill that connects directly to ratio, proportion, and percentage work.
Frequently asked questions
What is a lowest common denominator?
The lowest common denominator (LCD) is the smallest positive integer that both denominators divide into exactly. For denominators 4 and 6, the LCD is 12 (since 4 and 6 both divide 12, and no smaller number satisfies this). Using the LCD keeps the numbers as small as possible, which makes simplifying the final answer easier.
Do you always need to find the LCD, or can you use any common denominator?
Any common denominator will give the correct answer, but it may not be in its simplest form. For example, with 1/3 + 1/4, you could use a denominator of 24 (3 × 4): 8/24 + 6/24 = 14/24 = 7/12. You get 7/12 after simplifying. Using the LCD of 12 gives 7/12 directly, with no extra simplification needed. Either way works; the LCD just saves a step.
Can the answer to a fraction addition ever be a whole number?
Yes. If the numerator of the result equals the denominator, the fraction equals 1. More generally, if the numerator is an exact multiple of the denominator, the result is a whole number. For example, 3/8 + 5/8 = 8/8 = 1. At KS3 you should always simplify to the whole number in such cases.
How do I subtract a larger fraction from a smaller one?
The answer will be negative. Apply the same steps — find the LCD, convert, then subtract — and the result will have a negative sign. For example, 1/4 − 3/4 = (1 − 3)/4 = −2/4 = −1/2. Negative fractions follow the same arithmetic rules as negative integers.
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