Adding & Subtracting Fractions

Adding and subtracting fractions with the same denominator, with different denominators using the LCD, and with mixed numbers — shown with a bar diagram, exam-style worked examples, a worksheet with memo and a quiz.

You can only add or subtract things that are the same kind. Three apples plus two apples is five apples — but three apples plus two oranges is neither. Fractions work exactly the same way: the denominators must match before you can add or subtract.

1Same denominator

Add or subtract the numerators only. The denominator stays the same.

Worked ExampleCalculate: 37 + 27

1
The denominators already match (sevenths).
Both fractions describe pieces of the same size.
2
Add the numerators: 3 + 2 = 5.
Three sevenths plus two sevenths is five sevenths.
3
Answer: 57.
The denominator never changes when adding.

⚠️

Do not add the denominators. 37 + 27 is 57, not 514. The bottom number tells you the size of the pieces — it does not change just because you have more of them.

2Different denominators — use the LCD

When the denominators differ, rewrite both fractions with a common denominator first. The best one to use is the LCM of the denominators, called the lowest common denominator (LCD).

12 and 13 are different-sized pieces. Rewrite both as sixths and they add easily: 36 + 26 = 56.

Worked ExampleCalculate: 12 + 13

1
LCD = LCM of 2 and 3 = 6.
The smallest number both 2 and 3 divide into.
2
12 = 36 (×3 top and bottom) and 13 = 26 (×2 top and bottom).
Rewrite each fraction in sixths.
3
Add: 36 + 26 = 56.
Same denominator now, so add the numerators.

Worked ExampleCalculate: 34 − 25

1
LCD = LCM of 4 and 5 = 20.
4 and 5 share no factors, so multiply them.
2
34 = 1520 and 25 = 820.
×5 for the first, ×4 for the second.
3
Subtract: 1520 − 820 = 720.
7 and 20 share no factor, so it is already in lowest terms.

3Mixed numbers

A mixed number like 214 means 2 wholes plus 14. The safest exam method is to turn each mixed number into an improper fraction first, do the sum, then convert back at the end.

Worked ExampleCalculate: 214 + 113

1
Convert: 214 = 94 (2×4 + 1 = 9) and 113 = 43 (1×3 + 1 = 4).
Whole × denominator + numerator, over the same denominator.
2
LCD of 4 and 3 is 12: 94 = 2712 and 43 = 1612.
3
Add: 2712 + 1612 = 4312.
4
Convert back: 43 ÷ 12 = 3 remainder 7 → 3 712.
An exam usually wants a mixed number in the final answer.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
29 + 59
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Same denominator → 79
2
78 − 38
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48 = 12 (simplify)
3
12 + 14
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LCD 4 → 24 + 14 = 34
4
23 + 16
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LCD 6 → 46 + 16 = 56
5
56 − 12
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LCD 6 → 56 − 36 = 26 = 13
6
34 + 25
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LCD 20 → 1520 + 820 = 2320 = 1 320
7
112 + 214
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32 + 94 = 64 + 94 = 154 = 334
8
313 − 112
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103 − 32 = 206 − 96 = 116 = 1 56
9
A pipe is 34 m long and another is 23 m. What is their total length?
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LCD 12 → 912 + 812 = 1712 = 1 512 m