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.
Watch this lesson
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
- 1The denominators already match (sevenths).Both fractions describe pieces of the same size.
- 2Add the numerators: 3 + 2 = 5.Three sevenths plus two sevenths is five sevenths.
- 3Answer: 5/7.The denominator never changes when adding.
Do not add the denominators. 3/7 + 2/7 is 5/7, not 5/14. 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).
- 1LCD = LCM of 2 and 3 = 6.The smallest number both 2 and 3 divide into.
- 21/2 = 3/6 (×3 top and bottom) and 1/3 = 2/6 (×2 top and bottom).Rewrite each fraction in sixths.
- 3Add: 3/6 + 2/6 = 5/6.Same denominator now, so add the numerators.
- 1LCD = LCM of 4 and 5 = 20.4 and 5 share no factors, so multiply them.
- 23/4 = 15/20 and 2/5 = 8/20.×5 for the first, ×4 for the second.
- 3Subtract: 15/20 − 8/20 = 7/20.7 and 20 share no factor, so it is already in lowest terms.
3Mixed numbers
A mixed number like 21/4 means 2 wholes plus 1/4. The safest exam method is to turn each mixed number into an improper fraction first, do the sum, then convert back at the end.
- 1Convert: 21/4 = 9/4 (2 × 4 + 1 = 9) and 11/3 = 4/3 (1 × 3 + 1 = 4).Whole × denominator + numerator, over the same denominator.
- 2LCD of 4 and 3 is 12: 9/4 = 27/12 and 4/3 = 16/12.
- 3Add: 27/12 + 16/12 = 43/12.
- 4Convert back: 43 ÷ 12 = 3 remainder 7 → 3 7/12.An exam usually wants a mixed number in the final answer.
Practice exercises
Work each one out, then click to reveal the answer.
- 12/9 + 5/9
Show answer ▾Hide answer ▴
Same denominator → 7/9 - 27/8 − 3/8
Show answer ▾Hide answer ▴
4/8 = 1/2 (simplify) - 31/2 + 1/4
Show answer ▾Hide answer ▴
LCD 4 → 2/4 + 1/4 = 3/4 - 42/3 + 1/6
Show answer ▾Hide answer ▴
LCD 6 → 4/6 + 1/6 = 5/6 - 55/6 − 1/2
Show answer ▾Hide answer ▴
LCD 6 → 5/6 − 3/6 = 2/6 = 1/3 - 63/4 + 2/5
Show answer ▾Hide answer ▴
LCD 20 → 15/20 + 8/20 = 23/20 = 1 3/20 - 711/2 + 21/4
Show answer ▾Hide answer ▴
3/2 + 9/4 = 6/4 + 9/4 = 15/4 = 33/4 - 831/3 − 11/2
Show answer ▾Hide answer ▴
10/3 − 3/2 = 20/6 − 9/6 = 11/6 = 1 5/6 - 9A pipe is 3/4 m long and another is 2/3 m. What is their total length?
Show answer ▾Hide answer ▴
LCD 12 → 9/12 + 8/12 = 17/12 = 1 5/12 m
Quick Quiz
5 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 8 past papers and worksheets on this topic.