Fractions explained: simplify, add, multiply and divide
A fraction is a division that has not been carried out yet. Four rules cover almost everything you need, and only addition requires a little preparation.
Simplifying a fraction
Divide the top and the bottom by the same number. The value does not change. A fraction is fully simplified when the two numbers share no common factor other than 1.
Example: simplify 18/24
- Both are divisible by 6
- 18 ÷ 6 = 3 and 24 ÷ 6 = 4
18/24 = 3/4
Adding and subtracting
Fractions can only be added when they share a denominator. Use the smallest common multiple of the two denominators.
Example: 1/4 + 1/6
- The smallest common multiple of 4 and 6 is 12
- 1/4 = 3/12 and 1/6 = 2/12
- 3/12 + 2/12 = 5/12
1/4 + 1/6 = 5/12
Multiplying
a/b × c/d = (a × c) / (b × d)
Example: 2/3 × 9/10
- 2 × 9 = 18 and 3 × 10 = 30
- 18/30 simplifies by 6
2/3 × 9/10 = 3/5
Dividing
Dividing by a fraction means multiplying by its reciprocal, the same fraction turned upside down.
a/b ÷ c/d = a/b × d/c
Example: 3/4 ÷ 2/5
- 3/4 × 5/2
- 15/8
3/4 ÷ 2/5 = 15/8
Simplify before multiplying whenever you can: the numbers stay small and errors become rarer.
Common pitfalls
- Adding tops and bottoms directly: 1/2 + 1/3 is not 2/5 but 5/6.
- Inverting the wrong fraction in a division: only the second one is turned over.
- Cancelling terms in a sum: in (2 + 3)/2, the 2s cannot be crossed out.
Worked exercises
Try to find the answer first, then open the solution.
Calculate 5/6 − 1/4.
Show the solution
- Common denominator: 12
- 10/12 − 3/12 = 7/12
Answer: 7/12
Calculate 4/9 × 3/8.
Show the solution
- 12/72
- Simplify by 12
Answer: 1/6
Calculate 7/8 ÷ 7/16.
Show the solution
- 7/8 × 16/7
- 112/56 = 2
Answer: 2