Addition and subtraction (2)
Adding and subtracting fractions
In this section we add and subtract two fractions that have different denominators. The key is to change them so they share a common denominator — then the pieces are the same size and you can add or subtract the numerators.
Why we need a common denominator
You can only add or subtract fractions when the pieces are the same size — that means the denominators must match. So first change the fractions so they share a common denominator.
Finding a common denominator
List the multiples of each denominator and look for one they share. For 3/5 and 3/4: multiples of 5 are 5, 10, 15, 20 and multiples of 4 are 4, 8, 12, 16, 20 — so 20 is a common denominator.
Making equivalent fractions
Multiply the top and bottom of each fraction by the same number so it has the common denominator. For example, 3/4 = 15/20 and 9/5 = 36/20.
Adding and subtracting
Once the denominators match, add or subtract the numerators and keep the denominator the same.
Simplifying and mixed numbers
Simplify the answer if you can, and change an improper fraction (top heavier than bottom) into a mixed number. Estimate first so you know roughly what answer to expect.
Worked examples
Adding with different denominators
- To work out 1/2 + 1/3, find a common denominator — it is 6.
- Write 1/2 = 3/6 and 1/3 = 2/6.
- Add the numerators: 3/6 + 2/6.
Answer: 1/2 + 1/3 = 5/6
Subtracting with different denominators
- To work out 3/4 − 1/6, the common denominator is 12.
- Write 3/4 = 9/12 and 1/6 = 2/12.
- Subtract the numerators: 9/12 − 2/12.
Answer: 3/4 − 1/6 = 7/12
An answer that simplifies
- To work out 1/2 + 1/10, the common denominator is 10.
- Write 1/2 = 5/10, so 5/10 + 1/10 = 6/10.
- Simplify 6/10 by dividing top and bottom by 2.
Answer: 1/2 + 1/10 = 3/5
An improper answer (mixed number)
- To work out 9/5 + 3/4, the common denominator is 20.
- Write 9/5 = 36/20 and 3/4 = 15/20, so 36/20 + 15/20 = 51/20.
- Change the improper fraction to a mixed number: 51 ÷ 20 = 2 remainder 11.
Answer: 9/5 + 3/4 = 2 11/20
How much is left
- Leroy colours 1/6 and 1/4 of a circle.
- Total coloured: 2/12 + 3/12 = 5/12.
- The whole circle is 12/12, so subtract: 12/12 − 5/12.
Answer: Leroy leaves 7/12 of the circle white.
Where you see this in real life
- Sharing two pizzas that are cut into different numbers of slices.
- Adding up the fractions of a class who voted for different outings.
- Working out what fraction of a garden is left for another crop.
- Combining parts of a recipe measured in different fractions.
Key words
- Numerator
- The top number of a fraction — how many parts you have.
- Denominator
- The bottom number of a fraction — how many equal parts the whole is split into.
- Common denominator
- A denominator that two or more fractions can share, found from the multiples of each denominator.
- Equivalent fraction
- A fraction of equal value, such as 1/2 = 5/10.
- Improper fraction
- A fraction whose numerator is larger than its denominator, such as 51/20.
- Mixed number
- A whole number and a fraction together, such as 2 11/20.
- Simplify
- To write a fraction with the smallest possible numbers, such as 6/10 = 3/5.