## Adding, Subtracting, Multiplying and Dividing Fractions

Jump to Adding and Subtracting Fractions
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## Adding and Subtracting Fractions

When fractions have the same denominator we can add them together (or subtract one from the other).

If we add one fifth and add two fifths we will have three fifths

15 + 25 = 35

If we have 3 quarters and we take away 2 quarters we have 1 quarter left

3424 = 14

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When we do not have fractions with the same denominator we need to make the denominators the same before we can add them (or take them away).

We can make denominators the same using equivalent fractions.

Example: 13 + 25

To add these fractions we need to make the denominators the same.

To make the denominators the same we need to find a number that is in both the 3 and the 5 times tables. 15 is the lowest number in both the 3 and 5 times tables.

We need to multiply the denominator of 13 by 5 to make it 15. We need to multiply the numerator by 5 as well to keep the fraction equivalent to 13
We multiply the numerator and denominator of 25 by 3.

1 × 53 × 5 + 2 × 35 × 3

515 + 615

Now both fractions have the same denominators we can add them:

515 + 615 = 1115

Example: 3416

To subtract these fractions we need to make the denominators the same.

To make the denominators the same we need a number that is in the 4 and 6 times tables. The smallest number in the 4 and 6 times tables is 12 (If we used another number in both times tables the answer would still be correct, the working out would just be more difficult).

We need to multiply the numerator and denominator of 34 by 3
We need to multiply the numerator and denominator of 16 by 2

3 × 34 × 31 × 26 × 2

912212

Now both fractions have the same denominators we can subtract them:

912212 = 712

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All answers are given in their simplest form

When we have mixed numbers we can change the mixed numbers to improper (top heavy) fractions before adding (or subtracting) the fractions.

Example: 134 + 23

The mixed number we have here is 134

One whole is the same as 4 quarters.
Therefore we have: 44 + 34

44 + 34 = 74

We can change the question to: 74 + 23

To make the denominators the same we multiply the top and bottom of 74 by 3 and the top and bottom of 23 by 4

7 × 34 × 3 + 2 × 43 × 4

2112 + 812

2112 + 812 = 2912

We could leave our answer as an improper fraction or convert it back to a mixed number.

To convert 2912 to a mixed number we need to see how many times 12 goes into 29
12 goes into 29 2 times (with 5 left over)

2912 = 2512

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All answers are given in their simplest form

## Multiplying Fractions

To multiply fractions we multiply the numerators and multiply the denominators.

Example: 34 × 25

We multiply the numerators and multiply the denominators

3 × 24 × 5

620

We can simplify our answer by dividing the numerator and the denominator by 2

620 = 310

When we have mixed numbers we need to convert them to top heavy fractions (improper) before we can multiply them

Example: 123 × 27

One whole is the same as three thirds.
3 thirds and 2 thirds make 5 thirds.

123 = 33 + 23 = 53

53 × 27

We can now multiply the numerators and multiply the denominators

5 × 23 × 7 = 1021

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All answers are given in their simplest form

## Dividing Fractions

Division is the opposite operation to multiplication

Multiplying by 23 is the same as dividing by 32
Multiplying by 45 is the same as dividing by 54

We can divide fractions by multiplying the first fraction by the second fraction flipped over (the reciprocal of the second fraction).

Example: 25 ÷ 23

25 ÷ 23 is the same as 25 × 32

25 × 32 = 2 × 35 × 2 = 610

We can simplify the answer by dividing the top and bottom by 2

610 = 35

When we have mixed numbers we need to convert them to improper fractions before dividing the fractions

Example: 34 ÷ 215

We need to convert 215 to a top heavy fraction first
2 is the same as 105
105 + 15 = 115

We now have:

34 ÷ 115

Dividing by 115 is the same as multiplying by 511

34 ÷ 115 = 34 × 511

34 × 511 = 3 × 54 × 11 = 1544

Try these:
All answers are given in their simplest form