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

1⁄5 + 2⁄5 = 3⁄5

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

3⁄4 − 2⁄4 = 1⁄4

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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.

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Example: 1⁄3 + 2⁄5

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 1⁄3 by 5 to make it 15. We need to multiply the numerator by 5 as well to keep the fraction equivalent to 1⁄3
We multiply the numerator and denominator of 2⁄5 by 3.

1 × 5⁄3 × 5 + 2 × 3⁄5 × 3

5⁄15 + 6⁄15

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

5⁄15 + 6⁄15 = 11⁄15

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Example: 3⁄4 − 1⁄6

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 3⁄4 by 3
We need to multiply the numerator and denominator of 1⁄6 by 2

3 × 3⁄4 × 3 − 1 × 2⁄6 × 2

9⁄12 − 2⁄12

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

9⁄12 − 2⁄12 = 7⁄12

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All answers are given in their simplest form

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

Example: 13⁄4 + 2⁄3

The mixed number we have here is 13⁄4

One whole is the same as 4 quarters.
Therefore we have: 4⁄4 + 3⁄4

4⁄4 + 3⁄4 = 7⁄4

We can change the question to: 7⁄4 + 2⁄3

To make the denominators the same we multiply the top and bottom of 7⁄4 by 3 and the top and bottom of 2⁄3 by 4

7 × 3⁄4 × 3 + 2 × 4⁄3 × 4

21⁄12 + 8⁄12

21⁄12 + 8⁄12 = 29⁄12

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

To convert 29⁄12 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)

29⁄12 = 25⁄12

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All answers are given in their simplest form

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

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Example: 3⁄4 × 2⁄5

We multiply the numerators and multiply the denominators

3 × 2⁄4 × 5

6⁄20

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

6⁄20 = 3⁄10

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

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Example: 12⁄3 × 2⁄7

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

12⁄3 = 3⁄3 + 2⁄3 = 5⁄3

5⁄3 × 2⁄7

We can now multiply the numerators and multiply the denominators

5 × 2⁄3 × 7 = 10⁄21

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All answers are given in their simplest form

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Division is the opposite operation to multiplication

Multiplying by 2⁄3 is the same as dividing by 3⁄2
Multiplying by 4⁄5 is the same as dividing by 5⁄4

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

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Example: 2⁄5 ÷ 2⁄3

2⁄5 ÷ 2⁄3 is the same as 2⁄5 × 3⁄2

2⁄5 × 3⁄2 = 2 × 3⁄5 × 2 = 6⁄10

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

6⁄10 = 3⁄5

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

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Example: 3⁄4 ÷ 21⁄5

We need to convert 21⁄5 to a top heavy fraction first
2 is the same as 10⁄5
10⁄5 + 1⁄5 = 11⁄5

We now have:

3⁄4 ÷ 11⁄5

Dividing by 11⁄5 is the same as multiplying by 5⁄11

3⁄4 ÷ 11⁄5 = 3⁄4 × 5⁄11

3⁄4 × 5⁄11 = 3 × 5⁄4 × 11 = 15⁄44

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All answers are given in their simplest form