Fraction operations: Add, subtract, multiply, divide

​​In a nutshell

There are four fraction operations: adding, subtracting, multiplying and dividing. When adding and subtracting, make sure that the denominators are the same to simplify your operation. When multiplying and dividing, make sure to multiply the numerators and denominators separately.

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Addition and subtraction

Addition and subtraction with same denominators

When adding and subtracting it is often easier for both fractions to have the same denominator. 

As the denominators are the same in both examples, simply add or subtract the numerators together whilst keeping the same denominator to get your final answer.

Example 1

Work out the value of 

$$\dfrac{1}{6}+\dfrac{1}{6}$$

$$= \underline{\dfrac{2}{6}}$$​​​​

​​Example 2

Work out the value of 

$$\dfrac{3}{4}- \dfrac{2}{4}$$

​$$= \underline {\dfrac{1}{4}}$$​​

​​

Addition and subtraction with different denominators

If the denominator is not the same, it helps to find the lowest common multiple (LCM) and then use equivalent fractions.

PROCEDURE

1. ​​Find the LCM of both the different denominators.
   
2. Work out the equivalent fractions so that both denominators have the same LCM.
   
3. Now add/subtract numerators, keeping the denominator the same.
   

​ 

When the fractions are in mixed number form it is easier to change them into improper fractions first and then follow the same procedure.

Example 3

Work out the value of 

​

$$\dfrac{1}{3} + \dfrac{1}{6}$$

The LCM of both $$3$$​ and $$6$$​ is $$6$$.

So $$\dfrac{1}{3} = \dfrac{2}{6}$$ and $$\dfrac{1}{6}$$ remains the same.

​

$$\dfrac{2}{6} + \dfrac{1}{6}$$

​​​

​$$= \underline {\dfrac{3}{6}}$$​​

Note: In some cases you can simplify your final answer, in this case it would be $$\dfrac{1}{2}$$ in simplified form.

Multiplication and division

Multiplication

When multiplying fractions, multiply the numerators and denominators separately.

Example 1

Work out the value of 

$$\dfrac{2}{3}$$ x $$\dfrac{1}{4}$$

​

First multiply the numerators together: 

$$2$$ x $$1$$ = $$2$$

Then multiply the denominators together:

 $$3$$ x $$4$$ = $$12$$

This gives a final answer of:

$$\dfrac{2}{12} = \underline{\dfrac{1}{6}}$$

Division

When dividing fractions, it is best to think about how many fractions can fit into that whole number.

Example 2

How many $$\dfrac{1}{4}$$ are there in $$1$$​? This can also be written as $$1$$ ÷ $$\dfrac{1}{4}$$? 

From this example we can see that this can also be written as $$1$$​ x $$\dfrac{4}{1}$$ = $$4$$.

Flipping one fraction and multiplying instead of dividing makes it easier to divide fractions.

Example 3

Work out the value of 

$$\dfrac{2}{6}$$ ÷ $$\dfrac{1}{2}$$

​First flip​

$$\dfrac{1}{2}$$ to $$\dfrac{2}{1}$$

Then write out the question again as:

$$\dfrac{2}{6}$$ x $$\dfrac{2}{1}$$

Now multiply numerators and denominators together separately ($$2$$​ x $$2$$ for the numerators and $$6$$ x $$1$$ for the denominators).

This gives a final answer of:

$$\dfrac{4}{6}$$ = $$\underline{\dfrac{2}{3}}.$$