Add and subtract fractions: Denominators are multiples

In a nutshell

To add and subtract fractions, they must have the same denominator.  The lowest common denominator (LCD) is the smallest common multiple of the denominators of several fractions.  

Common denominators

procedure

Example 1

Which is larger $$\frac{3}{5}$$ or $$\frac{7}{10}$$?

Find the common denominator (the larger denominator of the pair):  

$$10$$​

Work out the multiplier for the fraction with the smaller denominator: 

$$5 \times 2 = 10$$​

Multiply the numerator and denominator of the fraction by the multiplier:

$$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$$​​

Compare the two fractions:

$$\frac{6}{10} < \frac{7}{10}$$​

​$$\frac{7}{10}$$ is larger than $$\frac{6}{10}$$ $$( = \frac{3}{5})$$.  Therefore $$\frac{7}{10}$$ is the larger fraction.

Adding and subtracting fractions

procedure

Example 2

What is $$\frac{7}{12}+\frac{1}{6}$$?

Multiply up the fraction with the smaller denominator ($$6$$) to match the lowest common denominator ($$12$$).

$$6 \times 2 = 12$$​

$$\frac{1}{6} \times 2 = \frac{1\times2}{6\times2} = \frac {2}{12}$$​​

Add the numerators together.

$$\frac{7}{12}+\frac{2}{12} = \frac{7+2}{12} = \frac{9}{12}$$​​

Simplify the fraction. The numerator and denominator are both multiples of 3.

$$\frac{9\div3}{12\div3} = \frac{3}{4}$$​​

Note: Not all fractions can be simplified.