Addition and subtraction of fractions

Numbers: Fractions

Addition and subtraction of fractions

We have seen how we can add fractions with like denominators. However, it can very often be useful to add fractions that do not have like denominators. In the action plan below we will show how this works.

	Action plan	Example
	We would like to add two fractions.	#\frac{3}{4} + \frac{1}{6}=\dots#
Step 1	We write the fractions with like denominators.	#\dfrac{3}{4} = \dfrac{18}{24} \text{ and }\dfrac{1}{6} = \dfrac{4}{24} #
Step 2	We add the fractions with like denominators.	#\dfrac{18}{24} + \dfrac{4}{24} = \dfrac{22}{24}#
Step 3	If possible, we simplify the answer.	#\dfrac{22}{24}=\dfrac{11}{12}#

We can also subtract fractions from each other in a similar way.

Example subtraction

# \begin{array}{rcl}\dfrac{1}{6} - \dfrac{3}{4} &=&\dfrac{4}{24} - \dfrac{18}{24} \\ &&\phantom{xxx}\blue{\text{written with like denominators}}\\ &=& \dfrac{-14}{24} \\ &&\phantom{xxx}\blue{\text{subtracted the fractions with like denominators}}\\ &=& -\dfrac{7}{12}\\ &&\phantom{xxx}\blue{\text{simplified}}\end {array} #

Calculate #{{8}\over{9}}+{{3}\over{5}}# and simplify as much as possible.

#{{8}\over{9}}+{{3}\over{5}}=# #{{67}\over{45}}#

#\begin{array}{rcl}\displaystyle {{8}\over{9}}+{{3}\over{5}}&=&\dfrac{40}{45}+\dfrac{27}{45} \\ &&\phantom{xxx}\blue{\text{created fractions with like denominators, new denominator is }9 \times 5=45} \\
&=& \dfrac{67}{45}\\
&&\phantom{xxx}\blue{\text{added the numerators while the denominators remain the same}} \\

\end{array}#

New example