Convenient addition and subtraction of fractions

Numbers: Fractions

Convenient addition and subtraction of fractions

We have seen how to add and subtract fractions. The most convenient way to do this, is to use the least common multiple. That way, fewer steps are needed to simplify the result.

	Action plan	Example
	We would like to add two fractions.	#\frac{2}{4} + \frac{2}{6}=\dots#
Step 1	We write the fractions with like denominators using the #\mathrm{lcm}#. #\mathrm{lcm}(4,6)=12#, so the new denominator is #12#.	# \begin{array}{rcl} \dfrac{2}{4} &=& \dfrac{6}{12} \\ \dfrac{2}{6} &=& \dfrac{4}{12} \ \end {array} #
Step 2	We add the fractions with like denominators.	#\dfrac{6}{12} + \dfrac{4}{12} = \dfrac{10}{12}#
Step 3	If possible, we simplify the answer.	#\dfrac{10}{12}=\dfrac{5}{6}#

In the same way, we can also subtract two fractions from each other using the #\mathrm{lcm}#.

Example subtraction

# \begin{array}{rcl}\dfrac{2}{6} - \dfrac{2}{4} &=&\dfrac{4}{12} - \dfrac{6}{12} \\ &&\phantom{xxx}\blue{\text{created fractions with like denominators using the }\mathrm{lcm}}\\ &=& \dfrac{-2}{12} \\ &&\phantom{xxx}\blue{\text{subtracted the fractions with like denominators}}\\&=&-\dfrac{1}{6}\\&&\phantom{xxx}\blue{\text{simplified}}\end {array} #

Calculate #{{5}\over{6}}+{{3}\over{4}}# and simplify as much as possible.

#{{5}\over{6}}+{{3}\over{4}}=# #{{19}\over{12}}#

#\begin{array}{rcl}\displaystyle {{5}\over{6}}+{{3}\over{4}}&=&\dfrac{10}{12}+\dfrac{9}{12} \\ &&\phantom{xxx}\blue{\text{created fractions with new like denominator }\mathrm{lcm}(6,4)=12} \\
&=& \dfrac{19}{12}\\
&&\phantom{xxx}\blue{\text{added the numerators while the denominators remain the same}} \\

\end{array}#

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