Sums in Math can often require you to add and subtract different fractions.

Fractions addition and subtraction is similar to adding and subtracting whole numbers, but a little bit more care does often need to be taken.

Adding and subtracting like fractions which have the same denominator is the easier case. But we can also solve sums that require adding and subtracting unlike fractions with different denominators, although these cases usually require a bit more work.

A page that may be handy to view also is the understanding fractions page.
Which shows an introduction to fractions, and what they really mean in mathematical terms.

Examples

(1.1)

\bf{\frac{3}{7}}  +  \bf{\frac{2}{7}}

Solution

A sum such as this one where we are adding like fractions with the same denominators. The adding only needs to be performed on the numbers on top.
The denominator below of  7  will stay the same.

\bf{\frac{3}{7}} + \bf{\frac{2}{7}}   =    \bf{\frac{3 \space + \space 2}{7}}   =    \bf{\frac{5}{7}}

(1.2)

\bf{\frac{8}{9}}  −  \bf{\frac{1}{9}}

Solution

\bf{\frac{8}{9}}  −  \bf{\frac{1}{9}}   =    \bf{\frac{8 \space - \space 1}{9}}   =    \bf{\frac{7}{9}}

## Adding and Subtracting Unlike Fractions

The fractions examples above were fairly simple as we were subtracting and adding like fractions which had the same denominator.
But sums can involve unlike fractions where the denominators below are not the same value.

We could have for example:

\bf{\frac{1}{3}}   +   \bf{\frac{5}{6}}   =   ?

We would ideally like to have both of the denominators be the same number.
An effective method to do this, is by multiplying the numerator and denominator of each fraction, by the denominator number of the other fraction.

With

\bf{\frac{1}{3}}  +  \bf{\frac{5}{6}}       This would be:     \boldsymbol{\frac{1 \space \times \space 6}{3 \space \times \space 6}}   +   \boldsymbol{\frac{5 \space \times \space 3}{6 \space \times \space 3}}    =    \bf{\frac{6}{18}}  +  \bf{\frac{15}{18}}    =    \bf{\frac{21}{18}}    =    \bf{\frac{7}{6}}

Examples

(2.1)

\bf{\frac{7}{9}}  +  \bf{\frac{5}{8}}

Solution

\boldsymbol{\frac{7 \space \times \space 8}{9 \space \times \space 8}}   +   \boldsymbol{\frac{5 \space \times \space 9}{8 \space \times \space 9}}    =    \bf{\frac{56}{72}}  +  \bf{\frac{40}{72}}    =    \bf{\frac{106}{72}}           This can be further simplified to   \bf{\frac{53}{36}}.

(2.2)

\bf{\frac{7}{10}}  −  \bf{\frac{4}{9}}

Solution

\boldsymbol{\frac{7 \space \times \space 9}{10 \space \times \space 9}}    \boldsymbol{\frac{4 \space \times \space 10}{9 \space \times \space 10}}    =    \bf{\frac{63}{90}}    \bf{\frac{40}{90}}    =    \bf{\frac{23}{90}}

(2.3)

\bf{\frac{2}{5}}  +  \bf{\frac{6}{7}}

Solution

( \boldsymbol{\frac{2 \space \times \space 7}{5 \space \times \space 7}} )  +  \boldsymbol{\frac{6 \space \times \space 5}{7 \space \times \space 5}}    =    – ( \bf{\frac{14}{35}} )  +  \bf{\frac{30}{35}}    =    \bf{\frac{16}{35}}

(2.4)

\bf{\frac{1}{4}}  +  \bf{\frac{5}{2}}  +  \bf{\frac{4}{7}}

Solution

When we are adding and subtracting unlike fractions and there are  3  of them in a sum.
We can do the operations in stages. Add or subtract he first  2  fractions together, then proceed with the addition or subtraction with that result and the  3rd  fraction.

1)   \bf{\frac{1}{4}}  +  \bf{\frac{5}{2}}    =    \boldsymbol{\frac{1 \space \times \space 2}{4 \space \times \space 2}}  +  \boldsymbol{\frac{5 \space \times \space 4}{2 \space \times \space 4}}    =    \bf{\frac{2}{8}}  +  \bf{\frac{20}{8}}    =    \bf{\frac{22}{8}}    =    \bf{\frac{11}{4}}

2)   \bf{\frac{11}{4}}  +  \bf{\frac{4}{7}}    =    \boldsymbol{\frac{11 \space \times \space 7}{4 \space \times \space 7}}  +  \boldsymbol{\frac{4 \space \times \space 4}{7 \space \times \space 4}}    =    \bf{\frac{77}{28}}  +  \bf{\frac{16}{28}}    =    \bf{\frac{83}{28}}

\bf{\frac{1}{4}}  +  \bf{\frac{5}{2}}  +  \bf{\frac{4}{7}}    =    \bf{\frac{83}{28}}

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