### Numbers: Fractions

### Writing fractions with like denominators

Writing fractions with like denominators

We have seen that a fraction does not change when we multiply both the numerator and the denominator by the same number. This can be used when we want to create fractions with equal denominators. This is very useful, since we can then add and subtract these fractions.

A new denominator for two fractions that do not have the same denominator can be found by taking a common multiple of the two denominators. The easiest way to find a common multiple is to multiply the two denominators.

To find the smallest common denominator we can use the least common multiple of the two denominators.

**Example**

\[\begin{array}{rclccl}

\dfrac{1}{6} &=& \dfrac{4\cdot1}{4\cdot6}&=&\dfrac{4}{24} \\[5pt] \dfrac{1}{4} &=& \dfrac{6\cdot1}{6\cdot4}&=&\dfrac{6}{24} \\ \\ \\ \dfrac{1}{6} &=& \dfrac{2\cdot 1}{2\cdot6}&=&\dfrac{2}{12} \\[5pt] \dfrac{1}{4} &=& \dfrac{3\cdot1}{3\cdot4}&=&\dfrac{3}{12}\end{array}\]

We take #8 \times 9=72# as our new denominator. In the fraction #{{3}\over{8}}# we now multiply both the numerator and the denominator by #9#. In the fraction #{{5}\over{9}}# we multiply both the numerator and the denominator by #8#. In doing so, both fractions have a denominator that is equal to #72#.

\[\begin{array}{rcl}\displaystyle {{3}\over{8}}&=&\dfrac{27}{72} \\ &&\phantom{xxx}\blue{\text{numerator and denominator multiplied by }9} \\

\displaystyle {{5}\over{9}}&=&\dfrac{40}{72} \\ &&\phantom{xxx}\blue{\text{numerator and denominator multiplied by }8} \end{array}\]

There are other answers possible, since each common multiple of #8# and #9# can be used as our new denominator.

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