Simplifying Fractions

Numbers: Fractions

Simplifying Fractions

Simplifying fractions

We already saw earlier that #\tfrac{2}{6}# can be written as #\tfrac{1}{3}#. The process of turning the numerator and denominator into smaller numbers without altering the value of the fraction, is called simplification.

The fractions on the right cannot be simplified further. When this is the case, and the fraction contains no minus signs, we call this fraction simplified.

When simplifying, we have to get rid of minus signs in the fraction. When the fraction contains an even number of minus signs, they cancel out. When the fraction contains an odd number of minus signs, there will be one left. We place this minus sign in front of the fraction.

Examples

\[\begin{array}{rclll} \require{color} \definecolor{blue}{RGB}{45, 112, 179}
\dfrac{15}{35} &=& \dfrac{3}{7} &\;\color{blue}{\small\text{numerator and denominator divided by \(5\)}} \\ \\
\dfrac{48}{60} &=& \dfrac{24}{30} &\; \color{blue}{\small{\text{numerator and denominator divided by \(2\)}}} \\
&=& \dfrac{12}{15} &\; \color{blue}{\small\text{numerator and denominator divided by \(2\)}} \\
&=& \dfrac{4}{5} &\; \color{blue}{\small\text{numerator and denominator divided by \(3\)}} \\
\end{array}\]

Minus signs in the fractionWhen simplifying, we have to get rid of minus signs in the fractions.

\[\begin{array}{rcrll}
\dfrac{4}{-6} &=& \dfrac{2}{-3} &\qqqquad\color{blue}{\text{numerator and denominator divided by \(2\) }} \\
&=& -\dfrac{2}{3} &\qqqquad\color{blue}{\text{moved the minus sign by dividing the numerator and the denominator by \(-1\)}}\\\\\hline\\
\dfrac{-3}{-7} &=& \dfrac{3}{7}&\qqqquad\color{blue}{\text{numerator and denominator divided by \(-1\)}}
\end{array}\]

Simplify the following fraction as much as possible.
\[ \dfrac{48}{60} \]

#\dfrac{48}{60}=# #{{4}\over{5}}#

We can simplify the fraction by dividing both the numerator and the denominator by the same number. In this case, the numerator and the denominator can both be divided by #12#. This gives:
\[\begin{array}{rcl}\dfrac{48}{60}=\displaystyle {{4}\over{5}} &\phantom{xxx}\blue{\text{numerator and denominator divided by }12}\end{array}\]

We can also do this in steps:
\[\begin{array}{rcl}\dfrac{48}{60}&=&\dfrac{24}{30}\\ &&\phantom{xxx}\blue{\text{numerator and denominator divided by }2} \\ &=& \dfrac{12}{15} \\ &&\phantom{xxx}\blue{\text{numerator and denominator divided by }2} \\ &=&\dfrac{4}{5} \\ &&\phantom{xxx}\blue{\text{numerator and denominator divided by }3}\\ \end{array}\]

The number #12# also happens to be the greatest common divisor of #48# and #60#.

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