Equivalent fractions

Numbers: Fractions

Equivalent fractions

Fractions

When we cut a pizza into $6$ slices of equal size and take $2$ of those slices, we have $\tfrac{2}{6}$ of the pizza.
If we cut the pizza into $3$ slices of equal size and take $1$ slice, we will have the same amount of pizza.
We therefore see that $\tfrac{2}{6}= \tfrac{1}{3}$ .
In the fraction $\require{color} \definecolor{blue}{RGB}{45, 112, 179}\tfrac{\orange{2}}{\blue{6}}$ , we can divide both the and by $2$ , and then we find $\tfrac{\orange{1}}{\blue{3}}$ .

In general, we can say:

A fraction does not change if we divide or multiply both the numerator and denominator by the same number.

Examples

$\begin{array}{rcl} \dfrac{\orange{2}}{\blue{6}} = \dfrac{\orange{1}}{\blue{3}} \\ \dfrac{\orange{3}}{\blue{6}} = \dfrac{\orange{1}}{\blue{2}} \\ \dfrac{\orange{4}}{\blue{6}} = \dfrac{\orange{2}}{\blue{3}} \end{array}$

Fractions on the number line

Which number should be in the empty space?

${{5}\over{12}}=\dfrac{\Box}{72}$

${{5}\over{12}}=\dfrac{30}{72}$

The value of a fraction does not change when we multiply the numerator and the denominator by the same number.

$\begin{array}{rcl} \dfrac{5}{12}&=&\dfrac{\box}{72} \\ &&\phantom{xxx}\blue{\text{denominator multiplied by } \dfrac{72}{12}=6 } \\ \dfrac{5}{12}&=&\dfrac{30}{72} \\ &&\phantom{xxx}\blue{\text{numerator also has to be multiplied by } 6 } \\ \end{array}$

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