 ### Simplest form

A fraction is said to be in simplest form or lowest terms if the only common factor of the numerator and the denominator is one.

For example, $$\dfrac{2}{15}$$ is in simplest form since the highest common factor of two and 15 is one.

The number $$\dfrac{6}{15}$$ is not in simplest form as the highest common factor of six and 15 is three.

Our knowledge of factors and multiples will assist us to reduce fractions to their simplest form.

#### Cancelling

When reducing $$\dfrac{6}{15}$$ to its simplest form, we consider all the factors of six and 15. The highest common factor of six and 15 is three.

We divide the numerator and the denominator by three.

$$\dfrac{6}{15}=\dfrac{6÷3}{15÷3}=\dfrac{2}{5}$$

We can use cancelling notation to increase efficiency.

$$\dfrac{6}{15}=\dfrac{\color{red}{\setminus}\hspace{-3mm}{6^2}}{\color{red}{\setminus}\hspace{-4mm}{15^5}}=\dfrac{2}{5}$$

Sometimes it is more convenient to cancel in steps, rather than finding the highest common factor. When using steps, keep going until the highest common factor of the numerator and the denominator is one.

$$\dfrac{\color{red}{\setminus}\hspace{-4mm}{30^{15}}}{\color{red}{\setminus}\hspace{-4mm}{84^{42}}}=\dfrac{\color{red}{\setminus}\hspace{-4mm}{15^{5}}}{\color{red}{\setminus}\hspace{-4mm}{42^{14}}}=\dfrac{5}{14}$$   