Question

How do you convert -5.7 (7 repeating) as a fraction?

Answer 1

Hint: To convert a repeating decimal to a fraction we need to write it as an equation in terms of x, then multiply it with ${{10}^{y}}$ ,in which y depends in the number of digits repeating or recurring, then we subtract to remove the repeating decimal, and solve the equation to get the decimal as a fraction.

Complete step by step answer:
First, let us locate the position of the repeating decimal in the equation,
$\Rightarrow x=-5.\overline{7}.....(i)$
Here, $\overline{7}$ means 7 is repeating or recurring. In this case, the recurring decimal is just behind the decimal point.
So, we will multiply 10 on both the sides of the equation (i) to get,
 $\begin{align}
  & \Rightarrow x\times 10=-5.\overline{7}\times 10 \\
 & \Rightarrow 10x=-57.\overline{7}......(ii) \\
\end{align}$
Here, y will be equal to 1 in ${{10}^{y}}$ , as there is only one digit 7 that is repeating itself after the decimal. The value of y can differ depending on the number of terms after the decimal that are repeating themselves. Like in decimal 6.232323…. , we would multiply 100, i.e. we would take y=2 to solve further.
Now, we will subtract equation (i) from equation (ii) so that the recurring digits would be removed, and we will be left with simple integers to solve the equation, It will be as follows-
$\begin{align}
  & \Rightarrow 10x=-57.\overline{7} \\
 & \text{ }\underline{-x=-\left( -5.\overline{7} \right)} \\
 & \text{ }9x=-52 \\
\end{align}$
$\therefore x=\dfrac{-52}{9}$
Hence, the required fraction of the given decimal is $\dfrac{-52}{9}$ .

Note:
Remember to check the position of the recurring digits. Here, it was just after the decimal so we multiplied it with 10, in other cases these recurring digits may come after one or more decimal places, in those cases be careful to choose the value of y in ${{10}^{y}}$ .
For example, in 2.4353535… . we will multiply ${{10}^{3}}$ to get 2435.3535…. to solve the equation further.