Question

How do you convert $0.\overline{45}$ (meaning the $45$ is being repeated) to a fraction?

Answer 1

Hint: We first need to equate the given decimal number to a variable, say x, so that we can write the equation $x=0.\overline{45}$. Then, we need to multiply the equation by ten two times in order to obtain the equation $100x=45.\overline{45}$. On subtracting the second equation from the first equation, we will obtain the given decimal number in the fractional form.

Complete step by step solution:
Let us write the decimal number given in the above question as
$\Rightarrow x=0.\overline{45}........\left( i \right)$
We know that the bar over a digit, or over a group of digits, simply means that it is being repeated infinitely many times. Therefore, we can also write the above number as
$\Rightarrow x=0.45\overline{45}$
Now, in order to convert the given decimal number to a fraction, we multiply both the sides of the above equation by $10$ to get
$\begin{align}
  & \Rightarrow 10x=0.45\overline{45}\times 10 \\
 & \Rightarrow 10x=4.5\overline{45} \\
\end{align}$
Now, let us again multiply the above equation by $10$ to get
$\begin{align}
  & \Rightarrow 100x=4.5\overline{45}\times 10 \\
 & \Rightarrow 100x=45.\overline{45}........\left( ii \right) \\
\end{align}$
Now, let us subtract the equation (ii) from the equation (i) to get
$\begin{align}
  & \Rightarrow 100x-x=45.\overline{45}-0.\overline{45} \\
 & \Rightarrow 99x=45 \\
\end{align}$
On dividing the above number by $99$, we get
$\begin{align}
  & \Rightarrow \dfrac{99x}{99}=\dfrac{45}{99} \\
 & \Rightarrow x=\dfrac{5}{11}.......\left( iii \right) \\
\end{align}$
From the equations (i) and (iii) we can write
$\Rightarrow 0.\overline{45}=\dfrac{5}{11}$

Hence, the fractional form of the given decimal number is $\dfrac{5}{11}$.

Note: After obtaining the fractional form of the given decimal number, do not forget to check whether the obtained fraction is giving back the original decimal number on division or not. This can be done by dividing the numerator by the denominator. Also, we must remember the steps followed in the above solution for converting the decimal number into a fraction, in order to solve these types of questions.