Question

How do you convert $0.41\overline{6}$ to a fraction?

Answer 1

Convert $0.41\overline{6}$ to fraction:

Let, $y=0.41+0.00\overline{6}$

Assume $x=0.00\overline{6}$. Find the value of $1000x$ and $100x$,

$\begin{array}{rcl}1000x& =& 1000\times \left(0.00\overline{6}\right)\\ & \Rightarrow & 1000x=6.\overline{6}\\ \Rightarrow 100x& =& 100\left(0.00\overline{6}\right)\\ \Rightarrow 100x& =& 0.\overline{6}\end{array}$

From the obtained values it is observed that, $1000x=6+100x$.

Simplify the obtained equation:

$\begin{array}{rcl}1000x& =& 6+100x\\ & \Rightarrow & 900x=6\\ & \Rightarrow & x=\frac{6}{900}\\ & \Rightarrow & x=\frac{1}{150}\end{array}$

Now,

$$\begin{gathered} y=0.41+0.00\overline{6} \\ =\frac{41}{100}+\frac{6}{900}\left[\because 0.00\overline{6}=\frac{6}{900}\right] \\ =\frac{375}{900} \\ =\frac{5}{12} \end{gathered}$$

Thus, fraction form of $0.41\overline{6}$ is $\frac{5}{12}$.