Question

How do you convert $7.12$ ($12$ being repeated) to a fraction?

Answer 1

Convert the given decimal number into fraction.

Let us suppose $x$ be $7.12121212....$.

i.e.

$x=7.12121212..........\left(1\right)$

Multiply $x$ by 100:

$100x=712.121212.......\left(2\right)$

Subtract the equation $\left(1\right)$ from the equation $\left(2\right)$ :

$\begin{array}{rcl}100x-x& =& 712.121212....-7.12121212....\\ & \Rightarrow & 99x=705\left[\text{Simplifying}\right]\\ & \Rightarrow & x=\frac{705}{99}\\ \therefore x& =& \frac{235}{33}\end{array}$

Hence, the fraction of $7.12$ ($12$ being repeated) is $\frac{\mathbf{235}}{\mathbf{33}}$.