Question

Express $15.7\overline{12}$ as a fraction in the simplest form.

Answer 1

Let $x=15.7\overline{12}$
i.e. $x=15.7121212...$(1)
Multiplying (1) by $10$ and $100$ successively, we get
$10x=\mathrm{157.121212...}$(2)
and $1000x=15712.1212...$(3)
Subtracting (2) from (3), we get
$1000x-10x=\left(\mathrm{15712.1212...}\right)–\left(\mathrm{157.1212...}\right)$
$\Rightarrow 990x=15555$
$\Rightarrow x=\frac{15555}{990}$
$\Rightarrow x=\frac{1037}{66}$
Hence, $15.7\overline{12}=\frac{1037}{66}$