Write the number $2\mathrm{.}713=2\mathrm{.}7131313$dots as a ratio of integers.

Solution

$2\mathrm{.}7131\frac{13}{{10}^3}\frac{13}{{\mathrm{.}10}^5}\frac{13}{{10}^7}$ cdots

After the first term we have a geometric series with $a=\frac{13}{{10}^3}$ and $r=.$

Therefore we have the following.

$\frac{2\mathrm{.}7}{b}ar(13)=2\mathrm{.}7\frac{\frac{13}{{10}^3}}{1-}$

$=2\mathrm{.}7\frac{\frac{13}{1000}}{\frac{99}{100}}$

$=27\frac{27}{990}$

$=$