Write the number $\frac{4\mathrm{.}1}{b}ar(13)=4\mathrm{.}1131313$dots as a ratio of integers.

Solution

$4\mathrm{.}1131313$dots$=4\mathrm{.}1\frac{13}{{10}^3}\frac{13}{{10}^5}\frac{13}{{10}^7}$ cdots

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

Therefore we have the following.

$\frac{4\mathrm{.}1}{b}ar(13)=4\mathrm{.}1\frac{\frac{13}{{10}^3}}{1-}$

$=4\mathrm{.}1\frac{\frac{13}{1000}}{\frac{99}{100}}$

$=41$

$=$