Chapter A.1, Problem 37E

If $c=\mathrm{0.151515...},$ then $100c=\mathrm{15.1515...}$ and

$100c-c=\mathrm{15.1515...}-\mathrm{0.151515...}$

$99c=15$

$c=\frac{15}{99}=\frac{5}{35}$

Proceeding similarly, convert the repeating decimal $\mathrm{0.090909...}$ into a fraction. (All repeating decimals are rational numbers, and all rational numbers have repeating decimal representations.)