Question

Express the number as a ratio of integers.

\(2.\overline {516} = 2.516516516...\)

Short answer

The ratio of integer \(2.\overline {516} = 2.516516516...\) is \(\frac{{838}}{{333}}\) .

Step-by-step solution

Definition of repeating series and geometric series

A recurring decimal or repeating decimal having digits periodic and the infinite repeated portion is not zero.

The series \(a + ar + a{r^2} + ...\) is infinite geometric series.

The formula of sum of Geometric Series of infinite terms is \({S_\infty } = \frac{a}{{1 - r}}\)

Where\(|r| < 1\)

\(a\)= first term

\(r\) =common ratio

Step 2

\(\begin{aligned}2.\overline {516} &= 2.516516516...\\ &= 2 + \frac{{516}}{{{{10}^3}}} + \frac{{516}}{{{{10}^6}}} + \frac{{516}}{{{{10}^9}}} + ...\\ &= 2 + \frac{{\frac{{516}}{{{{10}^3}}}}}{{1 - {{10}^3}}}\\ &= 2 + \frac{{516}}{{999}}\\ &= 2 + \frac{{172}}{{333}}\\ &= \frac{{2(333) + 172}}{{333}}\\ = \frac{{838}}{{333}}\end{aligned}\)

Hence the ratio of integer \(2.\overline {516} = 2.516516516...\) is \(\frac{{838}}{{333}}\)