​ Express 0.bar68 as a rational number. ​

Question

Express 0.\(\overline {68}\)as a rational number.

Answer 1

Let, x = 0.68686868… 

⇒ x = 0.68 + 0.0068 + 0.000068 + …∞ 

⇒ x = 68(0.01 + 0.0001 + …∞)

⇒ x = 68 \(\left(\frac{1}{10^2} + \frac{1}{10^4} + \frac{1}{10^6} + \frac{1}{10^9} + .... \infty\right)\)

Here, a = \(\frac{1}{10^2}\) and r = \(\frac{1}{10^2}\)

\(\Rightarrow\) x = \(\left(68\times \frac{1}{99}\right)\) = \(\frac{68}{999}\) 

\(0.\overline{68}\) = \(\frac{68}{999}\)