Find the rational number whose decimal expansion is given below : (i) 0.bar3 (ii) 0.bar(231) (iii) 3.bar(52)

Question

Find the rational number whose decimal expansion is given below :

(i) 0.\(\bar3\)

(ii) 0. \(\overline {231}\)

(iii) 3. \(\overline{52}\)

Answer 1

(i) Let, x=0.3333… 

⇒ x = 0.3+0.03+0.003+… 

⇒ x = 3(0.1+0.01+0.001+0.0001+…∞)

⇒ x = 3 (\(\frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \frac{1}{10000} + .... \infty\))

This is an infinite geometric series. 

Here, a = 1/10 and r = 1/10

\(\Rightarrow\) 0.\(\bar 3\) = \(\frac{1}{3}\)

(ii) Let, x=0.231231231…. 

⇒ x=0.231+0.000231+0.000000231+…∞ 

⇒ x=231(0.001+0.000001+0.000000001+…∞)

⇒ x = 231 (\(\frac{1}{10^3} + \frac{1}{10^6} + \frac{1}{10^9} + \frac{1}{10^{12}}\)+ ....... ∞)

This is an infinite geometric series.

Here, a = \(\frac{1}{10^3}\) and r = \(\frac{1}{10^3}\)

\(\Rightarrow\) 0.\(\overline {231}\) = \(\frac{231}{999}\)

(iii) Let, x=3.525252552… 

⇒ x=3+0.52+0.0052+0.000052+…∞ 

⇒ x=3+52(0.01+0.0001+…∞)

⇒ x=3+52 (\(\frac{1}{10^2} + \frac{1}{10^4} + \frac{1}{10^6} + \frac{1}{10^{8}}\)+....... ∞)

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

\(\Rightarrow\) 3.\(\overline{52}\) = \(\cfrac{349}{999}\)