​ Express the recurring decimal 0.125125125 …. = 0. bar(125) as a rational number. ​

Question

Express the recurring decimal 0.125125125 …. = 0. \(\overline{125}\) as a rational number.

Answer 1

Let, x=0.125125125… …(i) 

Multiplying this equation by 1000 on both the sides so that repetitive terms cancel out and we get: 

1000x =125.125125125… …(ii) 

Equation (ii) - (i), 

⇒ 1000x-x =125.125125125-0.125125125 =125 

⇒ 999x =125

⇒ x = \(\frac{125}{999}\)

⇒ 0.\(\overline{125}\) = \(\frac{125}{999}\)