Mathematics

Problem

Consider the recurring decimal expansion

x = 0.\overline{abc},

where

a,\,b,\,c

are digits and the bar indicates that the block

abc

repeats without end. Answer the following. a. (i) Prove that

x = \dfrac{abc}{999}

. b. (ii) Hence express

0.\overline{527}

as a fraction in its simplest form.