You know that $\dfrac{1}{7}$ = $0.\overline{142857}$. Can you predict what the decimal expansions of $\dfrac{2}{7}$, $\dfrac{3}{7}$, $\dfrac{4}{7}$, $\dfrac{5}{7}$, $\dfrac{6}{7}$ are, without actually doing the long division? If so, how?

[Hint : Study the remainders while finding the value of $\dfrac{1}{7}$ carefully.]

Given,

$\dfrac{1}{7}$ = $0.\overline{142857}$

(i) $\dfrac{2}{7}$ = $2 \times \dfrac{1}{7}$

= $2 \times 0.\overline{142857}$

= $0.\overline{285714}$

(ii) $\dfrac{3}{7}$ = $3 \times \dfrac{1}{7}$

= $3 \times 0.\overline{142857}$

= $0.\overline{428571}$

(iii) $\dfrac{4}{7}$ = $4 \times \dfrac{1}{7}$

= $4 \times 0.\overline{142857}$

= $0.\overline{571428}$

(iv) $\dfrac{5}{7}$ = $5 \times \dfrac{1}{7}$

= $5 \times 0.\overline{142857}$

= $0.\overline{714285}$

(v)$\dfrac{6}{7}$ = $6 \times \dfrac{1}{7}$

= $6 \times 0.\overline{142857}$

= $0.\overline{857142}$