Question

You know that $\frac{1}{7}=0.\overline{142587}$ .Can you predict the decimal expansion of $\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ without actually doing the long division? If so, how?

Answer 1

Yes, we can know the decimal expansions of the given numbers by doing the following :

$\frac{2}{7}=2\times \frac{1}{7}=2\times 0.\overline{142587}=0.\overline{285714}\frac{3}{7}=3\times \frac{1}{7}=3\times 0.\overline{142587}=0.\overline{428571}\frac{4}{7}=4\times \frac{1}{7}=4\times 0.\overline{142587}=0.\overline{571428}\frac{5}{7}=5\times \frac{1}{7}=5\times 0.\overline{142587}=0.\overline{714285}\frac{6}{7}=6\times \frac{1}{7}=6\times 0.\overline{142587}=0.\overline{857142}$