Question

You know that $\frac{1}{7}=0.\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ are, without actually doing the long division? If so, how? (Hint: Study the remainders while finding the value of $\frac{1}{7}$ carefully.).

Answer 1

Given, $\frac{1}{7}=0.\overline{142857}$.

Then,

$\frac{2}{7}=2\times \frac{1}{7}=2\times 0.\overline{142857}=0.\overline{285714}$

$\frac{3}{7}=3\times \frac{1}{7}=3\times 0.\overline{142857}=0.\overline{428571}$

$\frac{4}{7}=4\times \frac{1}{7}=4\times 0.\overline{142857}=0.\overline{571428}$

$\frac{5}{7}=5\times \frac{1}{7}=5\times 0.\overline{142857}=0.\overline{714285}$

$\frac{6}{7}=6\times \frac{1}{7}=6\times 0.\overline{142857}=0.\overline{857142}$

Here, the first digit in the decimal expansion continued in the repeated numbers afterwards.