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 \(\frac{1}{7} \) of carefully.]

We have,
\( \frac{1}{7} = 0\overline{.142857}\)


So we can say that,
\(\because \frac{2}{7} = 2 × \frac{1}{7} \)
\( \Rightarrow \frac{2}{7} = 2 × 0\overline{.142857}\)
\(\Rightarrow \frac{2}{7} = 0\overline{.285714}\)

\(\because \frac{3}{7} = 3 × \frac{1}{7} \)
\(\Rightarrow \frac{3}{7} = 3 × 0\overline{.142857}\)
\( \Rightarrow \frac{3}{7} = 0\overline{.428571}\)

\(\because \frac{4}{7} = 4 × \frac{1}{7} \)
\(\Rightarrow \frac{4}{7} = 4 × 0\overline{.142857}\)
\( \Rightarrow \frac{4}{7} = 0\overline{.571428}\)

\(\because \frac{5}{7} = 5 × \frac{1}{7} \)
\(\Rightarrow \frac{5}{7} = 5 × 0\overline{.142857}\)
\( \Rightarrow \frac{5}{7} = 0\overline{.714285}\)

\(\because \frac{6}{7} = 6 × \frac{1}{7} \)
\(\Rightarrow \frac{6}{7} = 6 ×0 \overline{.142857}\)
\(\Rightarrow \frac{6}{7} = 0\overline{.857142}\)