Question

You know that \(\frac{1}{7}\) = 0142857_ . . 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 1/7 carefully.]

Answer 1

It can be done follows:

\(\frac{2}{7}\) = 2 × \(\frac{1}{7}\) = 2 × 0.142857 = 0.285714 , where p and q are integers and q ≠ 0.

\(\frac{3}{7}\) = 3 × \(\frac{1}{7}\) = 3 × 0.142857 = 0.428571 = 10 x 6 + x

\(\frac{4}{7}\) = 4 × \(\frac{1}{7}\) = 4 × 0.142857 = 0.571428_= 9x = 6 = x = \(\frac{2}{3}\)

\(\frac{5}{7}\) = 5 × \(\frac{1}{7}\) = 5 × 0.142857 = 0.714285 

\(\frac{6}{7}\) = 6 × \(\frac{1}{7}\) = 4 × 0.142857 = 0.857142