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