# 2. You know that $\dpi{100} \frac{1}{7}=0.\overline{142857}$Can you predict what the decimal expansions of $\dpi{100} \frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ are, without actually doing the long division? If so, how?

G Gautam harsolia

It is given  that  $\frac{1}{7}=0.\overline{142857}$

Therefore,

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

Similarly,

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

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

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

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

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