2. 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?

Answers (1)

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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