You know that \( \frac{1}{7}\) = 0.142857....... 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 of \( \frac{1}{7}\) carefully.]

Asked by Vishal kumar | 1 year ago |  198

1 Answer

Solution :-

\( \frac{1}{7}=0.\overline{142857}\) or \( \frac{1}{7}\)= 0.142857.....

find the values of \( \frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7}\) and \( \frac{6}{7}\), without performing long divison.

\( \frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7}\) and \( \frac{6}{7}\) can be rewritten as \( 2\times \frac{1}{7},3\times \frac{1}{7},4\times \frac{1}{7}\)\( 5\times\frac{1}{7}\), and \( 6\times\frac{1}{7}\).

On substituting value of \( \frac{1}{7}\) = 0.142857..... , we get

  • \( 2\times\frac{1}{7}\) = 2 \( \times\) 0.142857.... 0.285714....
  • \(3\times\frac{1}{7}\) = 3 \( \times\) 0.142857.... 0.428571....
  • \(4\times\frac{1}{7}\) = 4 \( \times\) 0.142857.... 0.571428....
  • \(5\times\frac{1}{7}\) = 5 \( \times\) 0.142857.... 0.714285....
  • \(6\times\frac{1}{7}\) = 6 \( \times\) 0.142857.... 0.857142....

Therefore, we conclude that, the values of \( \frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7}\) and \( \frac{6}{7}\), without performing long division. we get

  • \( \frac{2}{7}=0.\overline{285714}\)
  • \( \frac{3}{7}=0.\overline{428571}\)
  • \( \frac{4}{7}=0.\overline{571428}\)
  • \( \frac{5}{7}=0.\overline{714285}\)
  • \( \frac{6}{7}=0.\overline{857142}\)

1 year ago

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