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

##### 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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