Question

You know that $\frac{1}{11}=0.\overline{09}$ . Can you predict what the decimal expansions of $\frac{2}{11},\frac{3}{11},\frac{4}{11},\frac{5}{11},\frac{6}{11}$ are, without actually doing the long division? If so, how?

Answer 1

Step 1-determine the decimal expansions of $\frac{2}{11}$:

Given that $\frac{1}{11}=0.\overline{09}$

$\Rightarrow \frac{1}{11}=0.090909....$

Multiply the decimal expansion of $\frac{1}{11}$by $2$

$\Rightarrow \frac{2}{11}=2\times \frac{1}{11}$

$\Rightarrow \frac{2}{11}=2\times 0.\overline{09}$

$\Rightarrow \frac{2}{11}=0.\overline{18}$

Therefore,$\frac{2}{11}=0.\overline{18}$

Step2- determine the decimal expansions of $\frac{3}{11}$:

Multiply the decimal expansion of $\frac{1}{11}$by $3$

$\Rightarrow \frac{3}{11}=3\times \frac{1}{11}$

$\Rightarrow \frac{3}{11}=3\times 0.\overline{09}$

$\Rightarrow \frac{3}{11}=0.\overline{27}$

Therefore,$\frac{3}{7}=0.\overline{27}$

Step3- determine the decimal expansions of $\frac{4}{11}$:

Multiply the decimal expansion of $\frac{1}{11}$by $4$

$\Rightarrow \frac{4}{11}=4\times \frac{1}{11}$

$\Rightarrow \frac{4}{11}=4\times 0.\overline{09}$

$\Rightarrow \frac{4}{11}=0.\overline{36}$

Therefore,$\frac{4}{7}=0.\overline{36}$

Step4- determine the decimal expansions of $\frac{5}{11}$:

Multiply the decimal expansion of $\frac{1}{11}$by $5$

$\Rightarrow \frac{5}{11}=5\times \frac{1}{11}$

$\Rightarrow \frac{5}{11}=5\times 0.\overline{09}$

$\Rightarrow \frac{5}{11}=0.\overline{45}$

Therefore,$\frac{5}{7}=0.\overline{45}$

Step5- determine the decimal expansions of $\frac{6}{11}$:

Multiply the decimal expansion of $\frac{1}{11}$by $6$

$\Rightarrow \frac{6}{11}=6\times \frac{1}{11}$

$\Rightarrow \frac{6}{11}=6\times 0.\overline{09}$

$\Rightarrow \frac{6}{11}=0.\overline{54}$

Therefore,$\frac{6}{7}=0.\overline{54}$

Hence, decimal expansions of $\frac{2}{11},\frac{3}{11},\frac{4}{11},\frac{5}{11},\frac{6}{11}$ are $0.\overline{18}$, $0.\overline{27}$,$0.\overline{36}$,$0.\overline{45}$ and $0.\overline{54}$ respectively