Question

What is the value of $2.\overline{6}-1.\overline{9}$

• A. $0.\overline{6}$
• B. $0.\overline{9}$
• C. $0.\overline{7}$
• D. $0.7$

Answer 1

Answer: (A)

Lets evaluate $2.\overline{6}$

$x=2.\overline{6}$

$x=\mathrm{2.6666....}$--(1)

Since, $1$ decimal is recurring, multi[ply by $10$ on both sides

$10x=10\times \mathrm{2.666....}$

$\Rightarrow 10x=\mathrm{26.666....}$...(2)

(2) $-$ (1)

$10x-x=\left(\mathrm{26.666....}\right)-\left(\mathrm{2.666....}\right)$

$\Rightarrow 9x=24$

$\Rightarrow x=\frac{24}{9}$

$\Rightarrow 2.\overline{6}=\frac{24}{9}$----(i)

Lets evaluate $1.\overline{9}$

$y=1.\overline{9}$

$\Rightarrow y=\mathrm{1.999.....}$---(3)

Since, $1$ decimal is recurring, multi[ply by $10$ on both sides

$10y=10\times \mathrm{1.999....}$

$\Rightarrow 10y=\mathrm{19.999....}$...(4)

(4) $-$ (3)

$10y-y=\left(\mathrm{19.999....}\right)-\left(\mathrm{1.999....}\right)$

$\Rightarrow 9y=18$

$\Rightarrow x=\frac{18}{9}$

$\Rightarrow x=2$

$\Rightarrow 1.\overline{9}=2$ ---(ii)

Now,

$2.\overline{6}-1.\overline{9}$

$=\frac{24}{9}-2$

$=\frac{24-18}{9}$

$=\frac{6}{9}$

$=\frac{2}{3}$

$=\mathrm{0.666...}$

$=\overline{6}$

$\therefore 2.\overline{6}-1.\overline{9}=\overline{6}$

Hence, Option A is correct.

(OR)

The value of $2.\overline{6}-1.\overline{9}$ is
$=(2+\frac{6}{9})-(1+\frac{9}{9})$

$=2+\frac{6}{9}-(1+1)$
$=\frac{6}{9}=0.\overline{6}$