Question

If $0.12\overline{3}$ can be written as $\frac{a}{b}$ in simplified fraction, then

• A. $a=101$
• B. $a=111$
• C. $b=900$
• D. $b=990$

Answer 1

Answer: (B), (C)

$b=900$

Let, $x=0.12\overline{3}$
$\Rightarrow x=\mathrm{0.123333...}\cdots \left(i\right)$

Here, the repeating digit is 3.

Step 1: Place 123 to the left of the decimal point by multiplying it by 1000 because the unit digit 3 (in 0.123) in the thousandths place
$\Rightarrow 1000\times x=1000\times \mathrm{0.123333...}$
$\Rightarrow 1000x=\mathrm{123.3333...}\cdots \left(ii\right)$

Step 2: Place the repeating digit to the right of the decimal point by multiplying it by 100.
$\Rightarrow 100\times x=100\times \mathrm{0.123333...}$
$\Rightarrow 100x=\mathrm{12.3333...}\cdots \left(iii\right)$

Now, from $\left(ii\right)-\left(iii\right),$ we get
$1000x-100x=\mathrm{123.3333...}-\mathrm{12.3333...}$
$\Rightarrow 900x=111$
Dividing both the sides by 900, we get
$\Rightarrow \frac{900x}{900}=\frac{111}{900}\Rightarrow x=\frac{111}{900}$