Question

Find the value of $p+q$ by expressing $0.\overline{21}$ in the form $\frac{p}{q}$.

• A. 7
• B. 20
• C. 40
• D. 140

Answer 1

Answer: (C)

Let$x=0.\overline{21}$

$\Rightarrow x=\mathrm{0.2121....}$ -----------(i)

Here, the two numbers in decimal part continuously recurring. so its periodicity is $2$.

i.e., need to multiply by $100$.
$\Rightarrow 100x=\mathrm{21.2121...}$ -----------(ii)
Subtracting (i) from (ii), we get,
$100x-x=\mathrm{21.2121...}-\mathrm{0.2121...}=21$
$\Rightarrow 99x=21$
$\Rightarrow x=\frac{21}{99}$

$\Rightarrow x=\frac{7}{33}$ ----------(iii)
Now, comparing the value of $x$ in eq. (iii) with $\frac{p}{q}$ from we get.
$p=7andq=33$
So, $p+q=7+33=40$