Question

When you express $0$.$\overline{21}$ in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$, which of the following are the possible values of $(p+q)$?

• A. 66
• B. 72
• C. 80
• D. 96

Answer 1

The correct option is C 80

Let $x=\mathrm{0.2121....}$ ------------(i)
$100x=\mathrm{21.2121....}$ ------------(ii) [Multiply equation (i) by 100]
Subtracting (i) from (ii), we get,
$100x-x=\mathrm{21.2121....}-\mathrm{0.2121....}$
$99x=21$
$x=\frac{21}{99}=\frac{7}{33}$ ------------(iii)
Now, comparing the value of $x$ in equation (iii) with $\frac{p}{q}$ form we get,
$p=7$ and $q=33$
So, $p+q=7+33=40$
Since, $40$ is not given in the options, so we look for multiples of $40$.
Hence, the possible value of $p+q$ is $80$.