Question

The number $0.121212...$ in the form $\dfrac{p}{q}$ is equal to
A. $\dfrac{4}{{11}}$
B. $\dfrac{2}{{11}}$
C. $\dfrac{4}{{33}}$
D. $\dfrac{2}{{33}}$

Answer 1

Hint:
Let the given decimal expansion be $x$. Then, multiply it by 100 on both sides. Subtract both the equations and solve the resultant equation to find the value of $x$. Also, reduce the fractional form in the simplest form.

Complete step by step solution:
Let the given decimal expansion be $x$
$x = 0.121212...$ eqn. (1)
Since, there are 2 digits that are repeating after decimal.
We will multiply both sides of equation (1) by 100
Then, we have
$100x = 12.121212...$ eqn. (2)
Subtract equation (1) from equation (1)
$
\Rightarrow 100x - x = 12.121212... - 0.121212... \\
\Rightarrow 99x = 12 \\
$
Divide both sides by 99
$x = \dfrac{{12}}{{99}}$
We want the answer in simplest form.
Divide both numerator and denominator by 3
$ \Rightarrow x = \dfrac{4}{{33}}$
Thus, the value of $0.121212...$ is $\dfrac{4}{{33}}$

Hence, option C is correct.

Note:
We can convert only non-terminating repeating decimal and terminating decimal in fractional form. We cannot convert non-terminating, non-repeating decimals into fractional form. The non-terminating, non-repeating decimal represents irrational numbers.