The number of decimal place after which the decimal expansion of the rational number
Question:

The number of decimal place after which the decimal expansion of the rational number $\frac{23}{2^{2} \times 5}$ will terminate, is

(a) 1

(b) 2

(c) 3

(d) 4

Solution:

We have,

$\frac{23}{2^{2} \times 5^{1}}$

Theorem states:

Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

This is given that the prime factorization of the denominator is of the form $2^{m} \times 5^{n}$.

Hence, it has terminating decimal expansion which terminates after 2 places of decimal.

Hence, the correct choice is (b).