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).