Question:
The decimal expansion of the rational number $\frac{14587}{1250}$ will terminate after
(a) one decimal place
(b) two decimal place
(c) three decimal place
(d) four decimal place
Solution:
We have,
$\frac{14587}{1250}=\frac{14587}{2^{1} \times 5^{4}}$
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 4 places of decimal.
Hence, the correct choice is (d).