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# The decimal expansion of the rational number

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).