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# Without actually performing the long division, find if $\frac{987}{10500}$ will have terminating or non-terminating(repeating) decimal expansion. Give reasons for your answer.

Given:

Given rational number is $\frac{987}{10500}$.

To do:

Here, we have to check without actually performing the long division, whether the given rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

Solution:

If we have a rational number $\frac{p}{q}$, where $p$ and $q$ are co-primes and the prime factorization of $q$ is of the form $2^n.5^m$, where $n$ and $m$ are non-negative integers, then $\frac{p}{q}$ has a terminating expansion.

Now,

$\frac{987}{10500}=\frac{21\times47}{21\times500}=\frac{47}{500}$

In $\frac{47}{500}$:

- $47$ and $500$ are co-primes.
- $500=2^2 \times 5^3$, which is in the form $2^n\ \times\ 5^m$.

So, $\frac{987}{10500}$ has a terminating decimal expansion.

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