Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.$\frac{213}{3125}$

Given: 

Given rational number is $\frac{213}{3125}$.

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,

$\frac{213}{3125}=\frac{213}{5\times5\times5\times5\times5}=\frac{213}{5^5}$

In $\frac{213}{3125}$:

• $213$ and $3125$ are co-primes.
• $3125=2^0\times5^5$, which is in the form $2^n\ \times\ 5^m$.

So, $\frac{213}{3125}$ has a terminating decimal expansion. 

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Updated on: 10-Oct-2022

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