Question

Justify whether $\dfrac{{14587}}{{1250}}$ has a terminating decimal expansion or not. If it has found after how many places it would get terminated without performing actual division.

Answer 1

Hint: Here, to check whether the rational number $\dfrac{p}{q}$ will have a terminating decimal expansion or not we have to reduce the denominator, $q$ into prime factors and identify whether it has prime factors other than 2 and 5. If $q$ is written as $q = {2^m}{5^n}$ where $m$ and $n$ are integers, we can say that $\dfrac{p}{q}$ will have a terminating decimal expansion. If $q$ can be written as $q = {2^m}{5^n}k$ where $m$ and $n$ integers and $k$ is any prime number other than 2 and 5, we can say that $\dfrac{p}{q}$ will have a non-terminating repeating decimal expansion.

Complete step-by-step answer:
Here, without performing the long division, we have to check whether the rational numbers have a terminating decimal expansion or a non-terminating decimal expansion.
To check whether the rational number $\dfrac{p}{q}$ will have a terminating decimal expansion or not we have to reduce the denominator, $q$ into prime factors.
If $q$ can be written as $q = {2^m}{5^n}$ where $m$ and $n$ are integers, we can say that $\dfrac{p}{q}$ will have a terminating decimal expansion.
Else, if $q$ can be written as $q = {2^m}{5^n}k$ where $m$ and $n$ integers and $k$ is any prime number other than 2 and 5, we can say that $\dfrac{p}{q}$ will have a non-terminating repeating decimal expansion.
Now, we need to check the same for the rational number above.
Here $q = 1250$, now reduce $q$ into its prime factors,
$ \Rightarrow 1250 = 2 \times 5 \times 5 \times 5 \times 5$
Simplify,
$ \Rightarrow 1250 = 2 \times {5^4}$
Since the denominator can be written as a power of 2 and 5.
Thus, $\dfrac{{14587}}{{1250}}$ will have a terminating decimal expansion.
The terminating decimal place will be the max power of either 2 or 5.
$ \Rightarrow $ Terminating place $ = \max \left( {m,n} \right)$
Here, $m = 1$ and $n = 4$.
Substitute the values,
$ \Rightarrow $ Terminating place $ = \max \left( {1,4} \right) = 4$

Hence, $\dfrac{{14587}}{{1250}}$ will have a terminating decimal expansion and terminates after 4 decimal places.

Note: Here, this method is applicable only for non-terminating recurring decimal expansion not for non-terminating non-recurring decimal expansion. Also here, instead of doing separately you can also create a table and write the prime factors of the denominators. From the table, you can identify whether the rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.