Question

Without actually performing the long division, find if $\dfrac{{987}}{{10500}}$ will have the terminating or the non-terminating (repeating) decimal expansion. Give reason.

Answer 1

Hint: Without division, we can say that the decimal is terminating or non-terminating by checking its denominator. If the denominator is divisible by $2$ only,$5$ only or by both of them, then it will be terminating otherwise non-terminating.

Complete step by step solution:
Now as we know that any rational number can be expressed in the decimal form and now it may be terminating or non-terminating decimal.
Now before solving this problem, you should be familiar with the two terms which are terminating and non-terminating decimals. Basically, terminating decimals are those decimals which terminate or come to an end.
For example$ \Rightarrow \dfrac{1}{5} = 0.2,\dfrac{{10}}{{20}} = 0.5$
So $0.2,0.5$ are the terminating decimals.
The decimals which do not terminate or keep on repeating after the decimal point are known as non-terminating decimals. This means that they have no specific ending.
For example$:\dfrac{1}{3} = 0.33333........{\text{and so on}}$
$\pi = 3.141592653..........{\text{so on}}$
These are the examples of the non-terminating decimals.
Now we can find without dividing whether the given decimal is terminating or non-terminating.
Any decimal in the form of $\dfrac{p}{{{2^n} \times {5^m}}}$where $p$ is any integer and $m,n$ are whole numbers, then the rational will be terminating.
For example: $\dfrac{5}{{100}}$, here $100$ can be written as $100 = {5^2} \times {2^2}$
So it is of the form $\dfrac{5}{{{5^2} \times {2^2}}}$
So it is terminating decimal. Now we are given to find whether $\dfrac{{987}}{{10500}}$ terminates or not.
First of all, cancel it with the common multiple
$\dfrac{{987}}{{10500}}$$ = \dfrac{{47}}{{500}}$
Now we are supposed to find the multiple of $500$.
$500 = 2 \times 2 \times 5 \times 5 \times 5$
So it can be written as ${5^3} \times {2^2}$.

Hence it is a terminating decimal as it is divisible by $2.5$ only.

Note: For applying $\dfrac{p}{{{2^m} \times {5^n}}}$formula to determine whether it is terminating or non terminating, first of all divide numerator and denominator with common multiples and write in simplest form for example $\dfrac{{500}}{{1550}}$is given then its simplest form would be $\dfrac{{10}}{{31}}$.