Question

The decimal expansion of the rational number $\dfrac{{23}}{{{2^3} \times {5^2}}}$, will terminate after how many places of decimals?

Answer 1

Hint: We will use the fact that every rational number with denominator of the form ${2^m} \times {5^n}$ will always be terminating. Then, we will try to make powers of 10 in the denominator by multiplying and dividing with some numbers, so that we can get a decimal easily and know its place.

Complete step-by-step answer:
Let us first understand the meaning of terminating rational numbers:
A terminating decimal is a decimal with a certain number of digits to the right of the decimal point. Examples include 3.2, 4.075, and -300.12002. All of these are rational. In simpler words, it means that if a number ends at some point after decimal, it is terminating.
Now, coming back to the question:
We have: $\dfrac{{23}}{{{2^3} \times {5^2}}}$. It has ${2^3} \times {5^2}$ in its denominator.
We know that every rational number with denominator of the form ${2^m} \times {5^n}$ will always be terminating.
Hence, $\dfrac{{23}}{{{2^3} \times {5^2}}}$ is terminating as well.
Now, consider the denominator for once.
Denominator = \[{2^3} \times {5^2} = ({2^2} \times {5^2}) \times 2 = {10^2} \times 2\].
Hence, the number given to us becomes $\dfrac{{23}}{{{2^3} \times {5^2}}} = \dfrac{{23}}{{2 \times {{10}^2}}}$.
We can multiply and divide the number by 5, it would not make any difference because we are eventually multiplying the number by 1 because of multiplying and dividing by the same number.
So, $\dfrac{{23}}{{{2^3} \times {5^2}}} = \dfrac{{23}}{{2 \times {{10}^2}}} \times \dfrac{5}{5}$.
Simplifying it, we will have with us:-
$\dfrac{{23}}{{{2^3} \times {5^2}}} = \dfrac{{23 \times 5}}{{{{10}^2} \times 2 \times 5}} = \dfrac{{115}}{{{{10}^2} \times 10}} = \dfrac{{115}}{{{{10}^3}}}$
Hence, $\dfrac{{23}}{{{2^3} \times {5^2}}} = \dfrac{{115}}{{1000}} = 0.115$.
Hence, it terminates after 3 digits.
Hence, the answer is 3 digits.

Note: The students might make the mistake of just dividing the number as per their requirements but forgetting to multiply it as well to nullify any changes in the number.
The students also might forget to check in the start whether the number given to them is actually terminating or it is just a trick question. So, remember to check the conditions for that as well.
Always remember that all rational numbers may not be terminated. For example: $\dfrac{1}{3}$ and our most known number $\pi $ that is equal to $\dfrac{{22}}{7}$.