Question

The decimal expansion of \[\dfrac{{23457}}{{{2^3} \times {5^4}}}\] will terminate after how many places of decimals?
A) 2
B) 3
C) 4
D) 5

Answer 1

Hint:
First we will multiply the numerator and denominator by 2 of the given numbers. Then we will simplify values to convert the fraction into a decimal number. Then we will find the number will terminate after decimal.

Complete step by step solution:
We are given that the rational number \[\dfrac{{23457}}{{{2^3} \times {5^4}}}\].
Multiplying the numerator and denominator by 2 of the given number, we get
\[
   \Rightarrow \dfrac{{23457}}{{{2^3} \times {5^4}}} \times \dfrac{2}{2} \\
   \Rightarrow \dfrac{{46914}}{{{2^4} \times {5^4}}} \\
 \]
Rewriting the denominator of the above rational number, we get
\[
   \Rightarrow \dfrac{{46914}}{{{{\left( {2 \times 5} \right)}^4}}} \\
   \Rightarrow \dfrac{{46914}}{{{{10}^4}}} \\
   \Rightarrow \dfrac{{46914}}{{10000}} \\
 \]
Converting the above expression into decimal number, we get
\[ \Rightarrow 4.6914\]

So, after 4 decimal places the number will terminate.
Hence, option C is correct.

Note:
In solving these types of questions, students must know that there are infinite irrational numbers between two rational numbers. Hence, one can find as many irrational numbers as we want between the rational numbers. The answer is just one among them. We know that the rational numbers are those numbers which can be written in the form of \[\dfrac{p}{q}\], where \[p\] is numerator, \[q\] is denominator, \[q \ne 0\] and both are integers.
We also know that irrational numbers are numbers that can not be represented in the rational number form, they are non-recurring and non-terminating decimal numbers.