Question

The decimal expansion for the rational number \[\dfrac{{11}}{{{2^3} \cdot {5^2}}}\] will terminate after:
A) One decimal place
B) Two decimal place
C) Three decimal place
D) More than 3 decimal places

Answer 1

Hint:
Here, we will first simplify the denominator of the given fraction by applying the exponent and then multiply the terms hence obtained. Then we will divide the numerator by the denominator to make the fraction into decimal form. We will then check after how many decimal places the rational number terminates.

Complete step by step solution:
We will first apply the exponent in the denominator to make it a proper whole number.
We know that the cube of 2 is 8 and the square of 5 is 25.
So, applying the exponent on the terms of the denominator, we get
 \[\dfrac{{11}}{{{2^3} \cdot {5^2}}} = \dfrac{{11}}{{8 \cdot 25}}\]
Now multiplying the terms in the denominator, we get
\[ \Rightarrow \dfrac{{11}}{{{2^3} \cdot {5^2}}} = \dfrac{{11}}{{200}}\]
On dividing 11 by 200, we get
\[ \Rightarrow \dfrac{{11}}{{{2^3} \cdot {5^2}}} = 0.055\]
The decimal expansion of the rational number is 0.055 and we can clearly see that it will terminate after three decimal places.

Hence, the correct option is C.

Note:
We can also vary our answer by using the following fact. We know that if a rational number is of form \[\dfrac{p}{q}\] and \[q\] can be expressed as \[{2^n}{5^m}\], then the rational number will terminate after \[n\] places, if \[n > m\] and the decimal expansion will terminate after \[m\] places, if \[m > n\].
Here the given rational number is \[\dfrac{{11}}{{{2^3} \cdot {5^2}}}\] and the denominator is of the form \[{2^n}{5^m}\]. We can see that the power of 2 is 3 and the power of 5 is 2. Therefore, it will terminate after three decimal places.