Question

The decimal expansion of the rational number \[\dfrac{14587}{1250}\] will terminate after
(A) one decimal place
(B) two decimal places
(C) three decimal places
(D) four decimal places

Answer 1

Hint: The given rational number is \[\dfrac{14587}{1250}\] . Now, convert the given number into a decimal value. We know that termination of digits in decimal expansion means that after a finite number of decimal places, all succeeding place values are zero. Now, check after which decimal places, the succeeding place values are equal to zero. At last, pick the correct option.

Complete step by step answer:
According to the question, we are given a rational number and we have to find the decimal place after which the decimal expansion will terminate.
The given rational number = \[\dfrac{14587}{1250}\] ……………………………………(1)
We know that a rational number is in the form of \[\dfrac{p}{q}\] where \[q\ne 0\] .
We can observe that the given number is also in the form of \[\dfrac{p}{q}\] and \[q\ne 0\] so, no doubt the given number is rational.
Now, let us calculate the decimal value of the given number.
On converting the given rational number into decimal form, we get
\[1250\overset{11.6696}{\overline{\left){\begin{align}
  & 14587 \\
 & 1250 \\
 & \overline{\,\,\,208}7 \\
 & \,\,\,1250 \\
 & \overline{\begin{align}
  & \,\,\,\,\,8370 \\
 & \,\,\,\,\,7500 \\
 & \,\,\,\,\,\overline{\,\,\,870}0 \\
 & \,\,\,\,\,\,\,\,7500 \\
 & \,\,\,\,\,\,\,\,\overline{\begin{align}
  & 12000 \\
 & 11250 \\
 & \overline{\begin{align}
  & \,\,\,\,7500 \\
 & \,\,\,\,7500 \\
 & \,\,\,\,\,\overline{0000} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}\]
……………………………….(2)
We know that termination of digits in decimal expansion means that after a finite number of decimal places, all succeeding place values are zero.
For instance, in equation (2), we can observe that after 6696 we don’t have any digits. In other words, we can say that all succeeding place values are equal to zero. So, the decimal expansion of the decimal expansion \[\dfrac{14587}{1250}\] is terminating after 4 decimal places.
Hence, the correct option is (D).

Note:
 In this question, one might get confused about the word terminating. Here, we have to take one point into our consideration. That is, a decimal numeral in which after a finite number of decimal places, all succeeding place values are zero.