Question

Nitin’s age was equal to square of some number last year and the following year it would be cube of a number. If again Nitin’s age has to be equal to the cube of some number, then for how long will he have to wait?
\[\begin{align}
  & A.10\text{ years} \\
 & \text{B}\text{.38 years} \\
 & \text{C}\text{.39 years} \\
 & \text{D}\text{.64 years} \\
\end{align}\]

Answer 1

Hint: To solve this question, we will consider the gap of years which is in between last year and following year that is equal to 2. Because age is always a possible integer, so we will try to form two numbers such that their difference is 2 and smaller is square of some number and larger is cube of some number. After finding these numbers we will find the present age of Nitin and then subtract it to the next cube to get the required answer.

Complete step by step answer:
Given that, the age of Nitin was square of some number last year and any information about present year is not given, then in the following year his age is cube of a number.
Basically we have a 2 year gap in between as the age of last year and following year is known. So, observing this, we need 2 numbers whose difference is 2 and one smaller number is a perfect square and another bigger number is a perfect cube.
Difference of two is required as the 2 year gap is in between. Also, as age is always a positive integer, so we will search for a number in positive integer such that it is a perfect square and adding 2 to that number will give a perfect cube.
\[\begin{align}
  & \text{Consider }4={{\left( 2 \right)}^{2}}\text{ but }4+2=6\text{ not a perfect cube} \\
 & \text{Consider }9={{\left( 3 \right)}^{2}}\text{ but }9+2=11\text{ not a perfect cube} \\
 & \text{Consider }16={{\left( 4 \right)}^{2}}\text{ but }16+2=18\text{ not a perfect cube} \\
 & \text{Consider }25={{\left( 5 \right)}^{2}}\text{ and }25+2=27\text{ which is cube of 3 as }{{\left( 3 \right)}^{3}}=27 \\
\end{align}\]
So, finally we have two numbers whose difference is 2 and smaller number 25 is square of 5 and greater number 27 is cube of 3.
25 was Nitin's age last year and 27 is the age of Nitin in the following year.
Therefore, 26 is Nitin's present age.
Since, the next perfect cube after 27 is $64={{\left( 4 \right)}^{3}}$
Then he has to wait $\text{64}-\text{26 }=\text{ 38 years more}$
Therefore, Nitin has to wait for 38 years.

So, the correct answer is “Option B”.

Note: The possibility of mistake in this question can be considering the year gap between last year and following year as 1, this would be wrong. This can be confusing as any theory about the present age of Nitin is not given in the question. So this mistake should be avoided. Another mistake is not subtracting 26 from 64 at the end, this would be wrong because we need the next age when it is a perfect cube so the present age should not be avoided.