Question

Out of a group of swans, $\dfrac{7}{2}$ times the square root of the number of swans are playing on the shore of the tank. The two remaining are playing, with an amorous fight, in the water. What is the total number of swans?

Answer 1

Hint: Assume a variable x which will represent the total number of swans. It is given that the $\dfrac{7}{2}$ times the square root of the number of swans are playing on the shore of the tank. So, the number of swans at the shore of the tank is $\dfrac{7\sqrt{x}}{2}$. Summation of the swans playing at the shore and in water will be equal to the total number of swans.

Complete step by step solution:
In the question, it is given that out of a group of swans, $\dfrac{7}{2}$ times the square root of the number of swans are playing on the shore of the tank. The two remaining are playing, with an amorous fight, in the water. We are required to find the total number of swans.
Let us assume that there are x swans in total. Since $\dfrac{7}{2}$ times the square root of the number of swans are playing on the shore of the tank, the number of swans at the shore of the tank is $\dfrac{7\sqrt{x}}{2}$. Also, there are two more swans that are there in the water. This means that the total number of swans are also equal to $\dfrac{7\sqrt{x}}{2}$ + 2. Hence, we can say that,
$\begin{align}
  & \dfrac{7\sqrt{x}}{2}+2=x \\
 & \Rightarrow x-2=\dfrac{7\sqrt{x}}{2} \\
 & \Rightarrow 2x-4=7\sqrt{x} \\
\end{align}$
Squaring both the sides, we have,
$\begin{align}
  & {{\left( 2x-4 \right)}^{2}}={{\left( 7\sqrt{x} \right)}^{2}} \\
 & \Rightarrow 4{{x}^{2}}+16-16x=49x \\
 & \Rightarrow 4{{x}^{2}}-65x+16=0 \\
 & \Rightarrow 4{{x}^{2}}-64x-x+16=0 \\
 & \Rightarrow 4x\left( x-16 \right)-1\left( x-16 \right)=0 \\
 & \Rightarrow \left( 4x-1 \right)\left( x-16 \right)=0 \\
 & \Rightarrow x=\dfrac{1}{4},x=16 \\
\end{align}$
Since x represents the number of swans, it must be a natural number.
So, x = 16.
Hence, there are 16 swans in total.

Note: This is an easy question which can be easily done if one reads each line of the question carefully. So, there is a possibility of making a mistake when one does not read the question carefully or one may commit a mistake while doing calculations.