Out of a group of swans, $\frac{7}{2}$ times the square root of the total number are playing on the share of a pond. The two remaining ones are swinging in water. Find the total number of swans.

Given:

Out of a group of swans, $\frac{7}{2}$ times the square root of the total number are playing on the share of a pond. The two remaining ones are swinging in water.

To do:

We have to find the total number of swans.

Solution:

Let the total number of swans be $x$.

This implies,

Number of swans playing on the share of a pond$=\frac{7}{2}\times \sqrt{x}$

Number of swans swinging in water$=2$.

Therefore,

$x=2+\frac{7}{2}\times \sqrt{x}$

$x-2=\frac{7}{2}\times \sqrt{x}$

$(x-2)^2=(\frac{7}{2}\times \sqrt{x})^2$   (Squaring on both sides)

$x^2-4x+4=\frac{49}{4}x$

$4(x^2-4x+4)=49x$

$4x^2-16x+16-49x=0$

$4x^2-65x+16=0$

Solving for $x$ by factorization method, we get,

$4x^2-64x-x+16=0$

$4x(x-16)-1(x-16)=0$

$(4x-1)(x-16)=0$

$4x-1=0$ or $x-16=0$

$4x=1$ or $x=16$

$x=\frac{1}{4}$ or $x=16$

Therefore, the value of $x$ is $16$.   ($x$ cannot be a fraction)

The total number of swans is $16$.