The total area of a rectangular plot is 720 square meters, and the ratio of its length to breadth is 5 : 4. If the diameter of a circular fountain is $1\frac{1}{6}\text{th}$ of the breadth of the rectangular plot, what is the difference between the area of the rectangular plot and the circular fountain (in square meters)?

The correct answer is 104.

Step 1: Determine the dimensions of the rectangular plot.

The ratio of the length to the breadth of the rectangular plot is given as 5 : 4.

Let the length of the plot be $5x$ meters and the breadth be $4x$ meters.

The formula for the area of a rectangle is:

$\text{Area}=\text{length}\times \text{breadth}$

Given that the total area of the rectangular plot is 720 square meters:

$5x\times 4x=720$

$20x^2=720$

$x^2=\frac{720}{20}=36$

$x=\sqrt{36}=6$

Now, calculate the length and breadth of the rectangular plot:

$\text{Length}=5\times 6=30\text{meters}$

$\text{Breadth}=4\times 6=24\text{meters}$

Step 2: Find the radius of the circular fountain.

The diameter of the circular fountain is $1\frac{1}{6}$ times the breadth of the rectangular plot.

Convert the mixed fraction to an improper fraction:

$1\frac{1}{6}=\frac{7}{6}$

Calculate the diameter of the circular fountain:

$\text{Diameter}=\frac{7}{6}\times 24=28\text{meters}$

Calculate the radius ($r$) of the circular fountain:

$\text{Radius (r)}=\frac{\text{Diameter}}{2}=\frac{28}{2}=14\text{meters}$

Step 3: Calculate the area of the circular fountain.

The area of a circle is given by:

$\text{Area of fountain}=\pi r^2$

Using $\pi =\frac{22}{7}$:

$\text{Area of fountain}=\frac{22}{7}\times 14\times 14$

$\text{Area of fountain}=22\times 2\times 14=616\text{square meters}$

Step 4: Find the difference between the areas.

$\text{Difference}=\text{Area of rectangular plot}-\text{Area of circular fountain}$

$\text{Difference}=720-616=104\text{square meters}$

Therefore, the difference between the area of the rectangular plot and the circular fountain is 104 square meters.