Question

The cost of fencing of circular ground is 25 paisa per meter is Rs. 440. The cost of reaping the grass at 60 paisa per 100 square meters is
(a) Rs. 1790.00
(b) Rs. 1887.40
(c) Rs. 1478.40
(d) Rs. 1646.80

Answer 1

Hint: let us take a rough figure of the circular ground as follows

For fencing the circular ground we use the circumference to find the radius of the circular ground.
That is if the cost of fencing is 25 paisa for one meter then we take the cost of fencing of the whole circumference as Rs. 440
The formula of the circumference of a circle of radius \['r'\] is given as
\[C=2\pi r\]
Then we can find the cost of reaping taking the condition that it takes 60 paise for 100 square meters considering the area of the circle.
The formula of the area of a circle is given as
\[\Rightarrow A=\pi {{r}^{2}}\]
We use the conversion of rupee to paisa as
\[1rupee=100paisa\]
Complete step by step answer:
We are given that the cost of fencing of the circular ground is 25 paisa per meter is Rs. 440
Let us assume that the circumference of the given circular ground as \[C\]
We know that for the fencing we need to consider the circumference.
So, we can consider that the cost of fencing of 1 meter is 25 paisa and the cost of fencing of the circumference is Rs. 440.
We know that the conversion of rupee to paisa as
\[1rupee=100paisa\]
By converting the 25 paisa in above statement into rupee we get
\[\begin{align}
  & \Rightarrow 25paisa=\dfrac{25}{100}rupee \\
 & \Rightarrow 25paisa=\dfrac{1}{4}rupee \\
\end{align}\]
Now, by converting the given statement into mathematical equation we get
\[\begin{align}
  & \Rightarrow \dfrac{C}{1}=\dfrac{440}{\left( \dfrac{1}{4} \right)} \\
 & \Rightarrow C=440\times 4 \\
 & \Rightarrow C=1760 \\
\end{align}\]
We know that the formula of circumference of circle of radius \['r'\] is given as
\[C=2\pi r\]
By substituting this formula in above equation we get
\[\begin{align}
  & \Rightarrow 2\pi r=1760 \\
 & \Rightarrow r=\dfrac{880}{\pi } \\
\end{align}\]
Therefore we can see that the radius of given circular ground is \[\dfrac{880}{\pi }\] meters.
Now, let us find the cost of reaping grass.
We know that for the reaping of grass we need to consider the area of the circle.
Now, let us assume that the cost of reaping the grass as \[X\] for \[A\] square meters
We are given that the cost of reaping the grass is 60 paisa per 100 square meters
Now, by converting the 60 paisa to rupee we get
\[\begin{align}
  & \Rightarrow 60paisa=\dfrac{60}{100}rupee \\
 & \Rightarrow 60paisa=\dfrac{3}{5}rupee \\
\end{align}\]
Now, by converting the above statements into mathematical equation we get
\[\begin{align}
  & \Rightarrow \dfrac{X}{\left( \dfrac{3}{5} \right)}=\dfrac{A}{100} \\
 & \Rightarrow X=\dfrac{3A}{500}.......equation(i) \\
\end{align}\]
We know that formula of area of circle is given as
\[\Rightarrow A=\pi {{r}^{2}}\]
By using this formula we get the area for radius \[\dfrac{880}{\pi }\] meters as
\[\begin{align}
  & \Rightarrow A=\pi {{\left( \dfrac{880}{\pi } \right)}^{2}} \\
 & \Rightarrow A=\dfrac{880\times 880}{\pi } \\
\end{align}\]
By substituting this area value in equation (i) we get
\[\begin{align}
  & \Rightarrow X=\dfrac{3}{500}\times \dfrac{880\times 880}{\pi } \\
 & \Rightarrow X=1478.39 \\
 & \Rightarrow X\simeq 1478.40 \\
\end{align}\]
Therefore we can conclude that the cost of reaping of grass of given circular field is Rs. 1478.40
So, option (c) is correct answer.
Note:
Students may make mistakes in consideration of circumference and area in the required conditions.
The fencing is done along the circular field which involves circumference whereas the reaping of grass is done everywhere inside the field which includes the area.
Students may get confused about these two points and may consider it reverse order.
We also need to convert all the paise to rupee because we know that we can add or subtract or multiply or divide only the terms with the same units.