The cost of levelling a circular field at a rate of 50 paise per square metre is Rs 7700. Using π = 22/7, what will be the cost (in rupees) of putting up a fence all around the boundary of this field at Rs 1.20 per metre?

Introduction / Context:
This question combines the concepts of area and circumference of a circle with unit cost calculations. First, we use the total cost and rate per square metre to determine the area of the circular field. From the area, we determine the radius of the circle using the formula for the area of a circle. Once we know the radius, we calculate the circumference, which represents the length of fencing required. Finally, we multiply this length by the cost per metre to obtain the total fencing cost.

Given Data / Assumptions:

• Cost of levelling = Rs 7700.
• Rate of levelling = 50 paise per square metre = Rs 0.50 per square metre.
• The field is circular.
• Use π = 22/7.
• Cost of fencing = Rs 1.20 per metre.
• No wastage or additional costs are considered.

Concept / Approach:
The area of the circular field is given by area = cost of levelling / rate per square metre. Once we have the area, we use the formula area = π * r^2 to solve for r, the radius. After finding the radius, we find the circumference using C = 2 * π * r. The total cost of fencing is then the circumference multiplied by the rate per metre of fencing. All steps require careful handling of units and substitution of π as 22/7.

Step-by-Step Solution:
Rate of levelling = Rs 0.50 per m^2.Total cost = Rs 7700.Area of the circular field = 7700 / 0.50 = 15400 m^2.Use area formula: π * r^2 = 15400.With π = 22/7, we have (22 / 7) * r^2 = 15400.Multiply both sides by 7: 22 * r^2 = 15400 * 7.Compute 15400 / 22 = 700 (since 22 * 700 = 15400).Thus r^2 = 700 * 7 = 4900, so r = √4900 = 70 m.Circumference of the field = 2 * π * r = 2 * (22 / 7) * 70.Compute: (22 / 7) * 70 = 22 * 10 = 220, so circumference = 2 * 220 = 440 m.Fencing cost per metre = Rs 1.20.Total fencing cost = 440 * 1.20 = Rs 528.

Verification / Alternative check:
We can quickly recompute the intermediate values. Area: 7700 at Rs 0.50 per m^2 gives 7700 / 0.5 = 15400 m^2, correct. Radius: area = (22 / 7) * r^2 and r^2 = 15400 * 7 / 22 = 700 * 7 = 4900, hence r = 70 m. Circumference: 2 * (22 / 7) * 70 = 440 m, and 440 * 1.20 = 440 * (6 / 5) = 528, confirming the arithmetic. Every step checks out and supports the final answer.

Why Other Options Are Wrong:
Rs 132 and Rs 264 correspond to using a much smaller radius or miscalculating the circumference. Rs 1056 is exactly double the correct answer, which might come from mistakenly doubling the circumference or cost. Rs 880 would require a circumference or cost rate inconsistent with the given data. Only Rs 528 matches the correct calculation based on the actual radius and fencing rate.

Common Pitfalls:
Common errors include forgetting to convert 50 paise into Rs 0.50, miscomputing r^2 from the area, or using an approximate value for π instead of the specified 22/7. Some candidates also mistakenly treat 7700 as the area instead of converting it through the rate. Careful handling of units and stepwise substitution avoids these errors and leads directly to the correct fencing cost.

Final Answer:
The cost of putting up a fence all around the field is Rs 528.