The ratio between the length and the breadth of a rectangular park is 3 : 2. A man cycles along the boundary of the park at a speed of 12 kilometres per hour and completes one round in 8 minutes. What is the area of the park in square metres?

Introduction / Context:
This question combines speed, time and distance concepts with basic geometry of a rectangle. The man cycling around the boundary covers the perimeter of the park. Once we know the perimeter and the ratio of the sides, we can find the actual length and breadth and then compute the area. Such mixed concept questions are very common in aptitude exams.

Given Data / Assumptions:
- Ratio of length to breadth of the rectangular park is 3 : 2.
- Cycling speed of the man is 12 km per hour.
- Time to complete one round along the boundary is 8 minutes.
- He travels exactly one full perimeter in that time.
- We need to find the area of the park in square metres.

Concept / Approach:
The distance travelled when moving at constant speed is:
Distance = speed * time
Here, the distance for one round equals the perimeter of the rectangle. Once we find the perimeter, say 2(L + B), we use the given ratio L : B = 3 : 2 to express L and B in terms of a common factor k. Then we solve for k and compute the area L * B in square metres.

Step-by-Step Solution:
Step 1: Convert cycling speed into metres per minute. Speed is 12 km per hour = 12 * 1000 metres per 60 minutes = 12000 / 60 = 200 metres per minute. Step 2: Time for one round is 8 minutes, so distance travelled in one round is 200 * 8 = 1600 metres. Step 3: This distance is the perimeter of the park, so 2(L + B) = 1600, which gives L + B = 800 metres. Step 4: Using the ratio L : B = 3 : 2, let L = 3k and B = 2k. Then L + B = 3k + 2k = 5k = 800, so k = 160. Step 5: Therefore length L = 3 * 160 = 480 metres and breadth B = 2 * 160 = 320 metres. Area of the park is L * B = 480 * 320 = 153600 square metres.

Verification / Alternative check:
We can quickly verify the perimeter: 2(L + B) = 2(480 + 320) = 2 * 800 = 1600 metres, which matches the distance cycled in 8 minutes at 200 metres per minute. This confirms that the side lengths and thus the area are consistent with the speed and time data.

Why Other Options Are Wrong:
15360 and 30720 are too small by factors of 10 or 5, arising from errors in unit conversion or forgetting that both length and breadth are large values.
307200 is double the correct area, often obtained when someone uses the perimeter directly in the area formula incorrectly.
76800 is exactly half the correct area and may appear when one of the side lengths is mistakenly taken as k rather than 3k or 2k.

Common Pitfalls:
A typical mistake is to neglect converting time into hours or minutes consistently, leading to wrong perimeter. Another common error is misusing the 3 : 2 ratio and assigning the wrong multiples to length and breadth. Ensure proper unit conversion and carefully apply the ratio to avoid these issues.

Final Answer:
The area of the park is 153600 square metres.