In a rectangle, the ratio of its perimeter to its breadth is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

Introduction / Context:
This problem is a standard mensuration question on rectangles. It tests how well you can connect perimeter, area, length and breadth, and form simple algebraic equations from the given ratio. Such questions are common in aptitude exams because they require both conceptual understanding and careful manipulation of formulas.

Given Data / Assumptions:

- The figure is a rectangle with length and breadth as its sides.
- Ratio of perimeter to breadth is 5 : 1.
- Area of the rectangle is 216 sq. cm.
- We are required to find the length of the rectangle in centimetres.
- All measurements are in centimetres and the rectangle is a normal plane figure with right angles.

Concept / Approach:
The key formulas for a rectangle are:
- Perimeter = 2 * (length + breadth).
- Area = length * breadth.
We use the ratio between perimeter and breadth to create one equation and the given area to create a second equation. Solving these simultaneously gives the actual numerical values of length and breadth.

Step-by-Step Solution:
Step 1: Let breadth = b cm and length = l cm.Step 2: Perimeter P = 2 * (l + b). The ratio P : b = 5 : 1, so 2 * (l + b) / b = 5.Step 3: This gives 2 * (l + b) = 5b, so 2l + 2b = 5b, hence 2l = 3b and l = 3b / 2.Step 4: Area A = l * b = 216 sq. cm. Substitute l = 3b / 2 to get (3b / 2) * b = 216.Step 5: This simplifies to 3b^2 / 2 = 216, so 3b^2 = 432 and b^2 = 144.Step 6: Therefore b = 12 cm (we take the positive value for a length).Step 7: Now l = 3b / 2 = 3 * 12 / 2 = 18 cm.

Verification / Alternative check:
Check perimeter: 2 * (18 + 12) = 2 * 30 = 60 cm. Ratio P : b = 60 : 12 = 5 : 1, so the ratio condition is satisfied. Area check: 18 * 12 = 216 sq. cm, which matches the given area. Both conditions hold, so the solution is consistent.

Why Other Options Are Wrong:
24 cm would give area 24 * 9 (if breadth were adjusted to keep ratio) which does not equal 216 sq. cm. 12 cm cannot be the length because it is actually the breadth obtained from the equations. 20 cm does not satisfy both the ratio and area conditions simultaneously. Data inadequate is incorrect because the given information is sufficient to determine unique values for length and breadth.

Common Pitfalls:
A common mistake is to treat the ratio 5 : 1 as length : breadth or perimeter : length instead of perimeter : breadth, which completely changes the equations. Another frequent error is mishandling the algebra when solving 2l + 2b = 5b. Some students also forget to verify both the area and the ratio, leading to incorrect answers that partially fit the data.

Final Answer:
Therefore, the length of the rectangle is 18 cm.