The ratio of the area of a rectangle to its perimeter is 60:11. The length and breadth of the rectangle are in the ratio 6:5. Find the length of the rectangle in units.

Introduction / Context:
This question links the concepts of area, perimeter, and ratios for a rectangle. Instead of giving direct dimensions, it provides two different ratios: the ratio of area to perimeter, and the ratio of length to breadth. By expressing length and breadth in terms of a single variable, we can use these ratios to solve for the actual dimensions of the rectangle, specifically its length.

Given Data / Assumptions:

• Area : Perimeter = 60 : 11.
• Length : Breadth = 6 : 5.
• Let length = 6x and breadth = 5x for some positive x.
• Area A = length * breadth = 6x * 5x.
• Perimeter P = 2(length + breadth) = 2(6x + 5x).

Concept / Approach:
Area and perimeter are both expressed in terms of x. Their ratio A:P must equal 60:11. By setting A / P = 60 / 11 and substituting A and P in terms of x, we obtain an equation that can be solved for x. Once x is found, the actual length 6x can be computed and compared with the options. This approach uses ratio algebra and basic mensuration formulas.

Step-by-Step Solution:
Let length L = 6x and breadth B = 5x.Area A = L * B = 6x * 5x = 30x^2.Perimeter P = 2(L + B) = 2(6x + 5x) = 2 * 11x = 22x.Given area to perimeter ratio A : P = 60 : 11, so A / P = 60 / 11.A / P in terms of x is (30x^2) / (22x) = (30x) / 22 = 15x / 11.Set 15x / 11 = 60 / 11.Therefore 15x = 60, so x = 60 / 15 = 4.Length L = 6x = 6 * 4 = 24 units.

Verification / Alternative check:
With x = 4, breadth B = 5 * 4 = 20 units. Area A = 24 * 20 = 480 square units. Perimeter P = 2(24 + 20) = 2 * 44 = 88 units. The ratio A : P = 480 : 88 simplifies by dividing both by 8, giving 60 : 11, exactly as given. This confirms that L = 24 units is consistent with all the information in the question.

Why Other Options Are Wrong:
If L = 40 units, breadth would be 40 * (5 / 6) ≈ 33.33 units, giving an area and perimeter ratio that does not simplify to 60:11. Lengths of 30 or 13 units also fail when the ratio of area to perimeter is computed. A length of 20 units would force breadth to be less than required by the 6:5 ratio. Only 24 units satisfies both given ratios simultaneously.

Common Pitfalls:
Some students misread the ratio 60:11 and set A = 60 and P = 11 directly, ignoring the scaling. Others mistake which ratio applies to which pair of quantities or forget to reduce A / P in terms of x properly. Carefully writing out A and P in terms of x and then equating the ratios ensures a systematic solution. Always verify the final dimensions by recomputing both area and perimeter ratios.

Final Answer:
24 units