The ratio between the perimeter and the breadth of a rectangle is 5:1. If the area of the rectangle is 216 cm^2, find the length of the rectangle in centimetres.

Introduction / Context:
This question tests translating a ratio involving perimeter into an algebraic relation between length and breadth. Perimeter is 2(l + b). When you are given perimeter:breadth = 5:1, you can form an equation linking l and b. Then the given area allows you to solve for the dimensions and identify the required length.

Given Data / Assumptions:

• Perimeter : breadth = 5 : 1
• Let length = l cm, breadth = b cm
• Perimeter P = 2(l + b)
• Area = l*b = 216 cm^2

Concept / Approach:
Convert the ratio to an equation: P/b = 5. Since P = 2(l + b), we get 2(l + b)/b = 5. Solve for l in terms of b, then use area to find b and finally l.

Step-by-Step Solution:
Given: P : b = 5 : 1 => P/b = 5 P = 2(l + b) So 2(l + b)/b = 5 2l + 2b = 5b 2l = 3b => l = (3/2)b = 1.5b Area: l*b = 216 => (1.5b)*b = 216 1.5b^2 = 216 => b^2 = 216/1.5 = 144 b = 12 cm (positive) l = 1.5*12 = 18 cm

Verification / Alternative check:
Perimeter = 2(18 + 12) = 60. Ratio P:b = 60:12 = 5:1, correct. Area = 18*12 = 216 cm^2, correct.

Why Other Options Are Wrong:
16 cm: with area 216, breadth would be 13.5, giving perimeter ratio not equal to 5:1. 20 cm: breadth would be 10.8, ratio fails. 22 cm: breadth would be 9.82, ratio fails. 24 cm: breadth would be 9, ratio P:b becomes 2(24+9)/9 = 66/9 = 7.33, not 5.

Common Pitfalls:
Using l:b = 5:1 instead of P:b = 5:1. Forgetting perimeter formula is 2(l + b), not l + b. Making an arithmetic mistake when converting the ratio to an equation.

Final Answer:
Length of the rectangle = 18 cm