The length, breadth, and height of a room are in the ratio 7:3:1. If the breadth and height are doubled while the length is halved, by what percent does the total area of the four walls increase?

Introduction / Context:
This question tests surface area of the four walls of a cuboid-shaped room (lateral surface area). The area of the four walls depends on height and the perimeter of the base: Area = 2*h*(l + b). When dimensions change, you should compute the new lateral area in terms of the old using ratios, not absolute measurements. Since only ratios are given, we represent dimensions using a common factor and compare the expressions. Finally, percent increase is computed as ((new - old) / old) * 100.

Given Data / Assumptions:

• Length:breadth:height = 7:3:1
• New length = half of old length
• New breadth = 2 times old breadth
• New height = 2 times old height
• Four walls area formula: 2*h*(l + b)

Concept / Approach:
Let l=7x, b=3x, h=x. Compute original lateral area A0. Update dimensions to l'=3.5x, b'=6x, h'=2x and compute new area A1. Then percent increase = ((A1-A0)/A0)*100.

Step-by-Step Solution:
Original: l = 7x, b = 3x, h = x A0 = 2*h*(l + b) = 2*x*(7x + 3x) = 2*x*(10x) = 20x^2 New: l' = (1/2)*7x = 3.5x, b' = 2*3x = 6x, h' = 2*x = 2x A1 = 2*h'*(l' + b') = 2*(2x)*(3.5x + 6x) = 4x*(9.5x) = 38x^2 Increase = 38x^2 - 20x^2 = 18x^2 Percent increase = (18x^2 / 20x^2)*100 = 90%

Verification / Alternative check:
The ratio A1/A0 = 38/20 = 1.9, meaning the new area is 190% of the old, so the increase is 90%. This matches the computed result.

Why Other Options Are Wrong:
88% and 84% come from adding changes separately instead of using the full formula. 85% and 80% occur when the halving/doubling is applied to the wrong dimension or missed.

Common Pitfalls:
Using total surface area instead of four walls area, forgetting the factor 2 in 2*h*(l+b), or mishandling 7/2 as 3 (instead of 3.5).

Final Answer:
The total area of the four walls increases by 90%.