Question

If the length, breadth and height of the room are in ratio 3:2:1. The breadth and height of the room are halved and length of the room is doubled. Then area of the four walls of the room will,

• A. decrease by 13.64%
• B. decrease by 15%
• C. decrease by 18.75%
• D. decrease by 30%

Hint:

Assign the ratio to a variable k, compute the original wall area with 2(L+B)H, then apply the given changes and compare the two areas.

Answer 1

The Correct Option is D

Step 1: Set up the original dimensions using the ratio.
Length : Breadth : Height = 3 : 2 : 1. Let Length \(= 3k\), Breadth \(= 2k\), Height \(= k\), for some positive constant \(k\).
Step 2: Recall the formula for the area of the four walls.
The area of the four walls of a room is \(2(\text{Length} + \text{Breadth}) \times \text{Height}\).
Step 3: Compute the original wall area.
Original area \(= 2(3k + 2k) \times k = 2 \times 5k \times k = 10k^2\).
Step 4: Apply the changes to get new dimensions.
Breadth and height are halved, and length is doubled: new Length \(= 6k\), new Breadth \(= k\), new Height \(= \frac{k}{2}\).
Step 5: Compute the new wall area.
New area \(= 2(6k + k) \times \frac{k}{2} = 2 \times 7k \times \frac{k}{2} = 7k^2\).
Step 6: Find the percentage change.
Change \(= \frac{7k^2 - 10k^2}{10k^2} \times 100 = \frac{-3k^2}{10k^2} \times 100 = -30\%\). This is a decrease of 30%, matching option (4). The other options come from arithmetic slips in applying the halving and doubling operations.