Question

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

• A. Remain the same
• B. Decrease by 13.64%
• C. Decrease by 15%
• D. Decrease by 18.75%
• E. Decrease by 30%

Hint:

For wall area changes, remember it depends only on perimeter $\times$ height, not on volume.

Answer 1

The Correct Option is D

Let the original dimensions be $3x$, $2x$, and $x$. Original four-wall area: \[ 2 \times (\text{length} + \text{breadth}) \times \text{height} = 2(3x+2x)(x) = 10x^2 \] New dimensions: length $= 6x$, breadth $= x$, height $= 0.5x$. New four-wall area: \[ 2 \times (6x + x) \times 0.5x = 7x^2 \] Percentage decrease: \[ \frac{10 - 7}{10} \times 100% = 30% \] Wait — that’s different from given options. Let's recheck: Original area should be \[ 2 \times (3x + 2x) \times x = 10x^2 \] New: \[ 2 \times (6x + x) \times 0.5x = 7x^2 \] Decrease = $3x^2$ out of $16x^2$? No, correction — initial perimeter $= 3x+2x = 5x$, height $= x$ → Area = $2 \times 5x \times x = 10x^2$. New: perimeter $= 6x + x = 7x$, height $= 0.5x$ → Area = $2 \times 7x \times 0.5x = 7x^2$. Decrease = $\frac{10-7}{10} \times 100 = 30%$. \[ \boxed{\text{30% decrease}} \]