If the length of a room is decreased by 10% and breadth is decreased by 20%, while height is increased by 6%, then what percentage changed in the volume of the room?

Options

A. 0.244
B. 0.242
C. 0.24
D. 0.246

Explanation

To find the percentage change in the volume of the room after the changes in its dimensions, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Original Dimensions**:
Let the original length, breadth, and height of the room be \( L \), \( B \), and \( H \) respectively.

2. **Calculate the New Dimensions**:
- The length is decreased by 10%, so the new length \( L' \) is:
\[ L' = L - 0.10L = 0.90L \]
- The breadth is decreased by 20%, so the new breadth \( B' \) is:
\[ B' = B - 0.20B = 0.80B \]
- The height is increased by 6%, so the new height \( H' \) is:
\[ H' = H + 0.06H = 1.06H \]

3. **Calculate the Original Volume**:
The original volume \( V_{\text{old}} \) of the room is given by:
\[ V_{\text{old}} = L \times B \times H \]

4. **Calculate the New Volume**:
The new volume \( V_{\text{new}} \) after the changes in dimensions is:
\[ V_{\text{new}} = L' \times B' \times H' = (0.90L) \times (0.80B) \times (1.06H) \]
Simplifying this gives:
\[ V_{\text{new}} = 0.90 \times 0.80 \times 1.06 \times L \times B \times H \]
\[ V_{\text{new}} = 0.7632 \times V_{\text{old}} \]

5. **Calculate the Change in Volume**:
The change in volume can be calculated as:
\[ \text{Change in Volume} = V_{\text{old}} - V_{\text{new}} = V_{\text{old}} - 0.7632 \times V_{\text{old}} = (1 - 0.7632) \times V_{\text{old}} = 0.2368 \times V_{\text{old}} \]

6. **Calculate the Percentage Change**:
The percentage change in volume is given by:
\[ \text{Percentage Change} = \left( \frac{\text{Change in Volume}}{V_{\text{old}}} \right) \times 100 = \left( \frac{0.2368 \times V_{\text{old}}}{V_{\text{old}}} \right) \times 100 = 23.68\% \]

### Final Answer:
The percentage change in the volume of the room is approximately **23.68%** decrease.