If the length of a rectangle is increased by 50%, by how much per cent its breadth should be reduced to keep the area same ?

Options

A. 50
B. `33(1)/(3)`
C. `66(1)/(6)`
D. `37(1)/(6)`

Answer

To solve the problem step by step, we need to determine how much the breadth of a rectangle should be reduced when its length is increased by 50% while keeping the area the same.

### Step-by-Step Solution:

1. **Define the Original Dimensions**:
Let the original length of the rectangle be \( L \) and the original breadth be \( B \).

2. **Calculate the Original Area**:
The area \( A \) of the rectangle is given by:
\[ A = L \times B \]

3. **Increase the Length by 50%**:
If the length is increased by 50%, the new length \( L' \) becomes:
\[ L' = L + 0.5L = 1.5L \]

4. **Set Up the Equation for the New Area**:
To keep the area the same after increasing the length, we need to find the new breadth \( B' \) such that:
\[ A = L' \times B' \]
This can be rewritten as:
\[ L \times B = 1.5L \times B' \]

5. **Cancel Out the Length**:
Since \( L \) is common on both sides, we can divide both sides by \( L \) (assuming \( L \neq 0 \)):
\[ B = 1.5 \times B' \]

6. **Solve for the New Breadth**:
Rearranging the equation gives:
\[ B' = \frac{B}{1.5} = \frac{B}{\frac{3}{2}} = \frac{2B}{3} \]

7. **Determine the Reduction in Breadth**:
The original breadth is \( B \) and the new breadth is \( \frac{2B}{3} \). The reduction in breadth is:
\[ \text{Reduction} = B - B' = B - \frac{2B}{3} = \frac{B}{3} \]

8. **Calculate the Percentage Reduction**:
The percentage reduction in breadth is given by:
\[ \text{Percentage Reduction} = \left(\frac{\text{Reduction}}{\text{Original Breadth}}\right) \times 100 = \left(\frac{\frac{B}{3}}{B}\right) \times 100 = \frac{1}{3} \times 100 = 33.33\% \]

### Final Answer:
The breadth should be reduced by **33.33%** to keep the area the same.