`l_1, b_1` and `l_2, b_2` are the lenghts and breadths of the two rectangles respectively, but the areas of the rectangles are same. `l_1` is increased by `25%` and `b_1` is decreased by `25%`. Similarly `l_2` is decreased by `25%` and `b_2` is increased by `25%`. If `A_1` and `A_2` is the new areas of the two rectangles respectively, then:

Options

A. `A_1 gt A_2`
B. `A_1 lt A_2`
C. `A_1 = A_2`
D. can't be determined

Answer

To solve the problem, we need to analyze the changes in the dimensions of the two rectangles and calculate their new areas.

### Step-by-Step Solution:

1. **Define the Areas of the Rectangles**:
Let the lengths and breadths of the two rectangles be:
- Rectangle 1: Length = \( l_1 \), Breadth = \( b_1 \)
- Rectangle 2: Length = \( l_2 \), Breadth = \( b_2 \)

Since the areas of the rectangles are the same, we have:
\[ l_1 \times b_1 = l_2 \times b_2 \]

2. **Calculate the New Dimensions for Rectangle 1**:
- The length \( l_1 \) is increased by 25%:
\[ \text{New Length} = l_1 + 0.25l_1 = 1.25l_1 \]
- The breadth \( b_1 \) is decreased by 25%:
\[ \text{New Breadth} = b_1 - 0.25b_1 = 0.75b_1 \]
- Therefore, the new area \( A_1 \) of Rectangle 1 is:
\[ A_1 = \text{New Length} \times \text{New Breadth} = (1.25l_1) \times (0.75b_1) = 0.9375(l_1 \times b_1) \]

3. **Calculate the New Dimensions for Rectangle 2**:
- The length \( l_2 \) is decreased by 25%:
\[ \text{New Length} = l_2 - 0.25l_2 = 0.75l_2 \]
- The breadth \( b_2 \) is increased by 25%:
\[ \text{New Breadth} = b_2 + 0.25b_2 = 1.25b_2 \]
- Therefore, the new area \( A_2 \) of Rectangle 2 is:
\[ A_2 = \text{New Length} \times \text{New Breadth} = (0.75l_2) \times (1.25b_2) = 0.9375(l_2 \times b_2) \]

4. **Compare the New Areas**:
Since we know that \( l_1 \times b_1 = l_2 \times b_2 \), we can substitute:
\[ A_1 = 0.9375(l_1 \times b_1) = 0.9375(l_2 \times b_2) = A_2 \]

5. **Conclusion**:
Thus, we find that:
\[ A_1 = A_2 \]

### Final Answer:
The correct option is:
- **C: \( A_1 = A_2 \)**