If the length of a rectangle increases by 50%and the breadth decreases by 25%, then what will be the percent increase in its area?

Options

A. `15%`
B. `17.5%`
C. `12.5%`
D. `25%`

Answer

To solve the problem step by step, we will first define the original dimensions of the rectangle and then calculate the new dimensions after the changes. Finally, we will compute the area before and after the changes and find the percentage increase in the area.

### Step 1: Define the original dimensions
Let the original length of the rectangle be \( L \) and the original breadth be \( B \).

### Step 2: Calculate the new dimensions
- The length of the rectangle increases by 50%. Therefore, the new length \( L' \) can be calculated as:
\[ L' = L + 0.5L = 1.5L = \frac{3L}{2} \]

- The breadth of the rectangle decreases by 25%. Therefore, the new breadth \( B' \) can be calculated as:
\[ B' = B - 0.25B = 0.75B = \frac{3B}{4} \]

### Step 3: Calculate the original area
The original area \( A \) of the rectangle is given by:
\[ A = L \times B \]

### Step 4: Calculate the new area
The new area \( A' \) of the rectangle with the new dimensions is:
\[ A' = L' \times B' = \left(\frac{3L}{2}\right) \times \left(\frac{3B}{4}\right) = \frac{9LB}{8} \]

### Step 5: Calculate the increase in area
The increase in area \( \Delta A \) is given by:
\[ \Delta A = A' - A = \frac{9LB}{8} - LB \]
To simplify this, we can express \( A \) in terms of a common denominator:
\[ \Delta A = \frac{9LB}{8} - \frac{8LB}{8} = \frac{LB}{8} \]

### Step 6: Calculate the percentage increase in area
The percentage increase in area can be calculated using the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta A}{A}\right) \times 100 = \left(\frac{\frac{LB}{8}}{LB}\right) \times 100 = \frac{1}{8} \times 100 = 12.5\% \]

### Final Answer
The percentage increase in the area of the rectangle is **12.5%**.

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