If the length of a rectangle increases by 10% and the breadth of the rectangle decreases by 12% then find the % change in area.

Answer

To find the percentage change in the area of a rectangle when the length increases by 10% and the breadth decreases by 12%, we can follow these steps:

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

### Step 3: Calculate the new length
The length increases by 10%, so the new length \( L' \) can be calculated as:
\[ L' = L + 0.10L = 1.10L = \frac{11L}{10} \]

### Step 4: Calculate the new breadth
The breadth decreases by 12%, so the new breadth \( B' \) can be calculated as:
\[ B' = B - 0.12B = 0.88B = \frac{88B}{100} \]

### Step 5: Calculate the new area
The new area \( A' \) of the rectangle is given by:
\[ A' = L' \times B' = \left(\frac{11L}{10}\right) \times \left(\frac{88B}{100}\right) \]
\[ A' = \frac{11 \times 88 \times LB}{1000} = \frac{968LB}{1000} \]

### Step 6: Calculate the change in area
The change in area is given by:
\[ \text{Change in Area} = A' - A = \frac{968LB}{1000} - LB \]
\[ = \frac{968LB}{1000} - \frac{1000LB}{1000} = \frac{968LB - 1000LB}{1000} = \frac{-32LB}{1000} \]

### Step 7: Calculate the percentage change in area
To find the percentage change in area, we use the formula:
\[ \text{Percentage Change} = \left(\frac{\text{Change in Area}}{\text{Original Area}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Change} = \left(\frac{-32LB/1000}{LB}\right) \times 100 \]
\[ = \left(\frac{-32}{1000}\right) \times 100 = -3.2\% \]

### Conclusion
The area of the rectangle decreases by \( 3.2\% \).