The length of a rectangle is decreased by 10% and its breadth is increased by 10%. By what percent is its area changed ?

Options

A. 0%
B. 1%
C. 5%
D. 10%

Answer

To solve the problem of how the area of a rectangle changes when its length is decreased by 10% and its breadth is increased by 10%, 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 \( B \). For simplicity, we can assume:
- \( L = 10 \) units
- \( B = 10 \) units

### Step 2: Calculate the New Length
Since the length is decreased by 10%, we calculate the new length as follows:
\[ \text{New Length} = L - 0.1L = L(1 - 0.1) = L \times 0.9 \]
Substituting the value of \( L \):
\[ \text{New Length} = 10 \times 0.9 = 9 \text{ units} \]

### Step 3: Calculate the New Breadth
Since the breadth is increased by 10%, we calculate the new breadth as follows:
\[ \text{New Breadth} = B + 0.1B = B(1 + 0.1) = B \times 1.1 \]
Substituting the value of \( B \):
\[ \text{New Breadth} = 10 \times 1.1 = 11 \text{ units} \]

### Step 4: Calculate the Original Area
The original area \( A_{\text{original}} \) of the rectangle is given by:
\[ A_{\text{original}} = L \times B = 10 \times 10 = 100 \text{ square units} \]

### Step 5: Calculate the New Area
The new area \( A_{\text{new}} \) of the rectangle after the changes is given by:
\[ A_{\text{new}} = \text{New Length} \times \text{New Breadth} = 9 \times 11 = 99 \text{ square units} \]

### Step 6: Calculate the Change in Area
The change in area can be calculated as:
\[ \text{Change in Area} = A_{\text{new}} - A_{\text{original}} = 99 - 100 = -1 \text{ square units} \]

### Step 7: Calculate the Percentage Change in Area
To find the percentage change in area, we use the formula:
\[ \text{Percentage Change} = \left( \frac{\text{Change in Area}}{A_{\text{original}}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Change} = \left( \frac{-1}{100} \right) \times 100 = -1\% \]

### Conclusion
The area of the rectangle has decreased by 1%.

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