the length of the rectangle is increased by 42 percent, by what percentage should the breadth be decreased so that there is no change in the area of the rectangle?

To solve this problem, we need to maintain the area of the rectangle while adjusting its dimensions.

### Step-by-Step Solution:

1. **Let the original length and breadth of the rectangle be \( L \) and \( B \) respectively.**

2. **Original Area:**  
   The area of the rectangle originally is given by:
   \[
   \text{Original Area} = L \times B
   \]

3. **New Length:**  
   The length is increased by 42%, so the new length becomes:
   \[
   L_{\text{new}} = L \times (1 + 0.42) = 1.42L
   \]

4. **New Breadth:**  
   Let the new breadth be \( B_{\text{new}} \). The area should remain unchanged, so:
   \[
   \text{New Area} = \text{Original Area}
   \]
   \[
   1.42L \times B_{\text{new}} = L \times B
   \]

5. **Solve for \( B_{\text{new}} \):**
   \[
   B_{\text{new}} = \frac{L \times B}{1.42L} = \frac{B}{1.42}
   \]

6. **Percentage Decrease in Breadth:**
   The percentage decrease in breadth is calculated as:
   \[
   \text{Percentage Decrease} = \left( \frac{B - B_{\text{new}}}{B} \right) \times 100
   \]
   \[
   = \left( \frac{B - \frac{B}{1.42}}{B} \right) \times 100
   \]
   \[
   = \left( 1 - \frac{1}{1.42} \right) \times 100
   \]
   \[
   = \left( 1 - 0.7042 \right) \times 100
   \]
   \[
   = 0.2958 \times 100 \approx 29.58\%
   \]

### Final Answer:
The breadth should be decreased by approximately **29.58%** to maintain the same area.

Would you like to explore this in more detail, or do you have any questions?

Here are some related questions you might find interesting:

1. What happens to the area if both dimensions are increased by a certain percentage?
2. How would a similar problem be solved if the area was required to increase by a certain percentage instead?
3. What if only the breadth was increased by a certain percentage? How would the length change?
4. How do these calculations change if the initial shape isn't a rectangle but a square?
5. Can this concept be applied to other shapes like circles or triangles?

**Tip:** When working with percentages, remember that increasing or decreasing a quantity by a percentage involves multiplying by \(1 + \frac{\text{percentage}}{100}\) or \(1 - \frac{\text{percentage}}{100}\) respectively.

The length of the rectangle is increased by 42 percent, by what percentage should the breadth be decreased so that there is no change in the area of the rectangle?

Find out how much the breadth of a rectangle should decrease when its length increases by 42% to maintain the same area. This geometry problem covers percentage changes and the formula for the area of a rectangle. Suitable for students in Grades 8-10.

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