One side of a square is increased by 30%. To maintain the same area, the other side will have to be decreased by

Options

A. `23(1)/(13)%`
B. `76(12)/(13)%`
C. `30%`
D. `15%`

Answer

To solve the problem, we need to determine how much the other side of the square must be decreased to maintain the same area after one side has been increased by 30%.

### Step-by-Step Solution:

1. **Let the original side length of the square be \( s \)**.
- The area of the square is given by the formula:
\[ \text{Area} = s^2 \]

2. **Increase one side by 30%**.
- The new length of the side after the increase will be:
\[ \text{New side} = s + 0.3s = 1.3s \]

3. **Set the area of the new square equal to the original area**.
- The area of the new square with the increased side is:
\[ \text{New Area} = (1.3s) \times x \]
- Here, \( x \) is the new length of the other side that we need to find.

4. **Equate the areas**.
- Since the area must remain the same, we set the areas equal to each other:
\[ s^2 = (1.3s) \times x \]

5. **Simplify the equation**.
- Dividing both sides by \( s \) (assuming \( s \neq 0 \)):
\[ s = 1.3x \]

6. **Solve for \( x \)**.
- Rearranging gives:
\[ x = \frac{s}{1.3} \]

7. **Calculate the percentage decrease**.
- The original side was \( s \) and the new side is \( \frac{s}{1.3} \). The decrease in the side length is:
\[ \text{Decrease} = s - \frac{s}{1.3} = s \left(1 - \frac{1}{1.3}\right) = s \left(\frac{1.3 - 1}{1.3}\right) = s \left(\frac{0.3}{1.3}\right) \]
- To find the percentage decrease:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{s}\right) \times 100 = \left(\frac{0.3}{1.3}\right) \times 100 \]

8. **Calculate the final percentage**.
- Performing the calculation:
\[ \text{Percentage Decrease} = \left(\frac{0.3}{1.3}\right) \times 100 \approx 23.08\% \]

### Final Answer:
The other side will have to be decreased by approximately **23.08%** to maintain the same area.