If each side of the square be increased by 50%, then find the percentage increase in the area :

Question

Accepted Answer

To determine the percentage increase in the area of a square when each side is increased by 50%, let's go through the calculations step by step.

1. **Original Area Calculation:**
   - Let the original side length of the square be $ s $.
   - The original area of the square is $ s^2 $.

2. **New Side Length Calculation:**
   - If each side of the square is increased by 50%, the new side length becomes $ s + 0.5s = 1.5s $.

3. **New Area Calculation:**
   - The new area of the square with the increased side length is $ (1.5s)^2 = 2.25s^2 $.

4. **Percentage Increase in Area:**
   - The increase in area is $ 2.25s^2 - s^2 = 1.25s^2 $.
   - To find the percentage increase, we use the formula:
     $$
     \text{Percentage Increase} = \left(\frac{\text{Increase in Area}}{\text{Original Area}}\right) \times 100\%
     $$
   - Substituting the values, we get:
     $$
     \text{Percentage Increase} = \left(\frac{1.25s^2}{s^2}\right) \times 100\% = 125\%
     $$

Therefore, the percentage increase in the area of the square is 125%, which corresponds to Option 3.

**Explanation for Incorrect Options:**

- **Option 1 (50%):** This option is incorrect because a 50% increase in the side length does not translate to a 50% increase in area. The area increases by a factor of the square of the increase in side length, not linearly.

- **Option 2 (100%):** This option is incorrect because a 100% increase would imply that the new area is double the original area. However, as calculated, the new area is 2.25 times the original area, not 2 times.

- **Option 4 (150%):** This option is incorrect because a 150% increase would mean the new area is 2.5 times the original area. The actual increase is 125%, making the new area 2.25 times the original area.

In conclusion, the correct answer is Option 3, which accurately reflects the 125% increase in the area when each side of the square is increased by 50%.