If each side of a square is increased by 25%, what is the percentage change in the area of the square (express your answer as the percentage increase in area)?

Introduction:
This question tests the relationship between linear change and area change. For squares and rectangles, area depends on the product of side lengths. If a side length increases by a certain percentage, the area changes by the square of the scale factor because area is a two-dimensional measure. Many students mistakenly add the percentage directly (like 25% + 25% = 50%), but that is not correct for area. The correct approach is to convert the percentage increase into a multiplication factor and then square it.

Given Data / Assumptions:

• Each side of a square is increased by 25%• Original side length = s (assume)• New side length = s increased by 25%

Concept / Approach:
If a quantity increases by p%, its new value is old * (1 + p/100). For p = 25, scale factor = 1.25. For a square, area is proportional to side^2, so the area scale factor becomes (1.25)^2. The percentage change is (new area - old area) / old area * 100.

Step-by-Step Solution:
Step 1: Let the original side be s.Original area = s^2Step 2: Increase side by 25%.New side = s * (1 + 25/100) = 1.25sStep 3: Compute new area.New area = (1.25s)^2 = (1.25)^2 * s^2Step 4: Evaluate (1.25)^2.1.25^2 = 1.5625So new area = 1.5625 * original areaStep 5: Convert this factor into percentage increase.Increase = (1.5625 - 1)*100 = 0.5625*100 = 56.25%

Verification / Alternative check:
Take s = 4 units. Original area = 16. New side = 5. New area = 25. Increase = 9. Percentage increase = 9/16*100 = 56.25%. This matches the formula result, confirming the answer.

Why Other Options Are Wrong:
50% is the common mistake of doubling 25% (linear thinking).56% and 65% are rounding or incorrect scaling errors.65.25% and 65% do not match the exact squared scale factor 1.5625.

Common Pitfalls:
• Adding percentages instead of using multiplication factors.• Forgetting that area depends on two dimensions.• Rounding too early and missing the exact 56.25%.

Final Answer:
The area increases by 56.25%.