The side length of a square is increased by k percent and as a result the area of the square increases by 69%. What is the value of k (the percentage increase in the side length)?

Introduction / Context:
This is a classic percentage and mensuration problem. For a square, area is proportional to the square of the side length. When the side increases by some percentage, the area increases by a larger percentage. The question gives the percentage increase in area and asks for the corresponding increase in the side length. This tests understanding of how linear and area measures scale.

Given Data / Assumptions:

• Let the original side length of the square be s.
• Original area is A₁ = s^2.
• Side is increased by k percent, so new side length is s * (1 + k / 100).
• New area A₂ is 69 percent more than original, so A₂ = 1.69 * A₁.
• We must find k.

Concept / Approach:
If linear dimension scales by a factor m, then the area of a square scales by factor m^2. In this case, the scale factor for side is (1 + k / 100) and the scale factor for area is 1.69. Therefore, (1 + k / 100)^2 = 1.69. Taking the positive square root gives 1 + k / 100. From this we solve for k. It turns out that the numbers here are chosen so that the square root is simple.

Step-by-Step Solution:
Step 1: Original area A₁ = s^2. Step 2: New side length is s₂ = s * (1 + k / 100). Step 3: New area A₂ = s₂^2 = s^2 * (1 + k / 100)^2. Step 4: Given that area increases by 69 percent, A₂ = 1.69 * A₁ = 1.69 * s^2. Step 5: Equate the expressions: s^2 * (1 + k / 100)^2 = 1.69 * s^2. Step 6: Cancel s^2 on both sides to get (1 + k / 100)^2 = 1.69. Step 7: Take the positive square root: 1 + k / 100 = √1.69 = 1.3. Step 8: Thus k / 100 = 1.3 − 1 = 0.3, so k = 0.3 * 100 = 30.

Verification / Alternative check:
We can check with a simple example. Let the original side be 10 units. Original area is 100 sq units. A 30 percent increase in side gives new side = 13 units. New area is 13^2 = 169 sq units. The increase in area is 169 − 100 = 69 sq units, which is 69 percent of 100. This matches the condition in the question, confirming k = 30 is correct.

Why Other Options Are Wrong:
If k = 33 percent, the scale factor would be 1.33 and the area factor would be approximately 1.7689, or a 76.89 percent increase, not 69 percent. For k = 34.5 or 35 percent, the area increases would be even larger. A 25 percent increase in side corresponds to an area factor of 1.25^2 = 1.5625, which is a 56.25 percent increase, still not 69 percent. Therefore only 30 percent gives the required area increase.

Common Pitfalls:
A frequent mistake is to assume that area increases by the same percentage as the side, which is incorrect for two dimensional shapes. Some learners also forget to convert k percent to a decimal before squaring, or they miscalculate the square root of 1.69. Always remember that area scales with the square of the linear factor, and do not skip the intermediate algebraic step of squaring and taking square roots.

Final Answer:
The value of k is 30 percent.