Problem: If the area of a square is increased by 69%, what is the percentage increase in the side of the square?

Solution:

To solve this problem, let's assume the original side of the square is 'x'.

Step 1: Calculating the original area of the square
The area of a square is given by the formula: A = side^2
So, the original area of the square is: A = x^2

Step 2: Calculating the increased area of the square
If the area of the square is increased by 69%, the new area can be calculated as follows:
Increased area = Original area + (Original area * Percentage increase)
Increased area = x^2 + (x^2 * 69/100)

Step 3: Calculating the increased side of the square
To find the increased side of the square, we need to take the square root of the increased area:
Increased side = √(Increased area)
Increased side = √(x^2 + (x^2 * 69/100))

Step 4: Calculating the percentage increase in the side of the square
The percentage increase in the side can be calculated as follows:
Percentage increase = ((Increased side - Original side) / Original side) * 100

Now, let's substitute the values and calculate the percentage increase:

Percentage increase = ((√(x^2 + (x^2 * 69/100)) - x) / x) * 100

Simplifying the equation:

Percentage increase = ((√(x^2(1 + 69/100)) - x) / x) * 100
Percentage increase = ((√(x^2(169/100)) - x) / x) * 100
Percentage increase = ((√(169/100) * x - x) / x) * 100
Percentage increase = ((13/10 * x - x) / x) * 100
Percentage increase = ((3/10 * x) / x) * 100
Percentage increase = (3/10) * 100
Percentage increase = 30%

Therefore, the side of the square increases by 30%. Hence, the correct answer is option 'B'.