The perimeter of a square and a rectangle is the same. If the rectangle is 12 cm by 10 cm, then by what percentage is the area of the square more than that of the rectangle?

### Concept & Percentage Change First, find the common perimeter, use it to find the side of the square, and compare their areas. The percentage increase is calculated as: $$ \text{Percentage Increase} = \frac{\text{Area of Square} - \text{Area of Rectangle}}{\text{Area of Rectangle}} \times 100 $$ ### Step-by-Step Solution - **Given**: Rectangle dimensions are $12$ cm and $10$ cm. - Calculate Rectangle Perimeter: $2 \times (12 + 10) = 2 \times 22 = 44$ cm. - Calculate Rectangle Area: $12 \times 10 = 120$ sq. cm. - The square has the same perimeter: $44$ cm. - Find the side of the square: $44 / 4 = 11$ cm. - Calculate Square Area: $11^2 = 121$ sq. cm. - Difference in area: $121 - 120 = 1$ sq. cm. - Percentage by which the square's area is more: $(1 / 120) \times 100 = 5 / 6 \%$. - Since $5/6$ is not among the given options, the correct choice is "None of these". ### Exam Strategy & Shortcut When a square and a rectangle have the same perimeter, the square always has the maximum area. Knowing the numbers (12, 10 -> perimeter 44 -> square side 11), the areas are 120 and 121. The difference is 1 over the base of 120. $1/120$ as a fraction is clearly less than 1%, so you can instantly scan the options, see they are all too large, and choose "None of these". ### Common Pitfall Calculating the difference (1) and mistakenly selecting option (b) "1", confusing the absolute difference with the percentage difference. ### Final Answer Therefore, the correct answer is **None of these**.