Final answer:
The area of a rectangle will be less than the original if its length increases by 25% and width decreases by 25%, due to the new area being multiplied by a factor of 0.9375. However, if the length increases by 25% and the width decreases by 20%, the area remains the same. Examples with specific measurements show that the new area can be smaller, equal, but not larger, demonstrating the effect of varying percentage changes.
Step-by-step explanation:
When we alter the dimensions of a rectangle, the new area can be calculated by multiplying the new length by the new width. The original area of the rectangle is L × W (Length multiplied by Width). If the length L increases by 25%, it becomes 1.25L. If the width W decreases by 25%, it becomes 0.75W. Therefore, the new area after these changes is (1.25L) × (0.75W) = 0.9375LW, which is less than the original area since 0.9375 is less than 1.
Now, if the length is increased by 25% to 1.25L and the width is decreased by only 20% to 0.8W, the new area will be (1.25L) × (0.8W) = LW, which is exactly equal to the original area. In this case, there will be no change in the area of the rectangle. Hence, for the first scenario, the new area is less than the original, and for the second scenario, the new area is equal to the original.
To further illustrate this concept, let's consider an example where the initial dimensions of the rectangle are 4 inches by 4 inches, so the original area is 16 square inches. In the first case, after the changes, the new dimensions are 5 inches (25% greater than 4 inches) by 3 inches (25% less than 4 inches), making the area 15 square inches which is less than the original area. In the second case, the new dimensions are 5 inches by 3.2 inches (20% less than 4 inches), resulting in an area of 16 square inches, equal to the original. These examples demonstrate that whether the area becomes larger, smaller, or remains the same depends on the percentage change in both length and width.