The area of a rectangle is 432 square centimeters. What would the new area be if the length of the rectangle was decreased by $10\%$ and the width of the rectangle was increased by $10\%$? Express your answer to the nearest whole number.
If the length of the rectangle is decreased by $10\%$, it will be $90\%$ of what it was. If the width is increased by $10\%$, it will be $110\%$ of what it was. The area will be $0.9 \times 1.1 = 0.99 = 99\%$ of what it was. Thus, $99\%$ of 432 is $0.99 \times 432 = 427.68$ or about $\boxed{428\text{ square centimeters}}$.