By what percentage will the area of a rectangle increase if its length is increased by 30% and its width by 20%.

We denote the lengths of the sides of the rectangle through x and y.
Then the area of this rectangle will be x * y.
If the length of this rectangle is increased by 30% and the width of this rectangle is increased by 20%, the area of the resulting rectangle will be:
(x + (30/100) * x) * (y + (20/100) * y) = (x + 0.3 * x) * (y + 0.2 * y) = 1.3 * x * 1.2 * y = 1.56 * x * y.
Compared to the area of the original rectangle, the area of the resulting rectangle will increase in percentage terms by:
100 * (1.56 * x * y – x * y) / (x * y) = 100 * (0.56 * x * y) / (x * y) = 100 * 0.56 = 56%.

Answer: the area of the rectangle will increase by 56%.