The figure below is made up of a rectangle and a square. The ratio of the area of the rectangle to the area of the square is 11 : 8. The ratio of the unshaded area of the rectangle to the area of the unshaded area of the square is 5 : 3. The length of the square is 12 cm. What is the area of the shaded part? Answer: ___

1. Calculate the area of the square.

$\text{Area of square} = 12 \times 12 = 144\text{ cm}^2$

2. Find the area of the rectangle using the given ratio.

$\text{Ratio of area of rectangle to square} = 11 : 8$
$\text{Area of rectangle} = \frac{11}{8} \times 144 = 198\text{ cm}^2$

3. Set up an equation for the unshaded areas.

Let the area of the shaded part be $S$.
$\text{Unshaded area of rectangle} = 198 - S$
$\text{Unshaded area of square} = 144 - S$
$\text{Ratio of unshaded areas} = 5 : 3$
$\frac{198 - S}{144 - S} = \frac{5}{3}$

4. Solve the equation to find the shaded area.

$3(198 - S) = 5(144 - S)$
$594 - 3S = 720 - 5S$
$2S = 720 - 594$
$2S = 126$
$S = 63\text{ cm}^2$