Three rectangles share a common width. The first rectangle has area 25, the third has area 32. The combined lengths of the first and second is 8, and the second and third is 9. Find the width.

Let $w$ be the common width of all three rectangles.
Let $l_1$ be the length of the first (leftmost) rectangle.
Let $l_2$ be the length of the second (shaded) rectangle.
Let $l_3$ be the length of the third (rightmost) rectangle.

The area of the first rectangle is $25 \text{ cm}^2$:

The area of the third rectangle is $32 \text{ cm}^2$:

The combined length of the first and second rectangles is $8 \text{ cm}$:

The combined length of the second and third rectangles is $9 \text{ cm}$:

From (1): $l_1 = \frac{25}{w}$
From (2): $l_3 = \frac{32}{w}$

Substitute $l_1$ into (3):

Substitute $l_3$ into (4):

From (5), express $l_2$:

Substitute this expression for $l_2$ into (6):

Subtract 8 from both sides:

Multiply by $w$:

So, the common width is $w = 7 \text{ cm}$.

Substitute $w = 7$ into the expression for $l_2$ from Step 5:

The area of the shaded region is $l_2 \times w$:

The area of the shaded region is $\boxed{\text{31 cm}^2}$.