b (3 pi)/(2)-(pi)/(3)=3 (1)/(2)

To solve the equation $b \frac{3 \pi}{2} - \frac{\pi}{3} = 3 \frac{1}{2}$, we need to isolate the variable $b$.

First, let's convert the mixed number $3 \frac{1}{2}$ to an improper fraction:
$$3 \frac{1}{2} = \frac{7}{2}$$

So the equation becomes:
$$b \frac{3 \pi}{2} - \frac{\pi}{3} = \frac{7}{2}$$

Next, we need to combine the terms on the left side of the equation. To do this, we need a common denominator. The common denominator for $\frac{3 \pi}{2}$ and $\frac{\pi}{3}$ is 6.

Rewrite each term with the common denominator:
$$b \frac{3 \pi}{2} = b \cdot \frac{3 \pi}{2} \cdot \frac{3}{3} = b \cdot \frac{9 \pi}{6}$$
$$\frac{\pi}{3} = \frac{\pi}{3} \cdot \frac{2}{2} = \frac{2 \pi}{6}$$

Now the equation is:
$$b \cdot \frac{9 \pi}{6} - \frac{2 \pi}{6} = \frac{7}{2}$$

Combine the fractions on the left side:
$$\frac{9 \pi b}{6} - \frac{2 \pi}{6} = \frac{7}{2}$$
$$\frac{9 \pi b - 2 \pi}{6} = \frac{7}{2}$$

To eliminate the fraction, multiply both sides by 6:
$$9 \pi b - 2 \pi = 21$$

Now, isolate $b$ by moving the constant term to the other side:
$$9 \pi b = 21 + 2 \pi$$

Finally, divide both sides by $9 \pi$ to solve for $b$:
$$b = \frac{21 + 2 \pi}{9 \pi}$$

So the solution to the equation is:
$$b = \frac{21 + 2 \pi}{9 \pi}$$