1. **State the problem:** We are given a right triangle with legs 32 and 60, hypotenuse 68, and we need to find $\tan B$ where $B$ is the angle opposite the leg of length 32. 2. **Recall the definition of tangent:** For an angle in a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$. 3. **Identify sides relative to angle B:** Opposite side to angle $B$ is 32, adjacent side to angle $B$ is 60. 4. **Calculate $\tan B$:** $$\tan B = \frac{32}{60}$$ 5. **Simplify the fraction:** $$\tan B = \frac{\cancel{32}}{\cancel{60}} = \frac{16}{30} = \frac{8}{15}$$ 6. **Final answer:** $$\tan B = \frac{8}{15}$$