1. **Problem statement:** We need to express $\tan C$ as a fraction in simplest terms given a right triangle $CDE$ with right angle at $D$. The side opposite angle $C$ is $DE=12$, the side adjacent to angle $C$ is $CD=16$, and the hypotenuse is $CE=20$. 2. **Formula:** Recall that $\tan$ of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle: $$\tan C = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Substitute values:** $$\tan C = \frac{12}{16}$$ 4. **Simplify the fraction:** $$\tan C = \frac{\cancel{12}^{3}}{\cancel{16}^{4}} = \frac{3}{4}$$ 5. **Answer:** The simplest fractional form of $\tan C$ is $$\boxed{\frac{3}{4}}$$