1. **State the problem:** Simplify the expression $\frac{3v^4}{4v}$.\n\n2. **Recall the rule for dividing powers with the same base:** When dividing terms with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}.$$\n\n3. **Apply the rule to the variable part:** $$\frac{v^4}{v} = v^{4-1} = v^3.$$\n\n4. **Rewrite the expression with simplified variable:** $$\frac{3v^4}{4v} = \frac{3}{4} \times v^3.$$\n\n5. **Final simplified expression:** $$\frac{3}{4}v^3.$$\n\nThis means the original expression simplifies to three-fourths times $v$ cubed.