1. **State the problem:** Mindy and Troy together ate 9 pieces of cake. Mindy ate 3 pieces, and Troy ate \(\frac{1}{4}\) of the total cake. We need to find the total number of pieces of cake.\n\n2. **Define variables:** Let \(T\) be the total number of pieces of cake.\n\n3. **Write the equation:** Troy ate \(\frac{1}{4}T\) pieces, Mindy ate 3 pieces, and together they ate 9 pieces. So, \n\n$$3 + \frac{1}{4}T = 9$$\n\n4. **Solve for \(T\):**\n\nSubtract 3 from both sides:\n\n$$\frac{1}{4}T = 9 - 3 = 6$$\n\nMultiply both sides by 4 to isolate \(T\):\n\n$$T = 6 \times 4 = 24$$\n\n5. **Interpretation:** The total cake had 24 pieces.\n\n**Final answer:** \(\boxed{24}\) pieces of cake in total.