1. **State the problem:** Add the mixed numbers $6 \frac{12}{13}$ and $9 \frac{4}{13}$.\n\n2. **Recall the formula:** To add mixed numbers, add the whole numbers and the fractions separately: $$a \frac{b}{c} + d \frac{e}{c} = (a + d) + \frac{b + e}{c}$$\n\n3. **Add the whole numbers:** $$6 + 9 = 15$$\n\n4. **Add the fractions:** $$\frac{12}{13} + \frac{4}{13} = \frac{12 + 4}{13} = \frac{16}{13}$$\n\n5. **Simplify the improper fraction:** $$\frac{16}{13} = 1 \frac{3}{13}$$ because $16 = 13 + 3$.\n\n6. **Combine the whole numbers:** $$15 + 1 \frac{3}{13} = 16 \frac{3}{13}$$\n\n**Final answer:** $$16 \frac{3}{13}$$