1. We are asked to add the fractions $\frac{16}{7}$ and $\frac{27}{15}$.\n\n2. To add fractions, we need a common denominator. The denominators are 7 and 15. The least common denominator (LCD) is the least common multiple (LCM) of 7 and 15.\n\n3. Calculate LCM of 7 and 15:\n$$\text{LCM}(7,15) = 105$$\n\n4. Convert each fraction to an equivalent fraction with denominator 105:\n$$\frac{16}{7} = \frac{16 \times 15}{7 \times 15} = \frac{240}{105}$$\n$$\frac{27}{15} = \frac{27 \times 7}{15 \times 7} = \frac{189}{105}$$\n\n5. Add the fractions now that they have the same denominator:\n$$\frac{240}{105} + \frac{189}{105} = \frac{240 + 189}{105} = \frac{429}{105}$$\n\n6. Simplify the fraction $\frac{429}{105}$. Find the greatest common divisor (GCD) of 429 and 105.\n\n7. Calculate GCD(429, 105):\n- 429 ÷ 3 = 143\n- 105 ÷ 3 = 35\nSo, GCD is 3.\n\n8. Simplify by dividing numerator and denominator by 3:\n$$\frac{\cancel{429}^ {143}}{\cancel{105}^{35}}$$\n\n9. Final simplified answer:\n$$\frac{143}{35}$$\n\nTherefore, $\frac{16}{7} + \frac{27}{15} = \frac{143}{35}$.