1. **State the problem:** We need to solve the expression $$\frac{1}{6} + \frac{11}{15} \div \frac{5}{2}$$. 2. **Recall the order of operations:** Division and multiplication are performed before addition. 3. **Rewrite the division as multiplication by the reciprocal:** $$\frac{11}{15} \div \frac{5}{2} = \frac{11}{15} \times \frac{2}{5}$$ 4. **Multiply the fractions:** $$\frac{11}{15} \times \frac{2}{5} = \frac{11 \times 2}{15 \times 5} = \frac{22}{75}$$ 5. **Now add the fractions:** $$\frac{1}{6} + \frac{22}{75}$$ 6. **Find the least common denominator (LCD):** The denominators are 6 and 75. Prime factors: - 6 = 2 \times 3 - 75 = 3 \times 5^2 LCD = 2 \times 3 \times 5^2 = 150 7. **Convert each fraction to have denominator 150:** $$\frac{1}{6} = \frac{1 \times 25}{6 \times 25} = \frac{25}{150}$$ $$\frac{22}{75} = \frac{22 \times 2}{75 \times 2} = \frac{44}{150}$$ 8. **Add the fractions:** $$\frac{25}{150} + \frac{44}{150} = \frac{25 + 44}{150} = \frac{69}{150}$$ 9. **Simplify the fraction:** The greatest common divisor (GCD) of 69 and 150 is 3. $$\frac{69}{150} = \frac{\cancel{3} \times 23}{\cancel{3} \times 50} = \frac{23}{50}$$ **Final answer:** $$\frac{23}{50}$$