1. **State the problem:** Simplify the expression $\frac{2}{5} + \frac{2}{9} + \frac{2}{11} + \frac{2}{3}$. 2. **Find a common denominator:** The denominators are 5, 9, 11, and 3. The least common multiple (LCM) of these is $\text{LCM}(5,9,11,3) = 495$. 3. **Rewrite each fraction with denominator 495:** - $\frac{2}{5} = \frac{2 \times 99}{5 \times 99} = \frac{198}{495}$ - $\frac{2}{9} = \frac{2 \times 55}{9 \times 55} = \frac{110}{495}$ - $\frac{2}{11} = \frac{2 \times 45}{11 \times 45} = \frac{90}{495}$ - $\frac{2}{3} = \frac{2 \times 165}{3 \times 165} = \frac{330}{495}$ 4. **Add the numerators:** $$198 + 110 + 90 + 330 = 728$$ 5. **Write the sum:** $$\frac{728}{495}$$ 6. **Simplify the fraction if possible:** The greatest common divisor (GCD) of 728 and 495 is 1, so the fraction is already in simplest form. **Final answer:** $$\frac{728}{495}$$