1. **State the problem:** Evaluate the expression $$\left(\frac{1}{4} + \frac{2}{9}\right) \cdot \frac{6}{7}$$ and write the answer in simplest form. 2. **Add the fractions inside the parentheses:** To add $$\frac{1}{4}$$ and $$\frac{2}{9}$$, find a common denominator. The least common denominator (LCD) of 4 and 9 is 36. Convert each fraction: $$\frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}$$ $$\frac{2}{9} = \frac{2 \times 4}{9 \times 4} = \frac{8}{36}$$ Add them: $$\frac{9}{36} + \frac{8}{36} = \frac{9 + 8}{36} = \frac{17}{36}$$ 3. **Multiply the sum by $$\frac{6}{7}$$:** $$\frac{17}{36} \cdot \frac{6}{7} = \frac{17 \times 6}{36 \times 7} = \frac{102}{252}$$ 4. **Simplify the fraction $$\frac{102}{252}$$:** Find the greatest common divisor (GCD) of 102 and 252. Prime factors: - 102 = 2 \times 3 \times 17 - 252 = 2^2 \times 3^2 \times 7 Common factors: 2 and 3 GCD = 2 \times 3 = 6 Divide numerator and denominator by 6: $$\frac{\cancel{6} \times 17}{\cancel{6} \times 42} = \frac{17}{42}$$ 5. **Final answer:** $$\boxed{\frac{17}{42}}$$