1. **State the problem:** Simplify the expression $$\frac{\frac{5}{6} \cdot \frac{7}{5}}{\frac{5}{6} \cdot \frac{7}{5} + \frac{1}{6} \cdot \frac{1}{7}}$$ as a quotient of integers reduced to lowest terms. 2. **Write the formula and rules:** To simplify a complex fraction, first simplify numerator and denominator separately, then divide numerator by denominator. 3. **Simplify numerator:** $$\frac{5}{6} \cdot \frac{7}{5} = \frac{5 \cdot 7}{6 \cdot 5} = \frac{\cancel{5} \cdot 7}{6 \cdot \cancel{5}} = \frac{7}{6}$$ 4. **Simplify denominator:** $$\frac{5}{6} \cdot \frac{7}{5} + \frac{1}{6} \cdot \frac{1}{7} = \frac{7}{6} + \frac{1}{42}$$ 5. **Find common denominator for denominator terms:** $$\text{LCM of } 6 \text{ and } 42 = 42$$ 6. **Rewrite terms with denominator 42:** $$\frac{7}{6} = \frac{7 \cdot 7}{6 \cdot 7} = \frac{49}{42}$$ 7. **Add the fractions:** $$\frac{49}{42} + \frac{1}{42} = \frac{49 + 1}{42} = \frac{50}{42}$$ 8. **Simplify denominator fraction:** $$\frac{50}{42} = \frac{\cancel{50} \div 2}{\cancel{42} \div 2} = \frac{25}{21}$$ 9. **Divide numerator by denominator:** $$\frac{7}{6} \div \frac{25}{21} = \frac{7}{6} \cdot \frac{21}{25} = \frac{7 \cdot 21}{6 \cdot 25}$$ 10. **Simplify multiplication:** $$\frac{7 \cdot 21}{6 \cdot 25} = \frac{7 \cdot \cancel{21}}{\cancel{6} \cdot 25} = \frac{7 \cdot 7}{1 \cdot 25} = \frac{49}{25}$$ **Final answer:** $$\frac{49}{25}$$