1. **State the problem:** We need to simplify the expression $$\frac{\frac{5}{6} \cdot \frac{7}{5}}{\frac{5}{6} \cdot \frac{7}{5} + \frac{1}{6} \cdot \frac{1}{7}}$$ and write it as a quotient of integers in lowest terms. 2. **Multiply the fractions in numerator and denominator:** Numerator: $$\frac{5}{6} \cdot \frac{7}{5} = \frac{5 \times 7}{6 \times 5} = \frac{35}{30}$$ Denominator: $$\frac{5}{6} \cdot \frac{7}{5} + \frac{1}{6} \cdot \frac{1}{7} = \frac{35}{30} + \frac{1}{42}$$ 3. **Simplify numerator:** $$\frac{35}{30} = \frac{\cancel{35}}{\cancel{30}} = \frac{7}{6}$$ (dividing numerator and denominator by 5) 4. **Find common denominator for denominator terms:** The denominators are 30 and 42. The least common denominator (LCD) is 210. Convert each fraction: $$\frac{35}{30} = \frac{35 \times 7}{30 \times 7} = \frac{245}{210}$$ $$\frac{1}{42} = \frac{1 \times 5}{42 \times 5} = \frac{5}{210}$$ 5. **Add the fractions in denominator:** $$\frac{245}{210} + \frac{5}{210} = \frac{245 + 5}{210} = \frac{250}{210}$$ 6. **Rewrite the entire expression:** $$\frac{\frac{7}{6}}{\frac{250}{210}} = \frac{7}{6} \div \frac{250}{210} = \frac{7}{6} \times \frac{210}{250}$$ 7. **Multiply and simplify:** $$\frac{7 \times 210}{6 \times 250} = \frac{1470}{1500}$$ Divide numerator and denominator by 30: $$\frac{\cancel{1470}^{49}}{\cancel{1500}^{50}} = \frac{49}{50}$$ 8. **Final answer:** The expression simplified to lowest terms is $$\boxed{\frac{49}{50}}$$.