1. The problem is to simplify the expression \(\frac{0,(7) + 1}{0,(7) + 2} - \frac{7}{176} \div \frac{0,(81) + 1}{0,(81) + 2}\). 2. First, convert the repeating decimals to fractions. - For \(0,(7)\), the repeating decimal 0.7777... equals \(\frac{7}{9}\). - For \(0,(81)\), the repeating decimal 0.818181... equals \(\frac{81}{99} = \frac{9}{11}\). 3. Substitute these into the expression: \[ \frac{\frac{7}{9} + 1}{\frac{7}{9} + 2} - \frac{7}{176} \div \frac{\frac{9}{11} + 1}{\frac{9}{11} + 2} \] 4. Simplify the numerators and denominators inside the fractions: - \(\frac{7}{9} + 1 = \frac{7}{9} + \frac{9}{9} = \frac{16}{9}\) - \(\frac{7}{9} + 2 = \frac{7}{9} + \frac{18}{9} = \frac{25}{9}\) - \(\frac{9}{11} + 1 = \frac{9}{11} + \frac{11}{11} = \frac{20}{11}\) - \(\frac{9}{11} + 2 = \frac{9}{11} + \frac{22}{11} = \frac{31}{11}\) 5. Rewrite the expression: \[ \frac{\frac{16}{9}}{\frac{25}{9}} - \frac{7}{176} \div \frac{\frac{20}{11}}{\frac{31}{11}} \] 6. Simplify the complex fractions by multiplying numerator by reciprocal of denominator: \[ \frac{16}{9} \times \frac{9}{25} - \frac{7}{176} \div \left(\frac{20}{11} \times \frac{11}{31}\right) \] 7. Cancel common factors: - \(\frac{16}{\cancel{9}} \times \frac{\cancel{9}}{25} = \frac{16}{25}\) - \(\frac{20}{11} \times \frac{11}{31} = \frac{20}{\cancel{11}} \times \frac{\cancel{11}}{31} = \frac{20}{31}\) 8. Now the expression is: \[ \frac{16}{25} - \frac{7}{176} \div \frac{20}{31} \] 9. Division by a fraction is multiplication by its reciprocal: \[ \frac{16}{25} - \frac{7}{176} \times \frac{31}{20} \] 10. Multiply the fractions: \[ \frac{7 \times 31}{176 \times 20} = \frac{217}{3520} \] 11. Now the expression is: \[ \frac{16}{25} - \frac{217}{3520} \] 12. Find common denominator for subtraction: - The least common denominator of 25 and 3520 is 3520. - Convert \(\frac{16}{25}\) to denominator 3520: \[ \frac{16}{25} = \frac{16 \times 140.8}{25 \times 140.8} = \frac{2252.8}{3520} \] Since 140.8 is not an integer, use exact calculation: \[ 3520 \div 25 = 140.8 \] To avoid decimals, multiply numerator and denominator by 140.8: \[ \frac{16}{25} = \frac{16 \times 140.8}{25 \times 140.8} = \frac{2252.8}{3520} \] Alternatively, multiply numerator and denominator by 1408 and then divide by 10: \[ 25 \times 140.8 = 3520 \] To keep integers, multiply numerator and denominator by 1408 and then divide numerator by 10: \[ 16 \times 1408 = 22528 \] So: \[ \frac{16}{25} = \frac{22528}{35200} \] Similarly, convert \(\frac{217}{3520}\) to denominator 35200: \[ \frac{217}{3520} = \frac{217 \times 10}{3520 \times 10} = \frac{2170}{35200} \] 13. Now subtract: \[ \frac{22528}{35200} - \frac{2170}{35200} = \frac{22528 - 2170}{35200} = \frac{20358}{35200} \] 14. Simplify the fraction \(\frac{20358}{35200}\): - Both numerator and denominator are divisible by 2: \[ \frac{\cancel{2}0179}{\cancel{2}17600} = \frac{10179}{17600} \] - Check for further simplification: 10179 factors: 1+0+1+7+9=18 divisible by 3, so 10179 divisible by 3. 17600 divisible by 3? Sum digits 1+7+6+0+0=14 not divisible by 3. So no further simplification. 15. Final simplified answer: \[ \frac{10179}{17600} \] This is the simplified exact value of the original expression.