20. \( \frac{\left(\frac{12}{22}+\frac{1}{2}\right)+\left(\frac{3}{5}-\frac{9}{11}\right)}{\left(\frac{1}{6}+\frac{2}{3}\right)\left(\frac{1}{10}-\frac{2}{5}\right)}= \)

To solve this problem, we need to follow the order of operations, which is parentheses first, then exponents (if any), followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). This is often remembered by the acronym PEMDAS. Let's start by simplifying the expressions inside the parentheses: 1. Simplify the numerator: \[ \frac{12}{22} + \frac{1}{2} = \frac{6}{11} + \frac{1}{2} \] To add these fractions, we need a common denominator. The least common denominator (LCD) for 11 and 2 is 22. So we convert both fractions to have a denominator of 22: \[ \frac{6}{11} = \frac{6 \times 2}{11 \times 2} = \frac{12}{22} \] \[ \frac{1}{2} = \frac{1 \times 11}{2 \times 11} = \frac{11}{22} \] Now we can add them: \[ \frac{12}{22} + \frac{11}{22} = \frac{23}{22} \] 2. Simplify the second part of the numerator: \[ \frac{3}{5} - \frac{9}{11} \] Again, we need a common denominator. The LCD for 5 and 11 is 55. So we convert both fractions to have a denominator of 55: \[ \frac{3}{5} = \frac{3 \times 11}{5 \times 11} = \frac{33}{55} \] \[ \frac{9}{11} = \frac{9 \times 5}{11 \times 5} = \frac{45}{55} \] Now we can subtract them: \[ \frac{33}{55} - \frac{45}{55} = \frac{-12}{55} \] 3. Simplify the denominator: \[ \frac{1}{6} + \frac{2}{3} \] The LCD for 6 and 3 is 6. So we convert the second fraction to have a denominator of 6: \[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \] Now we can add them: \[ \frac{1}{6} + \frac{4}{6} = \frac{5}{6} \] 4. Simplify the second part of the denominator: \[ \frac{1}{10} - \frac{2}{5} \] The LCD for 10 and 5 is 10. So we convert the second fraction to have a denominator of 10: \[ \frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} \] Now we can subtract them: \[ \frac{1}{10} - \frac{4}{10} = \frac{-3}{10} \] Now we have the simplified expressions for the numerator and the denominator: \[ \text{Numerator: } \frac{23}{22} + \frac{-12}{55} \] \[ \text{Denominator: } \frac{5}{6} \times \frac{-3}{10} \] To add the fractions in the numerator, we need a common denominator. The LCD for 22 and 55 is 110. So we convert both fractions to have a denominator of 110: \[ \frac{23}{22} = \frac{23 \times 5}{22 \times 5} = \frac{115}{110} \] \[ \frac{-12}{55} = \frac{-12 \times 2}{55 \times 2} = \frac{-24}{110} \] Now we can add them: \[ \frac{115}{110} + \frac{-24}{110} = \frac{91}{110} \] For the denominator, we can multiply the fractions directly: \[ \frac{5}{6} \times \frac{-3}{10} = \frac{5 \times -3}{6 \times 10} = \frac{-15}{60} \] Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15: \[ \frac{-15}{60} = \frac{-1}{4} \] Now we have the simplified numerator and denominator: \[ \text{Numerator: } \frac{91}{110} \] \[ \text{Denominator: } \frac{-1}{4} \] Finally, we divide the numerator by