1 5/11: 1 1/11-1 1/9* 1 1/5-7 5/7

Let's simplify the expression step by step. 1. **Convert all mixed numbers to improper fractions**: - \(1 \frac{5}{11} = \frac{11 \cdot 1 + 5}{11} = \frac{16}{11}\) - \(1 \frac{1}{11} = \frac{11 \cdot 1 + 1}{11} = \frac{12}{11}\) - \(1 \frac{1}{9} = \frac{9 \cdot 1 + 1}{9} = \frac{10}{9}\) - \(1 \frac{1}{5} = \frac{5 \cdot 1 + 1}{5} = \frac{6}{5}\) - \(7 \frac{5}{7} = \frac{7 \cdot 7 + 5}{7} = \frac{54}{7}\) Now the expression looks like: \[ \frac{16}{11} : \left(\frac{12}{11} - \frac{10}{9} \cdot \frac{6}{5} - \frac{54}{7}\right) \] 2. **Calculate \( \frac{10}{9} \cdot \frac{6}{5} \)**: \[ \frac{10}{9} \cdot \frac{6}{5} = \frac{10 \cdot 6}{9 \cdot 5} = \frac{60}{45} = \frac{4}{3} \] 3. **Calculate \( \frac{12}{11} - \frac{4}{3} - \frac{54}{7} \)**. We need a common denominator to subtract these fractions. The least common multiple of \(11\), \(3\), and \(7\) is \(231\). - Convert each fraction to have a denominator of \(231\): - \(\frac{12}{11} = \frac{12 \cdot 21}{11 \cdot 21} = \frac{252}{231}\) - \(\frac{4}{3} = \frac{4 \cdot 77}{3 \cdot 77} = \frac{308}{231}\) - \(\frac{54}{7} = \frac{54 \cdot 33}{7 \cdot 33} = \frac{1782}{231}\) 4. **Now substitute the fractions in**: \[ \frac{252}{231} - \frac{308}{231} - \frac{1782}{231} = \frac{252 - 308 - 1782}{231} = \frac{-1838}{231} \] 5. **Now plug that back into the original equation**: \[ \frac{16}{11} : \frac{-1838}{231} \] This is equivalent to multiplying by the reciprocal: \[ \frac{16}{11} \cdot \frac{231}{-1838} \] 6. **Multiply the fractions**: \[ \frac{16 \cdot 231}{11 \cdot -1838} \] 7. **Calculate \(16 \cdot 231\)** and \(11 \cdot -1838\)**: - \(16 \cdot 231 = 3696\) - \(11 \cdot -1838 = -20218\) So we have: \[ \frac{3696}{-20218} = -\frac{3696}{20218} \] 8. **Now simplify if possible**: The greatest common divisor (GCD) of \(3696\) and \(20218\) can be found, but for simplicity, we note that they do not divide evenly by any common factor easily. Thus the final answer is: \[ -\frac{3696}{20218} \] This is the simplified solution for the original expression.