Evaluate the expression
\[
  \frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) 
          + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{
        11 \left( \frac{1}{13} - \frac{1}{17} \right) 
          + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)} \, .
\]
1. Start by simplifying the numerator and the denominator separately. The given expression is:
   \[
   \frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{11 \left( \frac{1}{13} - \frac{1}{17} \right) + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)}
   \]

2. Simplify each term in the numerator:
   \[
   121 \left( \frac{1}{13} - \frac{1}{17} \right) = 121 \left( \frac{17 - 13}{13 \cdot 17} \right) = 121 \left( \frac{4}{221} \right) = \frac{484}{221}
   \]
   \[
   169 \left( \frac{1}{17} - \frac{1}{11} \right) = 169 \left( \frac{11 - 17}{17 \cdot 11} \right) = 169 \left( \frac{-6}{187} \right) = \frac{-1014}{187}
   \]
   \[
   289 \left( \frac{1}{11} - \frac{1}{13} \right) = 289 \left( \frac{13 - 11}{11 \cdot 13} \right) = 289 \left( \frac{2}{143} \right) = \frac{578}{143}
   \]

3. Simplify each term in the denominator:
   \[
   11 \left( \frac{1}{13} - \frac{1}{17} \right) = 11 \left( \frac{17 - 13}{13 \cdot 17} \right) = 11 \left( \frac{4}{221} \right) = \frac{44}{221}
   \]
   \[
   13 \left( \frac{1}{17} - \frac{1}{11} \right) = 13 \left( \frac{11 - 17}{17 \cdot 11} \right) = 13 \left( \frac{-6}{187} \right) = \frac{-78}{187}
   \]
   \[
   17 \left( \frac{1}{11} - \frac{1}{13} \right) = 17 \left( \frac{13 - 11}{11 \cdot 13} \right) = 17 \left( \frac{2}{143} \right) = \frac{34}{143}
   \]

4. Combine the simplified terms in the numerator:
   \[
   \frac{484}{221} + \frac{-1014}{187} + \frac{578}{143}
   \]

5. Combine the simplified terms in the denominator:
   \[
   \frac{44}{221} + \frac{-78}{187} + \frac{34}{143}
   \]

6. To simplify the fractions, find a common denominator for each set of fractions. For the numerator, the common denominator is \(221 \cdot 187 \cdot 143\). For the denominator, the common denominator is the same.

7. Simplify the combined fractions in the numerator and denominator:
   \[
   \frac{484 \cdot 187 \cdot 143 + (-1014) \cdot 221 \cdot 143 + 578 \cdot 221 \cdot 187}{44 \cdot 187 \cdot 143 + (-78) \cdot 221 \cdot 143 + 34 \cdot 221 \cdot 187}
   \]

8. Notice that the numerator and denominator have the same structure, and the terms will cancel out, simplifying to:
   \[
   \frac{30 \cdot \frac{4}{11} + 28 \cdot \frac{-6}{13} + 24 \cdot \frac{2}{17}}{\frac{4}{11} + \frac{-6}{13} + \frac{2}{17}}
   \]

9. Simplify the expression further:
   \[
   = 24 + \frac{6 \cdot \frac{4}{11} + 4 \cdot \frac{-6}{13}}{\frac{4}{11} + \frac{-6}{13} + \frac{2}{17}}
   \]

10. Finally, simplify to get the final result:
   \[
   = 41 + \frac{-11 \cdot \frac{4}{11} + -13 \cdot \frac{-6}{13} - 17 \cdot \frac{2}{17}}{\frac{4}{11} + \frac{-6}{13} + \frac{2}{17}}
   \]

11. The final simplified result is:
   \[
   \boxed{41}
   \]