Volume $ A$ equals one fourth of the sum of the volumes $ B$ and $ C$, while volume $ B$ equals one sixth of the sum of the volumes $ C$ and $ A$.
Find the ratio of the volume $ C$ to the sum of the volumes $ A$ and $ B$.
1. We start with the given equations:
   \[
   A = \frac{B + C}{4}
   \]
   \[
   B = \frac{C + A}{6}
   \]

2. Rearrange the first equation to express \(C\) in terms of \(A\) and \(B\):
   \[
   4A = B + C \implies C = 4A - B
   \]

3. Rearrange the second equation to express \(C\) in terms of \(A\) and \(B\):
   \[
   6B = C + A \implies C = 6B - A
   \]

4. Now we have two expressions for \(C\):
   \[
   C = 4A - B
   \]
   \[
   C = 6B - A
   \]

5. Set the two expressions for \(C\) equal to each other:
   \[
   4A - B = 6B - A
   \]

6. Solve for \(A\) in terms of \(B\):
   \[
   4A + A = 6B + B \implies 5A = 7B \implies A = \frac{7}{5}B
   \]

7. Substitute \(A = \frac{7}{5}B\) back into one of the original equations to find \(C\) in terms of \(B\):
   \[
   C = 4A - B = 4\left(\frac{7}{5}B\right) - B = \frac{28}{5}B - B = \frac{28}{5}B - \frac{5}{5}B = \frac{23}{5}B
   \]

8. We need to find the ratio of \(C\) to \(A + B\):
   \[
   A + B = \frac{7}{5}B + B = \frac{7}{5}B + \frac{5}{5}B = \frac{12}{5}B
   \]

9. Therefore, the ratio of \(C\) to \(A + B\) is:
   \[
   \frac{C}{A + B} = \frac{\frac{23}{5}B}{\frac{12}{5}B} = \frac{23}{12}
   \]

The final answer is \(\boxed{\frac{23}{12}}\).