To solve this problem, let's set up equations based on the information provided.

Given:

1. A alone can do the work in the same time B and C together can do it.
2. A and B together can complete the work in 10 days.
3. C alone can do the work in 50 days.

Steps:

1. Define work rates:

   • Let the total work be $W = 1$ (for simplicity).
   • Let $A$’s rate = $a$, $B$’s rate = $b$, and $C$’s rate = $c$.

2. Form equations for their rates:

   • Since C can do the work in 50 days, $c = \frac{1}{50}$.
   • Since A and B together can complete the work in 10 days, $a + b = \frac{1}{10}$.

3. Using the condition involving A, B, and C:

   • A’s rate is equal to the combined rate of B and C. So, $a = b + c$.

4. Substitute $c = \frac{1}{50}$ into $a = b + c$:
   $a = b + \frac{1}{50}$

5. Substitute $a = b + \frac{1}{50}$ into $a + b = \frac{1}{10}$:
   $(b + \frac{1}{50}) + b = \frac{1}{10}$ $2b + \frac{1}{50} = \frac{1}{10}$

6. Solve for $b$:
   $2b = \frac{1}{10} - \frac{1}{50}$ Find a common denominator: $2b = \frac{5 - 1}{50} = \frac{4}{50} = \frac{2}{25}$ $b = \frac{1}{25}$

Conclusion:

B alone can complete the work in 25 days.

Answer: 25 days

Would you like more details or have any other questions?

Here are some related questions:

1. What if the time taken by C alone was different? How would that affect B's time?
2. How would the answer change if A and B together took 15 days instead of 10?
3. Can we find the time taken by A alone to do the work?
4. How would we calculate this if A, B, and C worked at different rates on different days?
5. What if B and C together took a different amount of time than given?

Tip: When solving work rate problems, it often helps to define the total work as 1 unit, then express individual rates as fractions of this unit.