A can do a certain work in the same time in which B and C together can do it. If A and B together could do it in 10 days and C alone in 50 days, then B alone could do it in?

To solve this problem, we need to determine the time B alone would take to complete the work. Let’s define the variables and work through the problem step-by-step.

### Step-by-step Solution:

1. **Define Variables:**
   - Let \( A \) take \( x \) days to complete the work alone.
   - Let \( B \) take \( y \) days to complete the work alone.
   - Let \( C \) take \( z = 50 \) days to complete the work alone (as given).

2. **Understanding the Given Information:**
   - \( A \) can do the work in the same time in which \( B \) and \( C \) together can do it, i.e., \( \frac{1}{x} = \frac{1}{y} + \frac{1}{z} \).
   - \( A \) and \( B \) together can do the work in 10 days, i.e., \( \frac{1}{x} + \frac{1}{y} = \frac{1}{10} \).
   - \( C \) alone takes 50 days, so \( \frac{1}{z} = \frac{1}{50} \).

3. **Set Up Equations:**
   - From the third point, we know \( z = 50 \), so \( \frac{1}{z} = \frac{1}{50} \).
   - Substitute \( z = 50 \) into the first equation:  
     \[
     \frac{1}{x} = \frac{1}{y} + \frac{1}{50}
     \]
   - Also, we have the equation for \( A \) and \( B \) together:
     \[
     \frac{1}{x} + \frac{1}{y} = \frac{1}{10}
     \]

4. **Find \( \frac{1}{x} \):**
   - From \( \frac{1}{x} = \frac{1}{y} + \frac{1}{50} \):
     \[
     \frac{1}{x} = \frac{1}{y} + \frac{1}{50}
     \]
   - Substitute \( \frac{1}{x} \) in the second equation:
     \[
     \left(\frac{1}{y} + \frac{1}{50}\right) + \frac{1}{y} = \frac{1}{10}
     \]
   - Simplify the equation:
     \[
     \frac{2}{y} + \frac{1}{50} = \frac{1}{10}
     \]

5. **Solve for \( y \):**
   - Subtract \( \frac{1}{50} \) from both sides:
     \[
     \frac{2}{y} = \frac{1}{10} - \frac{1}{50}
     \]
   - Find a common denominator for the right-hand side:
     \[
     \frac{2}{y} = \frac{5}{50} - \frac{1}{50} = \frac{4}{50}
     \]
     \[
     \frac{2}{y} = \frac{2}{25}
     \]
   - Cross-multiply to solve for \( y \):
     \[
     2 \times 25 = 2y
     \]
     \[
     50 = 2y
     \]
     \[
     y = 25
     \]

### Conclusion:
\( B \) alone could complete the work in **25 days**.

Would you like more details or have any questions?

### Related Questions:
1. How long would it take for A alone to complete the work?
2. If all three, A, B, and C, work together, how many days would they need to complete the work?
3. If A and C work together, how long would it take them to complete the work?
4. How does the efficiency of each worker compare?
5. What is the total amount of work done by each worker if they all work together for a day?

**Tip:** To solve work problems, it’s often helpful to think in terms of rates of work per day and use the formula \(\text{Work} = \text{Rate} \times \text{Time}\).

Discover how to determine the time B alone takes to complete a work problem involving A, B, and C's work rates. Step-by-step solution using rate of work formulas and simultaneous work concepts. Suitable for high school students learning work rate problems.

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