A and B together can do a piece of work in 20 days, while B and C together can do the same work in 15 days. If A ,B and C together can finish this work in 10 days, then find the number of days in which B alone can finish the work.

Answer

To solve the problem step by step, we will use the information given about the work done by A, B, and C together, and then find out how long B alone would take to finish the work.

### Step 1: Determine the work done by A and B together
A and B can complete the work in 20 days. Therefore, the work done by A and B in one day is:
\[ \text{Work done by A and B in one day} = \frac{1}{20} \]

### Step 2: Determine the work done by B and C together
B and C can complete the work in 15 days. Therefore, the work done by B and C in one day is:
\[ \text{Work done by B and C in one day} = \frac{1}{15} \]

### Step 3: Determine the work done by A, B, and C together
A, B, and C can complete the work in 10 days. Therefore, the work done by A, B, and C in one day is:
\[ \text{Work done by A, B, and C in one day} = \frac{1}{10} \]

### Step 4: Set up the equations
Let:
- Work done by A in one day = \( A \)
- Work done by B in one day = \( B \)
- Work done by C in one day = \( C \)

From the information we have, we can write the following equations:
1. \( A + B = \frac{1}{20} \) (Equation 1)
2. \( B + C = \frac{1}{15} \) (Equation 2)
3. \( A + B + C = \frac{1}{10} \) (Equation 3)

### Step 5: Solve the equations
We can use Equations 1 and 3 to find A.

From Equation 3:
\[ A + B + C = \frac{1}{10} \]
Substituting \( A + B \) from Equation 1 into this:
\[ \frac{1}{20} + C = \frac{1}{10} \]
To isolate C, we can subtract \( \frac{1}{20} \) from both sides:
\[ C = \frac{1}{10} - \frac{1}{20} \]
Finding a common denominator (20):
\[ C = \frac{2}{20} - \frac{1}{20} = \frac{1}{20} \]

Now we have \( C = \frac{1}{20} \).

### Step 6: Substitute C back to find A and B
Now substitute \( C \) back into Equation 2:
\[ B + \frac{1}{20} = \frac{1}{15} \]
Subtract \( \frac{1}{20} \) from both sides:
\[ B = \frac{1}{15} - \frac{1}{20} \]
Finding a common denominator (60):
\[ B = \frac{4}{60} - \frac{3}{60} = \frac{1}{60} \]

### Step 7: Find A
Now substitute \( B \) back into Equation 1:
\[ A + \frac{1}{60} = \frac{1}{20} \]
Subtract \( \frac{1}{60} \) from both sides:
\[ A = \frac{1}{20} - \frac{1}{60} \]
Finding a common denominator (60):
\[ A = \frac{3}{60} - \frac{1}{60} = \frac{2}{60} = \frac{1}{30} \]

### Step 8: Conclusion
Now we have:
- \( A = \frac{1}{30} \)
- \( B = \frac{1}{60} \)
- \( C = \frac{1}{20} \)

Thus, B alone can finish the work in:
\[ \text{Days taken by B} = \frac{1}{B} = 60 \text{ days} \]

### Final Answer
B can finish the work alone in **60 days**.