In how many days will B alone complete the work?
I. A and B together can complete the work in 8 days. II. B and C together can complete the work in 10 days.
III. A and C together can complete the work in 12 days

Options

A. Only I and II
B. Only II and II
C. I, II and III
D. Data is insufficient

Answer

To determine how many days B alone will take to complete the work, we can analyze the information given in the statements step by step.

### Step 1: Understand the Work Rates
Let's denote:
- The total work as 1 unit.
- The rate of work done by A as \( A \) (units of work per day).
- The rate of work done by B as \( B \) (units of work per day).
- The rate of work done by C as \( C \) (units of work per day).

### Step 2: Analyze Statement I
From Statement I, we know:
- A and B together can complete the work in 8 days.
- Therefore, their combined work rate is:
\[ A + B = \frac{1}{8} \text{ (units of work per day)} \]

### Step 3: Analyze Statement II
From Statement II, we know:
- B and C together can complete the work in 10 days.
- Therefore, their combined work rate is:
\[ B + C = \frac{1}{10} \text{ (units of work per day)} \]

### Step 4: Analyze Statement III
From Statement III, we know:
- A and C together can complete the work in 12 days.
- Therefore, their combined work rate is:
\[ A + C = \frac{1}{12} \text{ (units of work per day)} \]

### Step 5: Set Up the Equations
Now we have three equations:
1. \( A + B = \frac{1}{8} \)
2. \( B + C = \frac{1}{10} \)
3. \( A + C = \frac{1}{12} \)

### Step 6: Solve the Equations
To find the individual work rates, we can manipulate these equations.

- From equations (1) and (2), we can express \( A \) and \( C \) in terms of \( B \):
- From (1): \( A = \frac{1}{8} - B \)
- From (2): \( C = \frac{1}{10} - B \)

Substituting these into equation (3):
\[ \left(\frac{1}{8} - B\right) + \left(\frac{1}{10} - B\right) = \frac{1}{12} \]

### Step 7: Combine and Solve
Combining the left side:
\[ \frac{1}{8} + \frac{1}{10} - 2B = \frac{1}{12} \]

Finding a common denominator (which is 120):
\[ \frac{15}{120} + \frac{12}{120} - 2B = \frac{10}{120} \]
\[ \frac{27}{120} - 2B = \frac{10}{120} \]

Rearranging gives:
\[ 2B = \frac{27}{120} - \frac{10}{120} = \frac{17}{120} \]
\[ B = \frac{17}{240} \]

### Step 8: Calculate Days for B Alone
To find the number of days B alone will take to complete the work:
\[ \text{Days for B} = \frac{1}{B} = \frac{1}{\frac{17}{240}} = \frac{240}{17} \approx 14.12 \text{ days} \]

Thus, B alone will complete the work in approximately 14.12 days.

### Conclusion
The answer is that B alone will complete the work in approximately 14.12 days.