A can build a wall in 15 days and B can build it in 10 days, while C can completely demolish the wall in 12 days. If they start working at the same time, in how many days will the work be completed.

Answer

To solve the problem of how many days A, B, and C will take to complete the work together, we can follow these steps:

### Step 1: Determine the work done by A, B, and C in one day.
- A can build the wall in 15 days, so A's work in one day is \( \frac{1}{15} \) of the wall.
- B can build the wall in 10 days, so B's work in one day is \( \frac{1}{10} \) of the wall.
- C can demolish the wall in 12 days, so C's work in one day is \( -\frac{1}{12} \) of the wall (negative because C is demolishing).

### Step 2: Find the combined work done by A, B, and C in one day.
To find the total work done in one day by A, B, and C together, we add their individual contributions:
\[ \text{Total work in one day} = \frac{1}{15} + \frac{1}{10} - \frac{1}{12} \]

### Step 3: Find a common denominator.
The least common multiple (LCM) of 15, 10, and 12 is 60. We will convert each fraction to have a denominator of 60:
- \( \frac{1}{15} = \frac{4}{60} \)
- \( \frac{1}{10} = \frac{6}{60} \)
- \( -\frac{1}{12} = -\frac{5}{60} \)

### Step 4: Combine the fractions.
Now, we can add the fractions:
\[ \text{Total work in one day} = \frac{4}{60} + \frac{6}{60} - \frac{5}{60} = \frac{4 + 6 - 5}{60} = \frac{5}{60} = \frac{1}{12} \]

### Step 5: Calculate the total time to complete the work.
If they can complete \( \frac{1}{12} \) of the wall in one day, then to complete the entire wall (which is 1 whole unit of work), we can find the total number of days by taking the reciprocal:
\[ \text{Total days} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

### Final Answer:
Thus, A, B, and C together will complete the work in **12 days**.

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