To solve the problem step by step, let's denote the work done by A, B, and C individually.
1. **Understanding the Work Rates**:
- A and B together can complete the task in 10 days. Therefore, their combined work rate is \( \frac{1}{10} \) of the work per day.
- B and C together can complete the task in 15 days. Hence, their combined work rate is \( \frac{1}{15} \) of the work per day.
- C and A together can complete the task in 12 days. Thus, their combined work rate is \( \frac{1}{12} \) of the work per day.
2. **Setting Up the Equations**:
- Let the work rates of A, B, and C be \( a \), \( b \), and \( c \) respectively.
- From the information given, we can write the following equations:
\[
a + b = \frac{1}{10} \quad \text{(1)}
\]
\[
b + c = \frac{1}{15} \quad \text{(2)}
\]
\[
c + a = \frac{1}{12} \quad \text{(3)}
\]
3. **Adding the Equations**:
- Adding equations (1), (2), and (3):
\[
(a + b) + (b + c) + (c + a) = \frac{1}{10} + \frac{1}{15} + \frac{1}{12}
\]
- This simplifies to:
\[
2a + 2b + 2c = \frac{1}{10} + \frac{1}{15} + \frac{1}{12}
\]
4. **Finding a Common Denominator**:
- The least common multiple of 10, 15, and 12 is 60. Thus, we convert the fractions:
\[
\frac{1}{10} = \frac{6}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{12} = \frac{5}{60}
\]
- Adding these gives:
\[
\frac{6}{60} + \frac{4}{60} + \frac{5}{60} = \frac{15}{60} = \frac{1}{4}
\]
5. **Solving for Total Work Rate**:
- Therefore, we have:
\[
2(a + b + c) = \frac{1}{4}
\]
- Dividing both sides by 2:
\[
a + b + c = \frac{1}{8}
\]
6. **Calculating Time Taken Together**:
- The combined work rate of A, B, and C is \( \frac{1}{8} \) of the work per day.
- To find the time taken to complete the entire work, we use the formula:
\[
\text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} = \frac{1}{\frac{1}{8}} = 8 \text{ days}
\]
Thus, A, B, and C together will take **8 days** to complete the task.
