To solve the problem, we need to determine how long A would take to complete the job alone, given the combined work rates of A, B, and C.
### Step-by-Step Solution:
1. **Identify the Work Rates**:
- A and B can complete the job in 12 days. Therefore, their combined work rate is:
\[
\text{Work rate of A and B} = \frac{1}{12} \text{ (jobs per day)}
\]
- B and C can complete the job in 15 days. Therefore, their combined work rate is:
\[
\text{Work rate of B and C} = \frac{1}{15} \text{ (jobs per day)}
\]
- C and A can complete the job in 20 days. Therefore, their combined work rate is:
\[
\text{Work rate of C and A} = \frac{1}{20} \text{ (jobs per day)}
\]
2. **Set Up the Equations**:
- Let the work rates of A, B, and C be represented as \(a\), \(b\), and \(c\) respectively. We can write the following equations based on the combined work rates:
\[
a + b = \frac{1}{12} \quad \text{(1)}
\]
\[
b + c = \frac{1}{15} \quad \text{(2)}
\]
\[
c + a = \frac{1}{20} \quad \text{(3)}
\]
3. **Add All Three Equations**:
- Adding equations (1), (2), and (3):
\[
(a + b) + (b + c) + (c + a) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}
\]
- This simplifies to:
\[
2a + 2b + 2c = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}
\]
- Dividing the entire equation by 2 gives:
\[
a + b + c = \frac{1}{24} \text{ (4)}
\]
4. **Calculate the Right Side**:
- To find the sum of the fractions on the right side, we need to find the LCM of 12, 15, and 20:
- LCM(12, 15, 20) = 60
- Convert each fraction:
\[
\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}
\]
- Adding these gives:
\[
\frac{5 + 4 + 3}{60} = \frac{12}{60} = \frac{1}{5}
\]
- Thus, from equation (4):
\[
a + b + c = \frac{1}{5}
\]
5. **Find Individual Work Rates**:
- Now we can find the work rate of A:
- From equation (1): \(b = \frac{1}{12} - a\)
- From equation (2): \(c = \frac{1}{15} - b\)
- Substitute \(b\) in terms of \(a\) into equation (2):
\[
c = \frac{1}{15} - \left(\frac{1}{12} - a\right) = a + \frac{1}{15} - \frac{1}{12}
\]
- Substitute \(c\) into equation (3) and solve for \(a\):
\[
c + a = \frac{1}{20}
\]
- After some algebra, we can isolate \(a\) and find:
\[
a = \frac{1}{30}
\]
6. **Calculate the Time Taken by A**:
- Since \(a\) represents the work rate of A, the time taken by A to complete the job alone is:
\[
\text{Time taken by A} = \frac{1}{a} = 30 \text{ days}
\]
### Final Answer:
A would take **30 days** to complete the job alone.
