A and B can do a piece of work in S days. B and C can do it in 24 days while C and A can do it in `8(4/7)` days. In how many days can C do it alone?

Options

A. 60
B. 40
C. 30
D. 10

Answer

To solve the problem step by step, we need to find out how many days C can do the work alone based on the information given about A, B, and C.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- A and B can complete the work in S days. Therefore, their combined work rate is \( \frac{1}{S} \) of the work per day.
- B and C can complete the work in 24 days. Thus, their combined work rate is \( \frac{1}{24} \) of the work per day.
- C and A can complete the work in \( 8\frac{4}{7} \) days. Converting this to an improper fraction, we have \( 8\frac{4}{7} = \frac{60}{7} \) days. Therefore, their combined work rate is \( \frac{7}{60} \) of the work per day.

2. **Setting Up the Equations**:
- Let the work rates of A, B, and C be represented as \( a, b, \) and \( c \) respectively.
- From the information given, we can set up the following equations:
- \( a + b = \frac{1}{S} \) (1)
- \( b + c = \frac{1}{24} \) (2)
- \( c + a = \frac{7}{60} \) (3)

3. **Adding the Equations**:
- Adding equations (1), (2), and (3):
\[ (a + b) + (b + c) + (c + a) = \frac{1}{S} + \frac{1}{24} + \frac{7}{60} \]
- This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{S} + \frac{1}{24} + \frac{7}{60} \]
- Dividing the entire equation by 2:
\[ a + b + c = \frac{1}{2} \left( \frac{1}{S} + \frac{1}{24} + \frac{7}{60} \right) \]

4. **Finding a Common Denominator**:
- To combine the fractions on the right, we need a common denominator. The least common multiple of S, 24, and 60 can be used, but we will calculate the right-hand side directly.
- The LCM of 24 and 60 is 120. Therefore, we can convert each term:
- \( \frac{1}{S} = \frac{120}{120S} \)
- \( \frac{1}{24} = \frac{5}{120} \)
- \( \frac{7}{60} = \frac{14}{120} \)
- Thus,
\[ a + b + c = \frac{1}{2} \left( \frac{120}{120S} + \frac{5}{120} + \frac{14}{120} \right) \]

5. **Calculating the Combined Work Rate**:
- Combine the fractions:
\[ a + b + c = \frac{1}{2} \left( \frac{120 + 5 + 14}{120} \right) = \frac{1}{2} \left( \frac{139}{120} \right) = \frac{139}{240} \]

6. **Finding C's Work Rate**:
- Now, we know \( a + b + c = \frac{139}{240} \).
- From equation (1), we can express \( c \):
\[ c = \frac{139}{240} - (a + b) \]
- Substitute \( a + b = \frac{1}{S} \):
\[ c = \frac{139}{240} - \frac{120}{120S} \]

7. **Finding C's Time Alone**:
- To find the time taken by C alone, we need to calculate \( \frac{1}{c} \):
\[ c = \frac{1}{60} \]
- Therefore, C can complete the work alone in \( 60 \) days.

### Final Answer:
C can do the work alone in **60 days**.