A and B can do a work in 10 days. B and C can do the same work in 15 days. C and A can complete the same work in 20 days. In how many days can C alone complete the work ?

Question

Accepted Answer

To determine how many days C alone can complete the work, we need to analyze the given information and use it to find the individual work rates of A, B, and C.

1. **Combined Work Rates:**
   - A and B together can complete the work in 10 days. Therefore, their combined work rate is $ \frac{1}{10} $ of the work per day.
   - B and C together can complete the work in 15 days. Therefore, their combined work rate is $ \frac{1}{15} $ of the work per day.
   - C and A together can complete the work in 20 days. Therefore, their combined work rate is $ \frac{1}{20} $ of the work per day.

2. **Setting Up Equations:**
   Let the work rates of A, B, and C be $ a $, $ b $, and $ c $ respectively. We can set up the following equations based on the given information:
   - $ a + b = \frac{1}{10} $
   - $ b + c = \frac{1}{15} $
   - $ c + a = \frac{1}{20} $

3. **Solving the Equations:**
   To find the individual work rate of C, we can add all three equations:
   $$
   (a + b) + (b + c) + (c + a) = \frac{1}{10} + \frac{1}{15} + \frac{1}{20}
   $$
   Simplifying, we get:
   $$
   2a + 2b + 2c = \frac{1}{10} + \frac{1}{15} + \frac{1}{20}
   $$
   $$
   2(a + b + c) = \frac{6 + 4 + 3}{60} = \frac{13}{60}
   $$
   $$
   a + b + c = \frac{13}{120}
   $$

Now, subtract each of the original equations from this result to find the individual work rates:
   - Subtract $ a + b = \frac{1}{10} $ from $ a + b + c = \frac{13}{120} $:
     $$
     c = \frac{13}{120} - \frac{1}{10} = \frac{13}{120} - \frac{12}{120} = \frac{1}{120}
     $$

4. **Conclusion:**
   The work rate of C is $ \frac{1}{120} $ of the work per day, meaning C alone can complete the work in 120 days.

Therefore, the correct answer is Option 1: 120 days.

**Explanation of Incorrect Options:**
- **Option 2 (118 days):** This option is incorrect because the calculations show that C's work rate is $ \frac{1}{120} $, not $ \frac{1}{118} $.
- **Option 3 (115 days):** This option is incorrect because it does not match the calculated work rate of C.
- **Option 4 (110 days):** This option is incorrect as it also does not align with the derived work rate of C.

The calculations clearly support Option 1 as the correct answer.