A' and 'B' can do a piece of work in 12 days, 'B' and 'C' can do in 15 days, 'C' and 'A' can do in 20 days. How many days 'A' alone will finish the same work?

Question

Accepted Answer

To determine how many days 'A' alone will take to finish the work, we need to analyze the information given about the combined work rates of 'A', 'B', and 'C'.

1. **Combined Work Rates:**
   - 'A' and 'B' together can complete the work in 12 days. Therefore, their combined work rate is $ \frac{1}{12} $ 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's denote the work rates of 'A', 'B', and 'C' as $ a $, $ b $, and $ c $ respectively. We can set up the following equations based on the given information:
   - $ a + b = \frac{1}{12} $
   - $ b + c = \frac{1}{15} $
   - $ c + a = \frac{1}{20} $

3. **Solving the Equations:**
   To find the individual work rates, we add all three equations:
   $$
   (a + b) + (b + c) + (c + a) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}
   $$
   Simplifying, we get:
   $$
   2a + 2b + 2c = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}
   $$
   $$
   2(a + b + c) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}
   $$

Finding a common denominator for the fractions, we have:
   $$
   \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}
   $$
   $$
   \frac{1}{12} + \frac{1}{15} + \frac{1}{20} = \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5}
   $$

Therefore:
   $$
   2(a + b + c) = \frac{1}{5}
   $$
   $$
   a + b + c = \frac{1}{10}
   $$

4. **Finding 'A's Work Rate:**
   Now, we can find 'A's work rate by substituting back into the equations:
   $$
   a = (a + b + c) - (b + c) = \frac{1}{10} - \frac{1}{15}
   $$

Converting to a common denominator:
   $$
   \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}
   $$
   $$
   a = \frac{3}{30} - \frac{2}{30} = \frac{1}{30}
   $$

Therefore, 'A' alone can complete the work in 30 days.

5. **Conclusion:**
   - **Option 1 (30 days)** is correct because we have calculated that 'A' alone can complete the work in 30 days.
   - **Option 2 (20 days)** is incorrect because it does not match the calculated work rate of 'A'.
   - **Option 3 (60 days)** is incorrect because it overestimates the time 'A' would take.
   - **Option 4 (40 days)** is incorrect because it also overestimates the time 'A' would take.

Thus, the correct answer is Option 1: 30 days.