A' can do a piece of work in 4 hours, 'B' and 'C' can do together this work in 3 hrs and 'A' and 'C' can do together in 2 hrs. In what time could 'B' do it alone ?

Question

Accepted Answer

To determine the time it would take for 'B' to complete the work alone, we need to analyze the information given and use it to find the individual work rates of 'A', 'B', and 'C'.

1. **Work Rates:**
   - 'A' can complete the work in 4 hours, so 'A's work rate is $ \frac{1}{4} $ of the work per hour.
   - 'B' and 'C' together can complete the work in 3 hours, so their combined work rate is $ \frac{1}{3} $ of the work per hour.
   - 'A' and 'C' together can complete the work in 2 hours, so their combined work rate is $ \frac{1}{2} $ of the work per hour.

2. **Setting Up Equations:**
   - Let the work rate of 'B' be $ B $ and the work rate of 'C' be $ C $.
   - From the given information, we have:
     - $ A + C = \frac{1}{2} $
     - $ B + C = \frac{1}{3} $
     - $ A = \frac{1}{4} $

3. **Solving for 'C':**
   - Substitute $ A = \frac{1}{4} $ into the equation $ A + C = \frac{1}{2} $:
     $$
     \frac{1}{4} + C = \frac{1}{2}
     $$
     $$
     C = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}
     $$

4. **Solving for 'B':**
   - Substitute $ C = \frac{1}{4} $ into the equation $ B + C = \frac{1}{3} $:
     $$
     B + \frac{1}{4} = \frac{1}{3}
     $$
     $$
     B = \frac{1}{3} - \frac{1}{4}
     $$
     - To subtract these fractions, find a common denominator (12):
     $$
     B = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}
     $$

5. **Time for 'B' to Complete the Work Alone:**
   - Since 'B's work rate is $ \frac{1}{12} $ of the work per hour, 'B' would take 12 hours to complete the work alone.

Therefore, the correct answer is Option 2: 12 hours.

**Explanation of Incorrect Options:**
- **Option 1 (10 hrs):** This option is incorrect because the calculated work rate for 'B' indicates that it would take longer than 10 hours.
- **Option 3 (8 hrs):** This option is incorrect because it underestimates the time required based on 'B's work rate.
- **Option 4 (24 hrs):** This option is incorrect because it overestimates the time required based on 'B's work rate.

The logical reasoning and calculations confirm that Option 2 is indeed the correct answer.