A and B together can do a work in 15 days, if B alone can do that work in 20 days. Then how many days will A take to do same work alone ?

Question

Accepted Answer

To determine how many days A will take to complete the work alone, we can use the concept of work rates. Let's break down the problem:

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

2. **Work Rate of B Alone:**
   - B alone can complete the work in 20 days.
   - Therefore, B's work rate is $ \frac{1}{20} $ of the work per day.

3. **Work Rate of A Alone:**
   - Let A's work rate be $ \frac{1}{x} $ of the work per day, where $ x $ is the number of days A takes to complete the work alone.

4. **Equation for Combined Work Rate:**
   - The combined work rate of A and B is the sum of their individual work rates:
     $$
     \frac{1}{x} + \frac{1}{20} = \frac{1}{15}
     $$

5. **Solving for A's Work Rate:**
   - Rearrange the equation to solve for $ \frac{1}{x} $:
     $$
     \frac{1}{x} = \frac{1}{15} - \frac{1}{20}
     $$
   - Find a common denominator (60) to subtract the fractions:
     $$
     \frac{1}{x} = \frac{4}{60} - \frac{3}{60} = \frac{1}{60}
     $$

6. **Conclusion:**
   - Therefore, $ x = 60 $.
   - A alone will take 60 days to complete the work.

**Explanation of Other Options:**

- **Option 2 (45 days):** If A took 45 days, the equation $ \frac{1}{45} + \frac{1}{20} $ would not equal $ \frac{1}{15} $. Therefore, this option is incorrect.
  
- **Option 3 (40 days):** Similarly, if A took 40 days, the equation $ \frac{1}{40} + \frac{1}{20} $ would not equal $ \frac{1}{15} $. Thus, this option is incorrect.

- **Option 4 (30 days):** If A took 30 days, the equation $ \frac{1}{30} + \frac{1}{20} $ would not equal $ \frac{1}{15} $. Hence, this option is incorrect.

In conclusion, the correct answer is Option 1, where A alone will take 60 days to complete the work.