A complete the work in 10 days and B in 15 days. In how many days `A+B` can complete the work?

Answer

To solve the problem of how many days A and B can complete the work together, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Work Done by A and B Individually:**
- A can complete the work in 10 days.
- B can complete the work in 15 days.

2. **Calculate the Total Work:**
- We can assume the total work to be the least common multiple (LCM) of the days taken by A and B.
- The LCM of 10 and 15 is 30.
- Thus, we can say the total work is 30 units.

3. **Calculate the Efficiency of A and B:**
- Efficiency is defined as the amount of work done in one day.
- For A:
\[ \text{Efficiency of A} = \frac{\text{Total Work}}{\text{Days taken by A}} = \frac{30}{10} = 3 \text{ units/day} \]
- For B:
\[ \text{Efficiency of B} = \frac{\text{Total Work}}{\text{Days taken by B}} = \frac{30}{15} = 2 \text{ units/day} \]

4. **Calculate the Combined Efficiency of A and B:**
- Combined efficiency of A and B:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = 3 + 2 = 5 \text{ units/day} \]

5. **Calculate the Time Taken by A and B Together:**
- Time taken by A and B together to complete the work:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{30}{5} = 6 \text{ days} \]

### Final Answer:
A and B together can complete the work in **6 days**.

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