A alone con finish a piece of work in 12 days and B alone can do it in 15 days. If both of them work at it together, how much time will they take to finish it ?

Answer

To solve the problem of how long A and B will take to finish the work together, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the work rates of A and B:**
- A can finish the work in 12 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{12} \]
- B can finish the work in 15 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{15} \]

2. **Calculate the combined work rate of A and B:**
- To find the total work done by A and B together in one day, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{15} \]

3. **Find a common denominator to add the fractions:**
- The least common multiple (LCM) of 12 and 15 is 60. We convert the fractions:
\[ \frac{1}{12} = \frac{5}{60} \quad \text{(since } 12 \times 5 = 60\text{)} \]
\[ \frac{1}{15} = \frac{4}{60} \quad \text{(since } 15 \times 4 = 60\text{)} \]

4. **Add the fractions:**
- Now we can add the two fractions:
\[ \text{Combined work rate} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} \]
- Simplifying this gives:
\[ \frac{9}{60} = \frac{3}{20} \]

5. **Calculate the time taken to finish the work together:**
- The time taken to complete the work when both A and B work together is the reciprocal of their combined work rate:
\[ \text{Time taken} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{3}{20}} = \frac{20}{3} \]

6. **Convert the time into days:**
- The fraction \(\frac{20}{3}\) can be expressed as a mixed number:
\[ \frac{20}{3} = 6 \frac{2}{3} \text{ days} \]

### Final Answer:
If A and B work together, they will take \(6 \frac{2}{3}\) days to finish the work.