A can do a piece of work in 6 days and B alone can do it in 8 days. How long would it take for both of them to finish.

Answer

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

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 6 days, so A's work rate is \( \frac{1}{6} \) of the work per day.
- B can complete the work in 8 days, so B's work rate is \( \frac{1}{8} \) of the work per day.

### Step 2: Calculate the combined work rate of A and B.
- The combined work rate of A and B is the sum of their individual work rates:
\[ \text{Combined work rate} = \frac{1}{6} + \frac{1}{8} \]

### Step 3: Find a common denominator to add the fractions.
- The least common multiple of 6 and 8 is 24. Therefore, we convert the fractions:
\[ \frac{1}{6} = \frac{4}{24} \quad \text{and} \quad \frac{1}{8} = \frac{3}{24} \]
- Now, adding these gives:
\[ \text{Combined work rate} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \]

### Step 4: Calculate the time taken to complete the work together.
- If A and B together can do \( \frac{7}{24} \) of the work in one day, then the time taken to complete the entire work (1 unit of work) is the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1}{\frac{7}{24}} = \frac{24}{7} \text{ days} \]

### Step 5: Convert the time into a mixed number.
- \( \frac{24}{7} \) can be expressed as:
\[ 3 \frac{3}{7} \text{ days} \]

### Final Answer:
A and B together can finish the work in \( 3 \frac{3}{7} \) days.

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