A alone can complete a work in 6 days and B alone can complete the same work in 8 days. In how many days both A and B together can complete the same work?

Options

A. `(48)/(13)` days
B. `(35)/(13)` days
C. `(26)/(7)` days
D. `(24)/(7)` days

Answer

To find out how many days A and B together can complete the work, we can use the formula for combined work rates. Here’s the step-by-step solution:

### Step 1: Determine the work rates of A and B
- A can complete the work in 6 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{6} \text{ (work per day)} \]
- B can complete the work in 8 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{8} \text{ (work per day)} \]

### Step 2: Combine the work rates
To find the combined work rate of A and B working together, we add their individual work rates:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} = \frac{1}{6} + \frac{1}{8} \]

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

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

### Final Answer
Thus, A and B together can complete the work in \(\frac{24}{7}\) days.

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