A alone can complete a work in 12 days and B alone can complete the same work in 24 days. In how many days can A and B together complete the same work?

Options

A. 6 days
B. 4 days
C. 10 days
D. 8 days

Answer

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

### Step 1: Determine the work done by A and B individually.
- A can complete the work in 12 days.
- B can complete the work in 24 days.

### Step 2: Calculate the amount of work done by A in one day.
- Since A completes the work in 12 days, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{12} \text{ of the work} \]

### Step 3: Calculate the amount of work done by B in one day.
- Since B completes the work in 24 days, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{24} \text{ of the work} \]

### Step 4: Calculate the combined work done by A and B in one day.
- The total work done by A and B together in one day is:
\[ \text{Combined work} = \frac{1}{12} + \frac{1}{24} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 24 is 24.
\[ \frac{1}{12} = \frac{2}{24} \]
- Thus,
\[ \text{Combined work} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \text{ of the work} \]

### Step 5: Calculate the total time taken by A and B together to complete the work.
- If A and B together complete \(\frac{1}{8}\) of the work in one day, then the total time taken to complete the entire work is:
\[ \text{Total time} = \frac{1}{\frac{1}{8}} = 8 \text{ days} \]

### Final Answer:
A and B together can complete the work in **8 days**.

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