A can complete a work in 12 days and B can complete the same work in 24 days. Find the time taken by both A and B together to complete the work.

Options

A. 8 days
B. 10 days
C. 6 days
D. 9 days

Answer

To find the time taken by both A and B together to 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. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{12} \]
- B can complete the work in 24 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{24} \]

### Step 2: Calculate the combined work done by A and B in one day.
- To find the total work done by both A and B together in one day, we add their individual work rates:
\[ \text{Combined work in one day} = \frac{1}{12} + \frac{1}{24} \]
- To add these fractions, we need a common denominator. The least common multiple of 12 and 24 is 24. Therefore:
\[ \frac{1}{12} = \frac{2}{24} \]
- Now, we can add:
\[ \text{Combined work in one day} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \]

### Step 3: Calculate the time taken by A and B together to complete the work.
- If A and B together can complete \(\frac{1}{8}\) of the work in one day, then the time taken to complete the entire work is the reciprocal of the combined work rate:
\[ \text{Time taken} = \frac{1}{\frac{1}{8}} = 8 \text{ days} \]

### Final Answer:
- The time taken by both A and B together to complete the work is **8 days**.

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