A and B separately can complete a work in 6 days and 3 days respectively. If they work together. Then in how many days will they complete the work ?

Options

A. 4 days
B. 3 days
C. 2 days
D. 5 days

Answer

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

### 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 3 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{3} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
- When A and B work together, their combined work rate is the sum of their individual work rates:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} = \frac{1}{6} + \frac{1}{3} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 3 is 6. Thus, we can rewrite \(\frac{1}{3}\) as \(\frac{2}{6}\):
\[ \text{Combined work rate} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \text{ (work per day)} \]

### Step 3: Calculate the time taken to complete the work together
- If A and B together can complete \(\frac{1}{2}\) of the work in one day, then to complete the entire work (1 unit of work), the time taken is:
\[ \text{Time taken} = \frac{1 \text{ (total work)}}{\frac{1}{2} \text{ (work per day)}} = 2 \text{ days} \]

### Final Answer
Thus, A and B together will complete the work in **2 days**.

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