A alone can complete a task in 3 days and B alone can complete the same task in 6 days. In how manny days can A and B complete it together?

Options

A. 2
B. 6
C. 1
D. 3

Answer

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

### Step 1: Determine the work rates of A and B
- A can complete the task in 3 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ task}}{3 \text{ days}} = \frac{1}{3} \text{ tasks per day} \]

- B can complete the task in 6 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1 \text{ task}}{6 \text{ days}} = \frac{1}{6} \text{ tasks per day} \]

### Step 2: Add the work rates of A and B
To find the combined work rate of A and B, we add their individual work rates:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} = \frac{1}{3} + \frac{1}{6} \]

### Step 3: Find a common denominator
To add the fractions, we need a common denominator. The least common multiple (LCM) of 3 and 6 is 6. We can rewrite \(\frac{1}{3}\) as \(\frac{2}{6}\):
\[ \frac{1}{3} = \frac{2}{6} \]
Now we can add:
\[ \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

### Step 4: Calculate the time taken to complete the task together
The combined work rate of A and B is \(\frac{1}{2}\) tasks per day. This means together they can complete \(\frac{1}{2}\) of the task in one day. To find out how many days it takes them to complete 1 whole task, we take the reciprocal of the combined work rate:
\[ \text{Time taken} = \frac{1 \text{ task}}{\frac{1}{2} \text{ tasks per day}} = 2 \text{ days} \]

### Final Answer
A and B together can complete the task in **2 days**.

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