A can knit a pair of socks in 3 days. B can knit the same thing in 6 days. If they are knitting together, in how many days will they knit two pairs of socks?
(a)4 days
(b)2 days
(c)`4 (1)/(2)` days
(d)3 days

Options

A. 4 days
B. 2 days
C. `4 (1)/(2)` days
D. 3 days

Answer

To solve the problem of how many days A and B will take to knit two pairs of socks together, we can follow these steps:

### Step 1: Determine the work done by A and B in one day.
- A can knit a pair of socks in 3 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ pair}}{3 \text{ days}} = \frac{1}{3} \text{ pairs per day} \]
- B can knit a pair of socks in 6 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1 \text{ pair}}{6 \text{ days}} = \frac{1}{6} \text{ pairs per day} \]

### Step 2: Calculate the combined work rate of A and B.
- To find the total work rate when A and B work together, we add their individual work rates:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} = \frac{1}{3} + \frac{1}{6} \]
- To add these fractions, we need a common denominator. The least common multiple of 3 and 6 is 6:
\[ \frac{1}{3} = \frac{2}{6} \]
- Now we can add:
\[ \text{Combined work rate} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \text{ pairs per day} \]

### Step 3: Determine the total work needed for 2 pairs of socks.
- Since we need to knit 2 pairs of socks, we can express this as:
\[ \text{Total work} = 2 \text{ pairs} \]

### Step 4: Calculate the time taken to knit 2 pairs of socks.
- We know the combined work rate is \(\frac{1}{2}\) pairs per day. To find the number of days required to complete 2 pairs, we use the formula:
\[ \text{Time} = \frac{\text{Total work}}{\text{Combined work rate}} = \frac{2 \text{ pairs}}{\frac{1}{2} \text{ pairs per day}} = 2 \times 2 = 4 \text{ days} \]

### Conclusion:
- Therefore, A and B together will take **4 days** to knit 2 pairs of socks.