A can do a work in 12 days. When he had worked for 3 days, B joined him. If they complete the work in 3 more days, in how many days can B alone finish the work?

Options

A. 6 days
B. 12 days
C. 4 days
D. 8 days

Answer

To solve the problem step by step, we will break down the information given and use it to find out how many days B alone can finish the work.

### Step 1: Determine A's work rate
A can complete the work in 12 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{12} \text{ (work per day)} \]

### Step 2: Calculate the work done by A in 3 days
In the first 3 days, A works alone. The amount of work done by A in 3 days is:
\[ \text{Work done by A in 3 days} = 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \]

### Step 3: Determine the remaining work
After A has worked for 3 days, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Calculate the work done by A and B together in the next 3 days
A and B work together for the next 3 days to complete the remaining work. Let B's work rate be \( \frac{1}{X} \) (where X is the number of days B takes to complete the work alone). Therefore, the combined work rate of A and B is:
\[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{X} \]

In 3 days, the amount of work they complete together is:
\[ 3 \left( \frac{1}{12} + \frac{1}{X} \right) \]

### Step 5: Set up the equation for the remaining work
Since they complete the remaining work of \( \frac{3}{4} \) in 3 days, we can set up the equation:
\[ 3 \left( \frac{1}{12} + \frac{1}{X} \right) = \frac{3}{4} \]

### Step 6: Simplify the equation
Dividing both sides by 3 gives:
\[ \frac{1}{12} + \frac{1}{X} = \frac{1}{4} \]

### Step 7: Solve for \( \frac{1}{X} \)
Subtract \( \frac{1}{12} \) from both sides:
\[ \frac{1}{X} = \frac{1}{4} - \frac{1}{12} \]

To subtract these fractions, find a common denominator (which is 12):
\[ \frac{1}{4} = \frac{3}{12} \]
Thus,
\[ \frac{1}{X} = \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \]

### Step 8: Find the value of X
Taking the reciprocal gives:
\[ X = 6 \]

### Conclusion
B can finish the work alone in **6 days**.