A and B can do a job in 16 days and 12 days respectively. B has started the work alone 4 days before finishing the job, A joins B. How many days has B worked alone?

Options

A. 6 days
B. 4 days
C. 5 days
D. 7 days

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the work done by A and B in one day.
- A can complete the job in 16 days, so A's work rate is \( \frac{1}{16} \) of the job per day.
- B can complete the job in 12 days, so B's work rate is \( \frac{1}{12} \) of the job per day.

### Step 2: Find the total work in terms of a common unit.
- The least common multiple (LCM) of 16 and 12 is 48. Therefore, we can consider the total work to be 48 units.
- A's work rate: \( \frac{48}{16} = 3 \) units per day.
- B's work rate: \( \frac{48}{12} = 4 \) units per day.

### Step 3: Calculate the work done by A and B together.
- When A joins B, their combined work rate is \( 3 + 4 = 7 \) units per day.

### Step 4: Determine the time they worked together.
- B started the work alone 4 days before A joined. Let’s denote the total time taken to complete the work as \( T \) days.
- Therefore, A and B worked together for 4 days, and B worked alone for \( T - 4 \) days.

### Step 5: Calculate the total work done.
- The work done by B alone in \( T - 4 \) days is \( 4(T - 4) \) units.
- The work done by A and B together in 4 days is \( 7 \times 4 = 28 \) units.
- The total work done can be expressed as:
\[ 4(T - 4) + 28 = 48 \]

### Step 6: Solve for \( T \).
- Expanding the equation:
\[ 4T - 16 + 28 = 48 \]
\[ 4T + 12 = 48 \]
\[ 4T = 48 - 12 \]
\[ 4T = 36 \]
\[ T = 9 \]

### Step 7: Calculate how many days B worked alone.
- Since \( T = 9 \), the number of days B worked alone is:
\[ T - 4 = 9 - 4 = 5 \text{ days} \]

### Conclusion:
B has worked alone for **5 days**.

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