(A + B) together can complete a job in 8 days. Both B and C working alone can finish the same job in 12 days. A and B commence work on the job and work for 4 days, where upon A leaves, B continues for 2 more days and then he leaves too, C now starts working and finishes the job, how many days will C require ?

Answer

To solve the problem step by step, we will first determine the work efficiencies of A, B, and C, and then calculate how much work is left after each phase of the work.

### Step 1: Determine the total work
Since A + B can complete the job in 8 days, we can express the total work in terms of work units. If we assume the total work is 24 units (which is a common multiple of 8 and 12), then:

- Total work = 24 units

### Step 2: Determine the efficiency of A + B
The efficiency of A + B can be calculated as follows:

- Efficiency of A + B = Total work / Time taken = 24 units / 8 days = 3 units/day

### Step 3: Determine the efficiency of B and C
Since B and C together can complete the job in 12 days, we can calculate their combined efficiency:

- Efficiency of B + C = Total work / Time taken = 24 units / 12 days = 2 units/day

### Step 4: Determine the efficiency of B
Now, we know that A + B's efficiency is 3 units/day and B + C's efficiency is 2 units/day. We can express A's efficiency as follows:

Let the efficiency of A be \( a \), the efficiency of B be \( b \), and the efficiency of C be \( c \).

From the equations:
1. \( a + b = 3 \) (1)
2. \( b + c = 2 \) (2)

### Step 5: Calculate the work done by A and B in the first 4 days
A and B work together for 4 days:

- Work done by A + B in 4 days = 4 days * 3 units/day = 12 units

### Step 6: Calculate the remaining work
After A and B have worked for 4 days, the remaining work is:

- Remaining work = Total work - Work done = 24 units - 12 units = 12 units

### Step 7: Calculate the work done by B in the next 2 days
B continues to work alone for 2 more days:

- Work done by B in 2 days = 2 days * b units/day

From equation (1), we can express \( b \):
- Since \( a + b = 3 \), we can substitute \( b = 3 - a \).

Now, we need to find \( b \) using equation (2):
- From \( b + c = 2 \), we can express \( c \) as \( c = 2 - b \).

### Step 8: Substitute and solve for B's efficiency
Assuming \( b = 1 \) (since \( b + c = 2 \) and it should be less than 2), we can find \( c \):

- If \( b = 1 \), then \( c = 2 - 1 = 1 \).

### Step 9: Work done by B in 2 days
Now, substituting \( b = 1 \):

- Work done by B in 2 days = 2 days * 1 unit/day = 2 units

### Step 10: Calculate the remaining work after B's contribution
After B works for 2 days, the remaining work is:

- Remaining work = 12 units - 2 units = 10 units

### Step 11: Calculate how long C will take to finish the remaining work
C now starts working to finish the remaining 10 units of work. Since C's efficiency is 1 unit/day:

- Days required by C = Remaining work / C's efficiency = 10 units / 1 unit/day = 10 days

### Final Answer
C will require **10 days** to finish the job.

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