A and 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, whereupon A leaves. B continues for 2 more days, and then the he leaves too. C now starts working days did C require?

Options

A. 5
B. 8
C. 3
D. 4

Answer

To solve the problem step by step, we will first determine the work rates of A, B, and C, and then calculate how many days C requires to finish the remaining work after A and B have worked.

### Step 1: Determine the work rates of A, B, and C

1. **A and B together can complete the job in 8 days.**
- Work done by A and B in one day = 1/8 of the job.

2. **B and C together can complete the job in 12 days.**
- Work done by B and C in one day = 1/12 of the job.

### Step 2: Express the work rates

Let:
- Work rate of A = \( a \)
- Work rate of B = \( b \)
- Work rate of C = \( c \)

From the above information:
- \( a + b = \frac{1}{8} \) (1)
- \( b + c = \frac{1}{12} \) (2)

### Step 3: Find the individual work rates

We can express \( c \) in terms of \( b \) using equation (2):
- \( c = \frac{1}{12} - b \)

Now substitute \( c \) into equation (1):
- \( a + b = \frac{1}{8} \)
- \( a + b + \left(\frac{1}{12} - b\right) = \frac{1}{8} + \frac{1}{12} \)

### Step 4: Find a common denominator and solve for \( a \)

The common denominator for 8 and 12 is 24:
- \( \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \)

So, we have:
- \( a + \left(\frac{1}{12} - b\right) = \frac{5}{24} \)
- This simplifies to \( a + \frac{1}{12} - b = \frac{5}{24} \)

Now, substituting \( b = \frac{1}{8} - a \):
- \( a + \frac{1}{12} - \left(\frac{1}{8} - a\right) = \frac{5}{24} \)

### Step 5: Solve for \( a \) and \( b \)

After substituting and simplifying, we can find the values of \( a \) and \( b \):
- \( a = \frac{1}{24} \)
- \( b = \frac{1}{8} - \frac{1}{24} = \frac{1}{12} \)

Now substituting \( b \) back to find \( c \):
- \( c = \frac{1}{12} - b = \frac{1}{12} - \frac{1}{12} = 0 \)

### Step 6: Calculate work done by A and B in 4 days

- Work done by A and B in 4 days:
- Work done = \( 4 \times (a + b) = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \)

### Step 7: Calculate work done by B in 2 more days

- Work done by B in 2 days:
- Work done = \( 2 \times b = 2 \times \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \)

### Step 8: Total work done and remaining work

Total work done by A and B:
- Total work done = \( \frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \)

Remaining work:
- Total work = 1 (whole job) - \( \frac{2}{3} = \frac{1}{3} \)

### Step 9: Calculate how many days C requires to finish the remaining work

Since C's work rate is \( c = \frac{1}{12} \):
- Days required by C to complete \( \frac{1}{3} \) of the work:
- Work done by C in one day = \( \frac{1}{12} \)
- Days required = Remaining work / C's work rate = \( \frac{1/3}{1/12} = \frac{1/3 \times 12/1} = 4 \) days.

### Final Answer

C requires **4 days** to complete the remaining work.

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