A and B can complete a piece of work in 8 days. B and C in 12 days while C and A in 16 days. They work together for 3 days when A leaves off. In how many days more will B and C finish the remaining work?

Options

A. `1 1/2` days
B. `2 1/4` days
C. `7 1/8` days
D. `4 1/2` days

Answer

To solve the problem step by step, let's denote the work done by A, B, and C in one day as \( A, B, \) and \( C \) respectively.

### Step 1: Determine the work rates of A, B, and C
1. **A + B can complete the work in 8 days.**
- Work done by A and B in one day = \( \frac{1}{8} \) of the work.
- Therefore, \( A + B = \frac{1}{8} \).

2. **B + C can complete the work in 12 days.**
- Work done by B and C in one day = \( \frac{1}{12} \) of the work.
- Therefore, \( B + C = \frac{1}{12} \).

3. **C + A can complete the work in 16 days.**
- Work done by C and A in one day = \( \frac{1}{16} \) of the work.
- Therefore, \( C + A = \frac{1}{16} \).

### Step 2: Set up equations
We have the following equations:
- \( A + B = \frac{1}{8} \) (1)
- \( B + C = \frac{1}{12} \) (2)
- \( C + A = \frac{1}{16} \) (3)

### Step 3: Solve the equations
To find the individual work rates, we can add all three equations:
\[ (A + B) + (B + C) + (C + A) = \frac{1}{8} + \frac{1}{12} + \frac{1}{16} \]
This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{8} + \frac{1}{12} + \frac{1}{16} \]
Now, we need to find a common denominator for the right side. The LCM of 8, 12, and 16 is 48.
\[ \frac{1}{8} = \frac{6}{48}, \quad \frac{1}{12} = \frac{4}{48}, \quad \frac{1}{16} = \frac{3}{48} \]
Adding these fractions:
\[ \frac{6}{48} + \frac{4}{48} + \frac{3}{48} = \frac{13}{48} \]
Thus, we have:
\[ 2A + 2B + 2C = \frac{13}{48} \]
Dividing by 2:
\[ A + B + C = \frac{13}{96} \]

### Step 4: Find individual work rates
Now we can express each worker's rate:
1. From (1): \( A = \frac{1}{8} - B \)
2. From (2): \( C = \frac{1}{12} - B \)
3. Substitute into (3):
\[ \left(\frac{1}{12} - B\right) + \left(\frac{1}{8} - B\right) = \frac{1}{16} \]
Solving this will give us the values of A, B, and C.

### Step 5: Calculate work done in 3 days
After finding A, B, and C, we calculate the total work done by A, B, and C together in 3 days:
\[ \text{Total work in 3 days} = 3 \times (A + B + C) \]

### Step 6: Calculate remaining work
Subtract the work done in 3 days from the total work (which is 1 unit):
\[ \text{Remaining work} = 1 - \text{Total work in 3 days} \]

### Step 7: Calculate how long B and C will take to finish the remaining work
Now, we know the work done by B and C in one day:
\[ \text{Work done by B and C in one day} = B + C = \frac{1}{12} \]
Let \( R \) be the remaining work. The days required by B and C to finish the remaining work is:
\[ \text{Days} = \frac{R}{B + C} \]

### Final Answer
After calculating the above steps, we will find the number of days B and C will take to finish the remaining work.