A, B and C can do job in 9, 12 and 36 days respectively if they worked alone. A leaves after they have worked together for 3 days. In how many days can B and C do the rest of the job?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem step by step, we will follow the process of calculating the work done by A, B, and C, and then determine how long it will take for B and C to finish the remaining work after A leaves.

### Step 1: Determine the work rates of A, B, and C
- A can complete the job in 9 days, so A's work rate is \( \frac{1}{9} \) 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.
- C can complete the job in 36 days, so C's work rate is \( \frac{1}{36} \) of the job per day.

### Step 2: Calculate the combined work rate of A, B, and C
To find the combined work rate when A, B, and C work together, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{9} + \frac{1}{12} + \frac{1}{36} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 9, 12, and 36 is 36. We can rewrite each fraction:
- \( \frac{1}{9} = \frac{4}{36} \)
- \( \frac{1}{12} = \frac{3}{36} \)
- \( \frac{1}{36} = \frac{1}{36} \)

Now, we can add them:
\[ \text{Combined work rate} = \frac{4}{36} + \frac{3}{36} + \frac{1}{36} = \frac{8}{36} = \frac{2}{9} \]

### Step 3: Calculate the work done in 3 days
Now that we have the combined work rate, we can find out how much work A, B, and C complete together in 3 days:
\[ \text{Work done in 3 days} = \text{Combined work rate} \times \text{Time} = \frac{2}{9} \times 3 = \frac{6}{9} = \frac{2}{3} \]

### Step 4: Determine the remaining work
The total work is considered as 1 (the whole job). After 3 days, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 5: Calculate the combined work rate of B and C
Next, we need to find out how fast B and C can complete the remaining work. Their combined work rate is:
\[ \text{Combined work rate of B and C} = \frac{1}{12} + \frac{1}{36} \]

Again, we find a common denominator. The LCM of 12 and 36 is 36:
- \( \frac{1}{12} = \frac{3}{36} \)
- \( \frac{1}{36} = \frac{1}{36} \)

Now, we can add them:
\[ \text{Combined work rate of B and C} = \frac{3}{36} + \frac{1}{36} = \frac{4}{36} = \frac{1}{9} \]

### Step 6: Calculate the time taken by B and C to finish the remaining work
Now we can find the time it takes for B and C to complete the remaining \( \frac{1}{3} \) of the work:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Combined work rate of B and C}} = \frac{\frac{1}{3}}{\frac{1}{9}} = \frac{1}{3} \times \frac{9}{1} = 3 \text{ days} \]

### Final Answer
B and C can complete the remaining work in **3 days**.

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