A, B and C, one can do a piece of work in 9, 12 and 18 days respectively. They all started the work together, but A left after 3 days. In how many days, was the remaining work completed?

Options

A. 2
B. `5/2`
C. `(11)/4`
D. `9/5`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the work done by A, B, and C in one day.
- A can complete the work in 9 days, so A's work in one day = \( \frac{1}{9} \).
- B can complete the work in 12 days, so B's work in one day = \( \frac{1}{12} \).
- C can complete the work in 18 days, so C's work in one day = \( \frac{1}{18} \).

### Step 2: Find the total work done by A, B, and C together in one day.
To find the total work done in one day, we add their individual work rates:
\[ \text{Total work in one day} = \frac{1}{9} + \frac{1}{12} + \frac{1}{18} \]

To add these fractions, we need to find a common denominator. The LCM of 9, 12, and 18 is 36.
- Convert each fraction:
- \( \frac{1}{9} = \frac{4}{36} \)
- \( \frac{1}{12} = \frac{3}{36} \)
- \( \frac{1}{18} = \frac{2}{36} \)

Now, adding them together:
\[ \text{Total work in one day} = \frac{4}{36} + \frac{3}{36} + \frac{2}{36} = \frac{9}{36} = \frac{1}{4} \]

### Step 3: Calculate the work done in the first 3 days.
Since they work together for 3 days, the total work done in 3 days is:
\[ \text{Work done in 3 days} = 3 \times \frac{1}{4} = \frac{3}{4} \]

### Step 4: Determine the remaining work.
The total work is considered as 1 (or 100%). Therefore, the remaining work after 3 days is:
\[ \text{Remaining work} = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 5: Calculate the work done by B and C together.
Now, we need to find the combined work rate of B and C:
- B's work in one day = \( \frac{1}{12} \)
- C's work in one day = \( \frac{1}{18} \)

Finding a common denominator (which is 36):
- \( \frac{1}{12} = \frac{3}{36} \)
- \( \frac{1}{18} = \frac{2}{36} \)

Thus, the combined work of B and C in one day is:
\[ \text{Combined work of B and C} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \]

### Step 6: Calculate the time taken by B and C to complete the remaining work.
To find out how many days it will take for B and C to complete the remaining \( \frac{1}{4} \) of the work:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Combined work of B and C}} = \frac{\frac{1}{4}}{\frac{5}{36}} = \frac{1}{4} \times \frac{36}{5} = \frac{36}{20} = \frac{9}{5} \text{ days} \]

### Final Answer:
The remaining work was completed in \( \frac{9}{5} \) days or 1.8 days.

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