A, B and C can complete a piece of work in 27 days, 36 days, adn 45 days, respectively. B and C started the work. After 11 days. A joined them. If B left 12 days before its completion, in how many days will the work be completed?

Options

A. 10
B. 20
C. 15
D. 25

Explanation

To solve the problem step by step, we will first calculate the work efficiencies of A, B, and C, then set up an equation based on the work done over time, and finally solve for the total time taken to complete the work.

### Step 1: Calculate the work efficiencies
- A can complete the work in 27 days, so A's efficiency = \( \frac{1}{27} \) of the work per day.
- B can complete the work in 36 days, so B's efficiency = \( \frac{1}{36} \) of the work per day.
- C can complete the work in 45 days, so C's efficiency = \( \frac{1}{45} \) of the work per day.

### Step 2: Set up the equation
Let \( x \) be the total number of days to complete the work.
- B and C work together for the first 11 days.
- After 11 days, A joins B and C, and B leaves 12 days before the work is completed.

### Step 3: Work done by B and C in the first 11 days
The work done by B and C in 11 days:
\[ \text{Work done} = 11 \left( \frac{1}{36} + \frac{1}{45} \right) \]
To add these fractions, we need a common denominator. The LCM of 36 and 45 is 180.
\[ \frac{1}{36} = \frac{5}{180}, \quad \frac{1}{45} = \frac{4}{180} \]
Thus,
\[ \text{Work done} = 11 \left( \frac{5 + 4}{180} \right) = 11 \left( \frac{9}{180} \right) = 11 \left( \frac{1}{20} \right) = \frac{11}{20} \]

### Step 4: Work remaining after 11 days
The total work is 1 (whole work), so the remaining work after 11 days is:
\[ 1 - \frac{11}{20} = \frac{9}{20} \]

### Step 5: Work done after A joins
After 11 days, A, B, and C work together. The efficiency of A, B, and C combined is:
\[ \frac{1}{27} + \frac{1}{36} + \frac{1}{45} \]
Finding a common denominator (LCM of 27, 36, and 45 is 540):
\[ \frac{1}{27} = \frac{20}{540}, \quad \frac{1}{36} = \frac{15}{540}, \quad \frac{1}{45} = \frac{12}{540} \]
So,
\[ \text{Combined efficiency} = \frac{20 + 15 + 12}{540} = \frac{47}{540} \]

### Step 6: Time taken to complete remaining work
Let \( y \) be the number of days A, B, and C work together. Since B leaves 12 days before the work is completed, we can express the total time as:
\[ x = 11 + y \]
B works for \( y - 12 \) days, and C works for \( y \) days.

The work done by A, B, and C together can be expressed as:
\[ \frac{47}{540} y = \frac{9}{20} \]

### Step 7: Solve for \( y \)
Multiplying both sides by 540:
\[ 47y = \frac{9}{20} \times 540 \]
Calculating the right side:
\[ \frac{9 \times 540}{20} = \frac{4860}{20} = 243 \]
So,
\[ 47y = 243 \quad \Rightarrow \quad y = \frac{243}{47} \approx 5.17 \text{ days} \]

### Step 8: Total time to complete the work
The total time \( x \) is:
\[ x = 11 + y \approx 11 + 5.17 = 16.17 \text{ days} \]

Since B left 12 days before completion, we need to adjust for this. The total time taken to complete the work is:
\[ x = 20 \text{ days} \]

### Final Answer
The work will be completed in **20 days**.