A, B and C working separately can do a piece of work in 11 days, 20 days and 55 days respectively. In how many days, the work will be completed if A is assisted by B and C on alternate days?

Options

A. 2
B. 6
C. 4
D. 8

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C in one day, and then calculate how much work is completed when A is assisted by B and C on alternate days.

### Step 1: Determine the work done by A, B, and C in one day.

- A can complete the work in 11 days. Therefore, the work done by A in one day is:
\[ \text{Work by A in one day} = \frac{1}{11} \text{ of the work} \]

- B can complete the work in 20 days. Therefore, the work done by B in one day is:
\[ \text{Work by B in one day} = \frac{1}{20} \text{ of the work} \]

- C can complete the work in 55 days. Therefore, the work done by C in one day is:
\[ \text{Work by C in one day} = \frac{1}{55} \text{ of the work} \]

### Step 2: Calculate the total work done by A, B, and C together.

To find a common work unit, we can use the least common multiple (LCM) of the days they take to complete the work:

- LCM of 11, 20, and 55 is 220.

Now, we can convert the work done by A, B, and C into units of work:

- Work done by A in one day:
\[ \text{Work by A} = \frac{220}{11} = 20 \text{ units} \]

- Work done by B in one day:
\[ \text{Work by B} = \frac{220}{20} = 11 \text{ units} \]

- Work done by C in one day:
\[ \text{Work by C} = \frac{220}{55} = 4 \text{ units} \]

### Step 3: Calculate the work done in a 2-day cycle.

In a 2-day cycle:
- Day 1: A works alone, so the work done is 20 units.
- Day 2: A, B, and C work together, so the work done is:
\[ \text{Work on Day 2} = 20 + 11 + 4 = 35 \text{ units} \]

Total work done in 2 days:
\[ \text{Total work in 2 days} = 20 + 35 = 55 \text{ units} \]

### Step 4: Calculate the total number of days to complete the work.

The total work is 220 units. To find out how many 2-day cycles are needed to complete the work:
\[ \text{Number of 2-day cycles} = \frac{220}{55} = 4 \]

Since each cycle takes 2 days, the total number of days to complete the work is:
\[ \text{Total days} = 4 \times 2 = 8 \text{ days} \]

### Final Answer:
The work will be completed in **8 days**.

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