A, B and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

Options

A. 10 days
B. 12 days
C. 15 days
D. 20 days

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C individually, and then calculate how much work is completed when A is assisted by B and C every third day.

### Step 1: Calculate the work done by A, B, and C in one day.
- A can complete the work in 20 days, so A's work in one day = \( \frac{1}{20} \).
- B can complete the work in 30 days, so B's work in one day = \( \frac{1}{30} \).
- C can complete the work in 60 days, so C's work in one day = \( \frac{1}{60} \).

### Step 2: Calculate the work done by A in the first two days.
- In two days, A will do:
\[ \text{Work done by A in 2 days} = 2 \times \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \]

### Step 3: Calculate the work done by A, B, and C together on the third day.
- On the third day, A is assisted by B and C. The total work done by A, B, and C together in one day is:
\[ \text{Work done by A, B, and C} = \frac{1}{20} + \frac{1}{30} + \frac{1}{60} \]
- To add these fractions, we need a common denominator. The least common multiple of 20, 30, and 60 is 60. So, we convert each fraction:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{60} = \frac{1}{60} \]
- Now, adding these fractions:
\[ \frac{3}{60} + \frac{2}{60} + \frac{1}{60} = \frac{6}{60} = \frac{1}{10} \]

### Step 4: Calculate the total work done in three days.
- In three days, the total work done is the sum of the work done in the first two days and the work done on the third day:
\[ \text{Total work in 3 days} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \]

### Step 5: Determine how many such cycles are needed to complete the work.
- Since \( \frac{1}{5} \) of the work is done in 3 days, we need to find out how many such cycles are required to complete the whole work (1 unit of work):
\[ \text{Number of cycles} = \frac{1}{\frac{1}{5}} = 5 \]
- Therefore, the total number of days required to complete the work is:
\[ \text{Total days} = 5 \times 3 = 15 \text{ days} \]

### Final Answer:
A can complete the work with the assistance of B and C on every third day in **15 days**.