A can do a work in 20 days. B can do a work in 30 days and C can do a work in 60 days. If 'B' and 'C' helps A every third day, then how many days after work will complete?

Options

A. 12 days
B. 15 days
C. 18 days
D. 10 days

Answer

To solve the problem step by step, we will calculate the work done by A, B, and C over a cycle of 3 days, where B and C help A every third day.

### Step 1: Determine the work rates of A, B, and C.
- A can complete the work in 20 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{20} \text{ (work per day)} \]
- B can complete the work in 30 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{30} \text{ (work per day)} \]
- C can complete the work in 60 days, so C's work rate is:
\[ \text{Work rate of C} = \frac{1}{60} \text{ (work per day)} \]

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

### Step 3: Calculate the work done on the 3rd day by A, B, and C together.
On the 3rd day, A, B, and C work together:
- Total work done by A, B, and C in 1 day:
\[ \text{Total work} = \frac{1}{20} + \frac{1}{30} + \frac{1}{60} \]
To add these fractions, we need a common denominator, which is 60:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{60} = \frac{1}{60} \]
So,
\[ \text{Total work} = \frac{3}{60} + \frac{2}{60} + \frac{1}{60} = \frac{6}{60} = \frac{1}{10} \]

### Step 4: Calculate the total work done in 3 days.
- Work done in 3 days (2 days by A + 1 day by A, B, and C):
\[ \text{Total work in 3 days} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \]

### Step 5: Determine how many cycles are needed to complete the work.
Since \(\frac{1}{5}\) of the work is completed in 3 days, we need to find out how many such cycles are required to complete the entire work (1 work):
\[ \text{Number of cycles required} = \frac{1}{\frac{1}{5}} = 5 \]
### Step 6: Calculate the total number of days to complete the work.
- Total days to complete the work:
\[ \text{Total days} = 5 \text{ cycles} \times 3 \text{ days per cycle} = 15 \text{ days} \]

### Final Answer:
The work will be completed in **15 days**.

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