Gaurav, Parit and Gaggi can do a work in 20,30 and 60 days respectively. How many days does it need to complete the work if Gaurav does the work and he is assisted by others on every alteNAte day?

Options

A. 10 days
B. 14 days
C. 15 days
D. 9 days

Answer

To solve the problem step by step, we will first determine the work done by Gaurav, Parit, and Gaggi individually and then calculate the total work done when Gaurav is assisted by Parit and Gaggi on alternate days.

### Step 1: Calculate the work done by each person in one day.
- Gaurav can complete the work in 20 days, so his work rate is:
\[ \text{Gaurav's work rate} = \frac{1}{20} \text{ of the work per day} \]

- Parit can complete the work in 30 days, so his work rate is:
\[ \text{Parit's work rate} = \frac{1}{30} \text{ of the work per day} \]

- Gaggi can complete the work in 60 days, so his work rate is:
\[ \text{Gaggi's work rate} = \frac{1}{60} \text{ of the work per day} \]

### Step 2: Find the total work done in one cycle of two days.
On the first day, Gaurav works alone:
- Work done by Gaurav on Day 1:
\[ \text{Work done} = \frac{1}{20} \]

On the second day, Gaurav is assisted by Parit:
- Total work done on Day 2:
\[ \text{Work done} = \frac{1}{20} + \frac{1}{30} \]
To add these fractions, we need a common denominator:
\[ \text{LCM of 20 and 30} = 60 \]
Thus,
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
Therefore,
\[ \text{Total work on Day 2} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]

### Step 3: Calculate total work done in two days.
- Total work done in two days (Day 1 + Day 2):
\[ \text{Total work in 2 days} = \frac{1}{20} + \frac{1}{12} \]
Finding a common denominator (LCM of 20 and 12 is 60):
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Thus,
\[ \text{Total work in 2 days} = \frac{3}{60} + \frac{5}{60} = \frac{8}{60} = \frac{2}{15} \]

### Step 4: Calculate how many such cycles are needed to complete the work.
- Total work to be done is 1 (the whole work).
- Work done in 2 days is \(\frac{2}{15}\).
- To find how many cycles (2-day periods) are needed:
\[ \text{Number of cycles} = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \]
This means 7 complete cycles (14 days) and then we need to find out how much work is left.

### Step 5: Calculate work done in 14 days.
- Work done in 14 days:
\[ 7 \times \frac{2}{15} = \frac{14}{15} \]
- Work remaining:
\[ 1 - \frac{14}{15} = \frac{1}{15} \]

### Step 6: Calculate how much time is needed to complete the remaining work.
- On the 15th day, Gaurav works alone:
\[ \text{Work done by Gaurav on Day 15} = \frac{1}{20} \]
- Since \(\frac{1}{20} > \frac{1}{15}\), Gaurav will complete the remaining work on the 15th day.

### Final Calculation of Total Days:
- Total days = 14 (from the cycles) + 1 (for the remaining work) = 15 days.

Thus, the total number of days required to complete the work is **15 days**.