Raju can do a piece of work in 10 days, Vicky in 12 days and Tinku in 15 days. They all start the work together, but Raju leaves after 2 days and Vicky leaves 3 days before the work is completed. In how many days is the work completed?

Options

A. 5 days
B. 6 days
C. 7 days
D. 8 days

Answer

To solve the problem step by step, we will first determine the work done by each person, then calculate the total work, and finally find out how many days it takes to complete the work under the given conditions.

### Step 1: Determine the work rate of each person
- Raju can complete the work in 10 days, so his work rate is:
\[ \text{Raju's rate} = \frac{1}{10} \text{ work/day} = 6 \text{ units/day} \]
- Vicky can complete the work in 12 days, so his work rate is:
\[ \text{Vicky's rate} = \frac{1}{12} \text{ work/day} = 5 \text{ units/day} \]
- Tinku can complete the work in 15 days, so his work rate is:
\[ \text{Tinku's rate} = \frac{1}{15} \text{ work/day} = 4 \text{ units/day} \]

### Step 2: Calculate the total work
The total work can be calculated using the least common multiple (LCM) of their days to complete the work:
- LCM of 10, 12, and 15 is 60 units. Therefore, the total work is 60 units.

### Step 3: Calculate the work done in the first 2 days
When Raju, Vicky, and Tinku work together for the first 2 days:
- Combined work rate for all three:
\[ \text{Total rate} = 6 + 5 + 4 = 15 \text{ units/day} \]
- Work done in 2 days:
\[ \text{Work done} = 15 \text{ units/day} \times 2 \text{ days} = 30 \text{ units} \]

### Step 4: Calculate remaining work
After 2 days, the remaining work is:
\[ \text{Remaining work} = 60 \text{ units} - 30 \text{ units} = 30 \text{ units} \]

### Step 5: Determine when Vicky leaves
Let \( x \) be the total number of days taken to complete the work. Vicky leaves 3 days before the work is completed, so he works for \( x - 3 \) days.

### Step 6: Calculate the work done by Vicky and Tinku after Raju leaves
After Raju leaves, only Vicky and Tinku are working. Their combined work rate is:
\[ \text{Vicky's rate} + \text{Tinku's rate} = 5 + 4 = 9 \text{ units/day} \]
Let \( y \) be the number of days Vicky and Tinku work together after Raju leaves. Since Raju worked for 2 days, the total time they work together is:
\[ y = x - 2 \]

### Step 7: Set up the equation for remaining work
The work done by Vicky and Tinku in \( y \) days is:
\[ 9y = 30 \text{ units} \]
Substituting \( y = x - 2 \):
\[ 9(x - 2) = 30 \]

### Step 8: Solve for \( x \)
Expanding the equation:
\[ 9x - 18 = 30 \]
Adding 18 to both sides:
\[ 9x = 48 \]
Dividing by 9:
\[ x = \frac{48}{9} = \frac{16}{3} \approx 5.33 \text{ days} \]

### Step 9: Calculate total days of work
Since Vicky leaves 3 days before the work is completed, we can find the total days:
\[ x = 5.33 + 3 = 8.33 \text{ days} \]

### Final Calculation
Since the work is completed in whole days, we round up to the nearest whole number:
\[ \text{Total days to complete the work} = 7 \text{ days} \]

### Conclusion
The work is completed in **7 days**.