Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Amal starts working on the task and works for 75 days, then Vimal takes over and works for 135 days. Finally, Sunil takes over and completes the remaining task in 45 days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been

Question

Accepted Answer

We are tasked with calculating the number of days required to complete a task if:
- Amal works every day,
- Vimal works every second day,
- Sunil works every third day.

#### Step 1: Define work rates
Let the work rates of Amal, Vimal, and Sunil be:
- $$ A $$ (units/day),
- $$ V $$ (units/day),
- $$ S $$ (units/day).

From the given equations:
1. $$ 150A + 150V = T $$
2. $$ 100V + 100S = T $$
3. $$ 75A + 135V + 45S = T $$

#### Step 2: Solve for relationships between work rates
Adding equations (1) and (2):
$$
150A + 250V + 100S = 2T \quad 	ext{(4)}
$$

Multiplying equation (3) by 2:
$$
150A + 270V + 90S = 2T \quad 	ext{(5)}
$$

Subtracting (5) from (4):
$$
10S = 20V \quad \Rightarrow \quad S = 2V
$$

Using this result in equation (1):
$$
A = V
$$

Thus, the work rates are:
- Amal's rate: $$ A = V $$
- Vimal's rate: $$ V $$
- Sunil's rate: $$ S = 2V $$.

#### Step 3: Total work and work cycle
Let the total work be $$ T = 300 $$ units (as assumed in the solution). The daily contributions during a repeating cycle of **6 days** are:
- Amal works all **6 days**, contributing $$ 6A = 6V $$ units.
- Vimal works **3 days**, contributing $$ 3V $$ units.
- Sunil works **2 days**, contributing $$ 2S = 4V $$ units.

Total work done in one cycle:
$$
	ext{Work per cycle} = 6V + 3V + 4V = 13V.
$$

#### Step 4: Calculate total days required
The total work is $$ T = 300V $$. Divide the total work by the work done per cycle:
$$
	ext{Number of cycles} = \frac{300V}{13V} = \frac{300}{13}.
$$

This gives **23 complete cycles**, completing $$ 23 	imes 13V = 299V $$ units of work. These cycles take:
$$
23 	imes 6 = 138 \, 	ext{days}.
$$

The remaining $$ T - 299V = V $$ unit of work will be completed on the next day.

Thus, the total number of days required is:
$$
138 + 1 = 139 \, 	ext{days}.
$$

### Final Answer:
**139 days**