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The problem you provided involves three workers—Pranjal, Sameer, and Arush—working together to complete a task. Let's break down the problem and solve it step by step:

### Given:
1. **Pranjal** takes 3 times as long as Arush and Sameer combined to complete the work.
2. **Sameer** takes twice as long as Arush and Pranjal together to complete the work.
3. All three together can complete the work in 30 days.

### Let:
- \( x \) be the number of days **Arush** takes to complete the work.
- \( y \) be the number of days **Sameer** takes to complete the work.
- \( z \) be the number of days **Pranjal** takes to complete the work.

From the problem, we can derive the following equations:

1. **Pranjal's work rate**:
   \[
   z = 3 \left( \frac{1}{x} + \frac{1}{y} \right)^{-1}
   \]
   This means that Pranjal takes 3 times as long as Arush and Sameer together.

2. **Sameer's work rate**:
   \[
   y = 2 \left( \frac{1}{x} + \frac{1}{z} \right)^{-1}
   \]
   This means Sameer takes twice as long as Arush and Pranjal together.

3. **Combined work rate**:
   Together, all three workers complete the job in 30 days, so the sum of their individual work rates equals:
   \[
   \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{30}
   \]

### Approach:
We will solve these three equations step-by-step to find the value of \( y \), which represents the number of days Sameer would take to complete the work alone.

Let me calculate this now.The solution to the system of equations gives the following:

- **Arush** takes 72 days to complete the work alone.
- **Sameer** takes 90 days to complete the work alone.
- **Pranjal** takes 120 days to complete the work alone.

Thus, **Sameer** would take **90 days** to complete the work by himself.

Do you need any further details or clarifications?

Here are 5 related questions you might find helpful:
1. How would the result change if the combined work rate changed?
2. What if Sameer took 3 times as long as Arush and Pranjal combined?
3. How can we generalize this problem for more workers?
4. Can this method be applied to different work rate problems involving machinery or groups of people?
5. What if the work durations were in hours instead of days?

**Tip:** Always express work rates as fractions of the total work done to simplify such problems.

To do a piece of work, Pranjal takes 3 times as long as Arush and Sameer together, and Sameer takes twice as long as Arush and Pranjal together. If the three together can complete the work in 30 days, how long would Sameer take to complete the work?

Solve a work rate problem involving Pranjal, Sameer, and Arush, where Pranjal takes 3 times as long as Arush and Sameer together, and Sameer takes twice as long as Arush and Pranjal together. The three can complete the work in 30 days. Suitable for students in Grades 10-12.

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