P, Q and R can do a job in 20, 30 and 60 days respectively. In how many days can P do the job if he is assisted by Q and R every third day?

We are given that:

- \( P \) can complete the job in 20 days.
- \( Q \) can complete the job in 30 days.
- \( R \) can complete the job in 60 days.
- \( P \) is assisted by \( Q \) and \( R \) every third day.

### Step 1: Calculate the work done by P, Q, and R per day
The work done per day by each worker is calculated as the reciprocal of the number of days each takes to complete the job.

- Work done by \( P \) per day: \( \frac{1}{20} \)
- Work done by \( Q \) per day: \( \frac{1}{30} \)
- Work done by \( R \) per day: \( \frac{1}{60} \)

### Step 2: Analyze the working pattern

For every three days:
- On the first two days, only \( P \) works.
- On the third day, \( P \) is assisted by both \( Q \) and \( R \).

### Step 3: Work done in a three-day cycle

- In the first two days, only \( P \) works, so the total work done by \( P \) in two days is:
  \[
  2 \times \frac{1}{20} = \frac{2}{20} = \frac{1}{10}
  \]

- On the third day, \( P \), \( Q \), and \( R \) all work together. The total work done on this day is:
  \[
  \frac{1}{20} + \frac{1}{30} + \frac{1}{60} = \frac{3}{60} + \frac{2}{60} + \frac{1}{60} = \frac{6}{60} = \frac{1}{10}
  \]

- So, the total work done in 3 days (two days by \( P \) and one day by \( P \), \( Q \), and \( R \)) is:
  \[
  \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5}
  \]

### Step 4: Calculate the total time to complete the job

Since \( \frac{1}{5} \) of the work is completed in 3 days, the total time to complete the entire job is:
\[
3 \times 5 = 15 \text{ days}
\]

Thus, \( P \), with the assistance of \( Q \) and \( R \) every third day, can complete the job in **15 days**.

---

Do you want further details or have any questions?  
Here are some related questions for deeper understanding:

1. How would the time change if \( Q \) assisted on every second day instead of the third?
2. What if \( P \), \( Q \), and \( R \) worked together continuously? How long would it take to finish the job?
3. How is the concept of least common multiple (LCM) relevant to solving work problems?
4. If \( P \)'s efficiency improved by 10%, how would that affect the completion time?
5. How can we represent this problem algebraically using work equations?

**Tip:** When analyzing shared tasks, converting days to work rates (like \( \frac{1}{x} \)) makes it easier to combine efforts across different time frames.

P, Q, and R can do a job in 20, 30, and 60 days respectively. In how many days can P do the job if he is assisted by Q and R every third day?

Explore how P, Q, and R can complete a job when P is assisted by Q and R every third day. Learn step-by-step how to calculate the total time needed, using work rate formulas and work per cycle analysis. Suitable for Grades 9-10.

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