Determining the Completion Time of a Job when Workers Work on Alternate Days

In this article, we will explore the method to determine the completion time of a job when two individuals, P and Q, work on alternate days. P and Q have different individual work rates, and we will break down the process step-by-step to find the total number of days required to complete the job.

Understanding the Problem

The problem states that P and Q can complete the job together in 8 days and 16 days respectively. The catch is that they are assigned to work on alternate days, with Q starting the job. We need to determine the total number of days taken to complete the job under this scenario. Let's delve into the solution.

Step-by-Step Solution

Step 1: Calculate Work Rates

First, we need to calculate the work rates of both P and Q.

P's work rate: P can complete the job in 8 days, hence the work rate of P is: Q's work rate: Q can complete the job in 16 days, hence the work rate of Q is:

Mathematically, we can express this as:

Work rate of P (frac{1}{8}) job per day

Work rate of Q (frac{1}{16}) job per day

Step 2: Work Done in Two Days

Since P and Q work on alternate days, we start with Q and then move to P. We can calculate the total work done in two days (one day by Q and one day by P).

Work done by Q in one day (frac{1}{16}) Work done by P in one day (frac{1}{8})

The total work done in two days:

Total work in 2 days (frac{1}{16} frac{1}{8} frac{1}{16} frac{2}{16} frac{3}{16})

Step 3: Calculate Total Days to Complete the Job

We need to determine how many complete two-day cycles are required to finish the job. Let n be the number of complete two-day cycles.

Total work done after n cycles (frac{3n}{16})

We need this to be equal to 1 (the whole job):

(frac{3n}{16} 1)

Solving for n:

3n 16 u2192 n (frac{16}{3}) u2192 n approx 5.33

This implies that after 5 complete two-day cycles (which is 10 days), the total work done is:

(frac{3 times 5}{16} frac{15}{16})

Step 4: Work Remaining

After 10 days, the remaining work is:

1 - (frac{15}{16}) (frac{1}{16})

Step 5: Completing the Remaining Work

On the 11th day, it is Q's turn to work. Q can complete (frac{1}{16}) of the job in one day, which is exactly the amount of work remaining.

Conclusion

Thus, the entire job will be completed in 11 days. Therefore, the job will be completed in 11 days.

Additional Insights

We can also derive the same result using a different method where every 2 days, P and Q together complete (frac{3}{16}) of the job:

To finish the job, it will take:

2 (times) (frac{16}{3}) (frac{32}{3}) days or 10 (frac{2}{3}) days

This confirms that the job will be completed in 11 days.