Solving a task, two agents P and Q can finish the project in 24 days, and the output rate of a third agent R is half of that of P. Working together, P, Q, and R can finish the project in 20 days. Determine the time needed to finish the project if Q and R work together?

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The correct option is 30 days.

Step-by-Step Explanation:

Step 1: Determine the work rate of agent R

Let the total work required to complete the project be equal to 1 unit.

We are given that agents P and Q working together can complete the project in 24 days. Therefore, their combined rate of work per day is:

$\text{Rate}\left(P+Q\right)=\frac{1}{24}$

We are also given that agents P, Q, and R working together can complete the project in 20 days. Therefore, their combined daily work rate is:

$\text{Rate}\left(P+Q+R\right)=\frac{1}{20}$

By subtracting the combined rate of P and Q from the combined rate of P, Q, and R, we can find the individual daily work rate of agent R:

$\text{Rate}\left(R\right)=\text{Rate}\left(P+Q+R\right)-\text{Rate}\left(P+Q\right)$

$\text{Rate}\left(R\right)=\frac{1}{20}-\frac{1}{24}=\frac{6-5}{120}=\frac{1}{120}$

So, agent R alone completes $\frac{1}{120}$ of the project per day.

Step 2: Determine the work rate of agent P

The problem states that the output rate of agent R is half of that of agent P:

$\text{Rate}\left(R\right)=\frac{1}{2}\times \text{Rate}\left(P\right)$

Therefore, the daily work rate of agent P is twice that of agent R:

$\text{Rate}\left(P\right)=2\times \text{Rate}\left(R\right)=2\times \frac{1}{120}=\frac{1}{60}$

Step 3: Determine the work rate of agent Q

Now, we can find the individual rate of agent Q by subtracting the rate of agent P from the combined rate of P and Q:

$\text{Rate}\left(Q\right)=\text{Rate}\left(P+Q\right)-\text{Rate}\left(P\right)$

$\text{Rate}\left(Q\right)=\frac{1}{24}-\frac{1}{60}=\frac{5-2}{120}=\frac{3}{120}=\frac{1}{40}$

Step 4: Calculate time needed for Q and R working together

Next, find the combined daily rate when agents Q and R work together:

$\text{Rate}\left(Q+R\right)=\text{Rate}\left(Q\right)+\text{Rate}\left(R\right)$

$\text{Rate}\left(Q+R\right)=\frac{1}{40}+\frac{1}{120}=\frac{3+1}{120}=\frac{4}{120}=\frac{1}{30}$

Finally, the total time required for Q and R to finish the project together is the reciprocal of their combined daily rate:

$\text{Time}=\frac{1}{\text{Rate}\left(Q+R\right)}=\frac{1}{1/30}=30\text{ days}$

Thus, Q and R working together will take 30 days to finish the project.