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?

Correct Answer: The time needed to finish the project if Q and R work together is 30 days.

Step-by-Step Explanation:

Let the total work to be completed be represented as 1 whole unit, or we can work in terms of daily work rates (work done per day).

Let $W_P$, $W_Q$, and $W_R$ be the work done in one day by agents P, Q, and R, respectively.

Step 1: Calculate the combined daily work rate of P and Q.

P and Q together can finish the project in 24 days. Therefore, their combined 1-day work rate is:

$W_P+W_Q=\frac{1}{24}$

Step 2: Calculate the combined daily work rate of P, Q, and R.

Working together, P, Q, and R can finish the project in 20 days. Their combined 1-day work rate is:

$W_P+W_Q+W_R=\frac{1}{20}$

Step 3: Find the daily work rate of agent R ($W_R$).

Subtracting the rate of (P + Q) from the rate of (P + Q + R):

$W_R=(W_P+W_Q+W_R)-(W_P+W_Q)$

$W_R=\frac{1}{20}-\frac{1}{24}$

Taking the LCM of 20 and 24, which is 120:

$W_R=\frac{6-5}{120}=\frac{1}{120}$

Step 4: Find the daily work rate of agent P ($W_P$).

We are given that the output rate of R is half of that of P ($W_R=\frac{1}{2}{W}_P$). Therefore:

$W_P=2\times W_R=2\times \frac{1}{120}=\frac{1}{60}$

Step 5: Find the daily work rate of agent Q ($W_Q$).

Using $W_P+W_Q=\frac{1}{24}$:

$W_Q=\frac{1}{24}-W_P=\frac{1}{24}-\frac{1}{60}$

Taking the LCM of 24 and 60, which is 120:

$W_Q=\frac{5-2}{120}=\frac{3}{120}=\frac{1}{40}$

Step 6: Calculate the combined daily work rate of Q and R ($W_Q+W_R$).

$W_Q+W_R=\frac{1}{40}+\frac{1}{120}$

Taking the common denominator 120:

$W_Q+W_R=\frac{3+1}{120}=\frac{4}{120}=\frac{1}{30}$

Conclusion:

Since Q and R together complete $\frac{1}{30}$ of the project per day, the time required for Q and R to finish the entire project working together is:

$\text{Total Days}=\frac{1}{1/30}=30\text{days}$