Step 1: Conceptual Foundation:
This problem falls under the category of work and time. The established method involves calculating the rate of work, defined as the proportion of a job completed per unit of time.
Rate of Work = \( \frac{1}{\text{Total Time to Complete Job}} \)
Step 2: Methodology Outline:
1. Determine the combined work rate of P and Q (\(R_{P+Q}\)).
2. Determine the individual work rate of P (\(R_P\)).
3. Compute the individual work rate of Q (\(R_Q\)) using the formula \(R_Q = R_{P+Q} - R_P\).
4. Calculate the total work completed by P and Q over 8 days.
5. Determine the amount of work remaining.
6. Calculate the duration required for Q to finish the remaining work.
Step 3: Detailed Calculation:
The combined rate of work for P and Q is \(R_{P+Q} = \frac{1}{24}\) of the job per day.
The individual rate of work for P is \(R_P = \frac{1}{32}\) of the job per day.
The individual rate of work for Q is calculated as \(R_Q = R_{P+Q} - R_P = \frac{1}{24} - \frac{1}{32}\).
To perform the subtraction, a common denominator is required. The least common multiple of 24 and 32 is 96.
Thus, \(R_Q = \frac{4}{96} - \frac{3}{96} = \frac{1}{96}\). This indicates that Q, working alone, can complete the entire job in 96 days.
In the 8 days that P and Q worked together, the work completed was \(R_{P+Q} \times 8 = \frac{1}{24} \times 8 = \frac{8}{24} = \frac{1}{3}\) of the job.
The remaining portion of the job is \(1 - \text{Work Done} = 1 - \frac{1}{3} = \frac{2}{3}\) of the job.
Following P's departure, Q must complete the remaining \( \frac{2}{3} \) of the job.
The time Q will take is computed using the formula: \( \frac{\text{Remaining Work}}{R_Q} \).
Time = \( \frac{2/3}{1/96} = \frac{2}{3} \times 96 \)
Time = \( 2 \times \frac{96}{3} = 2 \times 32 = 64 \) days.
Step 4: Conclusion:
Q will require 64 days to complete the remaining work.