P and Q can complete a job in 24 days working together. P can alone complete it in 32 days. Both of them worked together for 8 days and then P left. The number of days Q will take to complete the remaining job is:

The correct option is 64 days.

To find the number of days Q will take to complete the remaining job, we can break the problem down step-by-step using the concept of work rates.

Step 1: Determine the total work
Let us assume the total amount of work is the Least Common Multiple (LCM) of the days given for P and Q together (24 days) and P alone (32 days).
The LCM of 24 and 32 is 96.
So, let the total work be 96 units.

Step 2: Find the individual work rates per day
The combined daily work rate of P and Q working together is:

$$\text{Rate of (P + Q)}=\frac{96}{24}=4\text{ units per day}$$

The daily work rate of P working alone is:

$$\text{Rate of P}=\frac{96}{32}=3\text{ units per day}$$

Now, we can find the daily work rate of Q by subtracting P's rate from the combined rate:

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

$$\text{Rate of Q}=4-3=1\text{ unit per day}$$

Step 3: Calculate the work completed in the first 8 days
P and Q worked together for 8 days. The total work completed during this period is:

$$\text{Work completed}=8\text{ days}\times 4\text{ units per day}=32\text{ units}$$

Step 4: Find the remaining work
Subtract the completed work from the total work to find what remains:

$$\text{Remaining work}=96-32=64\text{ units}$$

Step 5: Calculate the time Q takes to finish the remaining work
Since P left, Q has to complete the remaining 64 units of work alone. At Q's rate of 1 unit per day, the time required is:

$$\text{Time taken by Q}=\frac{\text{Remaining work}}{\text{Rate of Q}}$$

$$\text{Time taken by Q}=\frac{64}{1}=64\text{ days}$$

Thus, Q will take 64 days to complete the remaining job.