P and Q together can complete a job in 24 days. P alone can complete the same job in 32 days. If they work together for 8 days and then P leaves, how many additional days will Q alone take to finish the remaining work?

Introduction / Context:
This is a standard work and time problem in which two people start a task together and later one person leaves. The aim is to determine how long the remaining person will need to finish the job. Questions like this appear frequently in banking, management, and government exams to test a candidate's ability to handle fractional work and sequential stages of a task. Correctly tracking how much work is completed at each stage is the core skill required here.

Given Data / Assumptions:

• P and Q together complete the job in 24 days.
• P alone completes the job in 32 days.
• P and Q work together for 8 days, and then P stops working.
• Q works alone after that until the job is completed.
• Both P and Q work at constant daily rates.

Concept / Approach:
The key idea is to calculate the daily work rates of P and Q using the total time they take individually and together. The combined rate gives the fraction of work done per day when both work together. We first compute how much of the job is finished in the initial 8 days when both are working. Subtracting this from the full job gives the fraction of work remaining. Then we divide the remaining work by Q's individual rate to obtain the time taken by Q alone to complete the task.

Step-by-Step Solution:
Step 1: Combined daily rate of P and Q is 1 / 24 of the job per day. Step 2: Daily rate of P alone is 1 / 32 of the job per day. Step 3: Therefore Q's daily rate is 1/24 minus 1/32 which is (4 minus 3) / 96 = 1 / 96 of the job per day. Step 4: In 8 days working together, P and Q complete 8 times 1/24 which equals 1 / 3 of the job. Step 5: Fraction of work remaining is 1 minus 1/3 which equals 2 / 3 of the job. Step 6: Time for Q alone to finish the remaining 2 / 3 at a rate of 1 / 96 is (2/3) divided by (1/96) which is (2/3) times 96 = 64 days.

Verification / Alternative check:
As a check, imagine the entire work as 96 equal units. P and Q together complete 96 / 24 = 4 units per day. P alone completes 96 / 32 = 3 units per day, so Q alone must complete 1 unit per day. In 8 days together, they complete 8 times 4 which equals 32 units. Remaining work is 64 units. Q alone at 1 unit per day will indeed take 64 days. This unit method gives the same answer and confirms our fraction based calculations.

Why Other Options Are Wrong:

• 56 days would correspond to a daily rate higher than Q's actual rate, which contradicts the derived value.
• 54 days and 60 days similarly do not match the required remaining work divided by Q's rate.
• Only 64 days matches exactly with the remaining two thirds of the job at Q's daily work rate.

Common Pitfalls:
Learners sometimes subtract times directly instead of subtracting rates, which is incorrect. Another pitfall is miscalculating the fraction of work completed in the initial days or forgetting to subtract it from the whole job. It is important to remember that work changes linearly with time only when the rate is constant, so we must always work with rates first and then compute fractions of work using multiplication, not by subtracting days.

Final Answer:
Once P leaves, Q alone will need 64 days to finish the remaining work.