P alone can complete a piece of work in 24 days. The time taken by P to finish one third of the work is equal to the time taken by Q to finish half of the same work. How many days will P and Q working together take to complete the entire work?

Introduction / Context:
This time and work question checks understanding of individual work rates and combined work rates. Many aptitude tests use such problems to test how well candidates convert verbal statements into algebraic relations. Here we compare how fast two workers, P and Q, can complete fractions of the same job, and then use that information to find how long they will take together to complete the entire work. The key skill is to translate the condition about one third and half of the work into daily work rates, then combine them in a clean and systematic way.

Given Data / Assumptions:

• P alone can complete the whole work in 24 days.
• Time taken by P to complete one third of the work equals time taken by Q to complete half of the work.
• We assume both P and Q work at constant rates every day.
• We are asked for total days when P and Q work together on the entire job.

Concept / Approach:
The basic concept is rate of work. If a person completes a job in D days, then the daily work rate is 1 / D of the job per day. Work done equals rate multiplied by time. We first find the time P needs for one third of the work using P's known rate. That same time is used to compute Q's rate for half of the work. Once both daily rates are known, we add the rates to obtain the combined rate when P and Q work together. Finally we invert the combined rate to get the total time required to finish one complete job.

Step-by-Step Solution:
Step 1: Rate of P is 1 / 24 of the work per day. Step 2: Time taken by P to complete one third of the work is (1/3) / (1/24) = 8 days. Step 3: In the same 8 days, Q completes half of the work, so Q's rate is (1/2) / 8 = 1 / 16 per day. Step 4: Combined daily rate of P and Q together is 1/24 + 1/16. Step 5: 1/24 + 1/16 = (2 + 3) / 48 = 5 / 48 of the work per day. Step 6: Total time to complete one full job together is 1 / (5/48) = 48 / 5 = 9.6 days, which is 9 3/5 days.

Verification / Alternative check:
We can verify by checking total work done in 9.6 days at the combined rate. Combined rate is 5 / 48 of the work per day. Multiplying 5 / 48 by 9.6 gives 5 / 48 times 48 / 5 which simplifies exactly to 1. This confirms that the entire work is completed in 9.6 days when both P and Q work together. The fractions match perfectly, so the calculation is consistent and there is no approximation involved in this solution.

Why Other Options Are Wrong:

• 7 days would require a much higher combined rate than 5 / 48, which is not supported by the given individual rates.
• 6 3/7 days is less than 7 days and is even more unrealistic for such slow individual rates.
• 8 days would mean they finish faster than the time P alone takes to finish one third of the work, which contradicts the given condition.

Common Pitfalls:
Many learners confuse time taken for a fraction of work with the daily rate directly. Another common mistake is to add the times taken by P and Q instead of adding their rates. Some students also misread the statement and assume one third and half refer to combined work instead of individual work. Carefully distinguishing between rate, time, and fraction of work is crucial for solving problems like this accurately in exams.

Final Answer:
The time required for P and Q working together to complete the entire work is 9 3/5 days.