P and Q can complete a project in 6 days and 12 days respectively when working alone. If P and Q work together, in how many days will they complete 25% of the entire project?

Introduction / Context:
This is another time and work question focused on partial completion of a project. Two workers P and Q can finish a project individually in different times. We are asked how long they will take to complete only 25 percent of the work when they collaborate. This type of problem frequently appears in quantitative aptitude sections and helps reinforce rate and fraction concepts.

Given Data / Assumptions:

• P alone can complete the project in 6 days.
• Q alone can complete the project in 12 days.
• P and Q now work together on the same project.
• We must find how many days they need to complete 25% of the project.
• Total project work is considered as 1 full job.

Concept / Approach:
If a worker completes a job in N days, the worker daily rate is 1/N of the job per day. When two workers collaborate, their rates add. Once we determine the combined rate, we multiply by time to find the fraction of work completed. Here, we invert that idea to find the time required to complete a given fraction, namely 1/4 of the job. Therefore, time = required work / combined rate.

Step-by-Step Solution:
Step 1: Assume total work is 1 job. Step 2: P rate = 1/6 job per day. Step 3: Q rate = 1/12 job per day. Step 4: Combined rate of P and Q = 1/6 + 1/12. Step 5: LCM of 6 and 12 is 12, so 1/6 = 2/12 and 1/12 = 1/12. Step 6: Combined rate = (2/12 + 1/12) = 3/12 = 1/4 job per day. Step 7: Required fraction of the project = 25% = 1/4 of the job. Step 8: Time taken to complete 1/4 of the job at a rate of 1/4 job per day = (1/4) / (1/4) = 1 day.

Verification / Alternative check:
We can verify by checking how much work is done in exactly 1 day. At 1/4 job per day, P and Q together complete 1/4 of the project in 1 day, which is exactly 25 percent. This matches the requirement and confirms that the time needed is indeed 1 day.

Why Other Options Are Wrong:
If 2 days were taken, they would complete 2 * (1/4) = 1/2 of the project, which is too much. Similarly, 3 days, 4 days, or 8 days would lead to even more work, well beyond the required 25 percent. Only 1 day produces exactly one quarter of the job and therefore is the correct choice.

Common Pitfalls:
A common mistake is to compute the time needed for full completion and then simply divide by four, which in this case still gives 1.5 days, a different value. That approach is incorrect unless the combined rate is very carefully used. Another pitfall is mishandling percentages and fractions, so always convert 25 percent to 1/4 before proceeding with calculations.

Final Answer:
P and Q together will complete 25% of the project in 1 day.