P and Q can complete a project in 20 days and 12 days respectively when working alone. If they work together on the project, in how many days will they complete 40% of the entire project?

Introduction / Context:
This question is about two workers P and Q who can each complete a project at different speeds. We are not asked for the full completion time, but instead for the time required to finish 40 percent of the project when both work together. Problems of partial work completion like 40 percent or 25 percent are very common in aptitude exams and help build speed with work rate calculations.

Given Data / Assumptions:

• P alone completes the project in 20 days.
• Q alone completes the project in 12 days.
• They work together on the same project.
• We must find the time taken to complete 40% (that is, 0.4) of the project.
• Total project work is considered as 1 full job.

Concept / Approach:
The fundamental approach uses work rate addition. If a person finishes a job in N days, their work rate is 1/N job per day. When two people work together, their rates add. After finding their combined rate, we compute the time needed to complete a given fraction of the job using time = required work / rate. For 40 percent, the required work is 2/5 or 0.4 of the job.

Step-by-Step Solution:
Step 1: Let the total work be 1 job. Step 2: Rate of P = 1/20 job per day. Step 3: Rate of Q = 1/12 job per day. Step 4: Combined rate of P and Q = 1/20 + 1/12. Step 5: The LCM of 20 and 12 is 60, so 1/20 = 3/60 and 1/12 = 5/60. Step 6: Combined rate = (3 + 5)/60 = 8/60 = 2/15 job per day. Step 7: Required fraction of work = 40% = 40/100 = 2/5 of the job. Step 8: Time taken = (required work) / (combined rate) = (2/5) / (2/15). Step 9: Simplify: (2/5) / (2/15) = (2/5) * (15/2) = 15/5 = 3 days.

Verification / Alternative check:
As a check, we can find how much work they do in 3 days at their combined rate of 2/15 job per day. In 3 days they complete 3 * (2/15) = 6/15 = 2/5 of the job, which is exactly 40 percent. This confirms that the calculated time of 3 days is accurate and consistent with the given data.

Why Other Options Are Wrong:
1.5 days and 2 days are too short, because in that time they would complete less than 40 percent of the project at the rate of 2/15 job per day. Six days and nine days are too long and would result in completing more than the required 40 percent. Only 3 days matches the correct fraction of the work.

Common Pitfalls:
Students sometimes mistakenly find the time for full completion and then take 40 percent of that time, which is incorrect because work rate is linear but time is inversely related to rate. It is important to find the fraction of work first and then divide by the combined rate. Also, care must be taken when working with fractions and LCMs to avoid arithmetic mistakes.

Final Answer:
P and Q together will complete 40% of the project in 3 days.