P and Q can complete a certain project in 150 days and 100 days respectively when working alone. If both P and Q work together on the project, in how many days will they complete 60 percent of the project?

Introduction / Context:

This problem focuses on combining work rates of two workers and calculating the time required to complete a specific fraction of a project. Instead of asking for the time to finish the entire job, it asks for 60 percent completion, which tests the ability to handle partial work and proportional reasoning in time and work problems.

Given Data / Assumptions:

• P alone can complete the project in 150 days.
• Q alone can complete the same project in 100 days.
• P and Q now work together on the project.
• We need to find the number of days required for them to complete 60 percent of the project.
• Total work is taken as one unit and work rates remain constant.

Concept / Approach:

The key idea is to find the combined daily work rate of P and Q by adding their individual rates. This combined rate tells us how much of the project they finish in one day. To find the time required to complete 60 percent of the project, we divide the required fraction of work by this combined rate. This is standard rate times time equals work reasoning applied to a partial completion objective.

Step-by-Step Solution:

Verification / Alternative Check:

In 36 days at a daily rate of 1/60, the work completed is 36 * 1/60 = 36/60 = 3/5, which is exactly 60 percent of the project. This confirms the correctness of the calculation. Additionally, if they wanted to complete the entire project, they would need 60 days at the same rate, which is consistent since 36 is 60 percent of 60.

Why Other Options Are Wrong:

A time of 12 days would yield only 12/60 = 1/5 of the work, far less than 60 percent. A time of 42 days produces 42/60 = 7/10, which is 70 percent, more than required. A time of 72 days would exceed the time needed to finish the whole project. Only 36 days gives exactly 60 percent completion.

Common Pitfalls:

Errors typically occur when students try to average the given times (150 and 100) or directly take 60 percent of the days instead of dealing with work rates. Another mistake is to miscalculate the combined rate, especially in the fraction addition step. Always convert times to rates, add rates, and then divide the required fraction of work by the combined rate.

Final Answer:

Working together, P and Q will complete 60 percent of the project in 36 days.