Solving the Project Completion Problem

This problem involves calculating the total time taken to complete a project when two people, A and B, work together for some time, and then one person, A, quits before the project finishes. We need to find the total number of days the project was completed.

Understanding Individual Work Rates

First, let's figure out how much work each person can do in a single day. This is their work rate.

• A can complete the project in 10 days. So, in 1 day, A completes \( \frac{1}{10} \) of the project.
• B can complete the project in 15 days. So, in 1 day, B completes \( \frac{1}{15} \) of the project.

Work Done in the Last Few Days

We are told that A quit 5 days before the project was completed. This means that during the last 5 days, only B was working on the project.

Let's calculate the amount of work B completed in these last 5 days:

Work done by B in 5 days = B's daily rate \(\times\) Number of days

Work done by B in 5 days = \( \frac{1}{15} \times 5 = \frac{5}{15} = \frac{1}{3} \) of the project.

Work Done by A and B Together

The project has a total work of '1' (representing the whole project). If B completed \( \frac{1}{3} \) of the work alone at the end, the remaining work must have been done by both A and B working together before A quit.

Remaining work = Total work - Work done by B alone

Remaining work = \( 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \) of the project.

Calculating Combined Work Rate

When A and B work together, their work rates add up to find their combined work rate per day.

Combined daily rate of A and B = A's daily rate + B's daily rate

Combined daily rate = \( \frac{1}{10} + \frac{1}{15} \)

To add these fractions, we find a common denominator, which is 30.

Combined daily rate = \( \frac{1 \times 3}{10 \times 3} + \frac{1 \times 2}{15 \times 2} = \frac{3}{30} + \frac{2}{30} = \frac{3+2}{30} = \frac{5}{30} = \frac{1}{6} \) of the project per day.

Finding the Time A and B Worked Together

Now we know that \( \frac{2}{3} \) of the project was completed by A and B working together at a combined rate of \( \frac{1}{6} \) per day. We can find the number of days they worked together.

Days A and B worked together = Remaining work / Combined daily rate

Days A and B worked together = \( \frac{2/3}{1/6} = \frac{2}{3} \div \frac{1}{6} = \frac{2}{3} \times \frac{6}{1} \)

Days A and B worked together = \( \frac{12}{3} = 4 \) days.

Total Project Completion Time

The total time for the project is the sum of the time A and B worked together and the time B worked alone.

Total time = Days A and B worked together + Days B worked alone

Total time = 4 days + 5 days = 9 days.

So, the project was completed in 9 days.

Revision Table: Project Time and Work

Additional Information: Time and Work Concepts

Time and work problems often involve calculating rates and combining them. Here are some key ideas:

• Efficiency: A person who takes less time to complete a project is more efficient and has a higher work rate.
• Inverse Relationship: Time taken to complete work is inversely proportional to the work rate. If the rate doubles, the time taken halves.
• Working Together: When people work together, their individual work rates add up to find the combined rate, assuming they don't hinder each other.
• Partial Work: If someone works for only a part of the total time, calculate the fraction of work they complete in that period. The remaining work is then completed by others.
• Left/Joined Scenarios: Problems where someone leaves or joins require breaking down the work into phases: the time when a certain group worked, and the time when a different group (or individual) worked.

Understanding these basic principles helps in tackling various types of time and work problems, including those where workers start or stop at different times.