Efficient Task Completion with A and B: A Mathematical Approach

In this article, we will explore a mathematical problem involving two workers, A and B, and how they can complete a task by working on alternate days. The problem is similar in nature to the classic work rate problems in mathematics. We will break down the solution step-by-step, using logical reasoning rather than complex calculations such as Least Common Multiple (LCM).

Understanding the Problem

A can complete a task in 12 days. B can complete the same task in 18 days. We need to determine how long it will take them to finish the entire task if they work on alternate days starting with A.

Solution with Logical Reasoning

Let's start with a simplified approach that leverages basic logic to find the answer.

Situation 1: A Starts

When A starts the work, we explore the possibility of completing the task in 17 days. Here, A works for 9 days, and B works for 8 days.

By breaking down the work as follows:

AB AB AB AB AB AB AB AB A

Here, A and B work in pairs for 8 days, and finally, A finishes the remaining part on the 9th day. This gives us the pattern and confirms that A and B together can complete the task in 17 days.

Situation 2: B Starts

If B starts, the task can be completed in 17.75 days. Here, B works for 9 days, and A works for 8.75 days.

Breaking it down as follows:

BA BA BA BA BA BA BA BA B 3/4A

The key here is that the total work done from Day 1 to Day 16 is the same in both situations. By comparing these days, we can see that the amount of work done by A and B in the final incomplete cycle balances out.

Reasoning and Conclusion

Let's use the information from both situations to derive the required work rates.

A  B   3/4A

This indicates that the work done by A in 1 day is equivalent to the work done by B in 4 days. Therefore, if B completes 1 unit of work in 1 day, A completes 4 units of work in 1 day.

From the first situation, we know that A works for 9 days and B works for 8 days. This gives us a total work done of 44 units in 17 days.

If they worked together for one day, they would complete 5 units of work. Therefore, in 8 days, they would complete 40 units of work. The remaining work is 4 units, which they can complete in 4/5 days.

Thus, the total time taken is 8.8 days, which is rounded to 9 days considering we are dealing with whole days in practical scenarios.

Using Mathematical Work Rates

To provide a more detailed mathematical approach:

Work Rates

A can complete the task in 12 days, so the work rate of A is:

Work rate of A  1/12 of the task per day

B can complete the task in 18 days, so the work rate of B is:

Work rate of B  1/18 of the task per day

Total Work Done in Two Days

On the first day, A works, and on the second day, B works. The total work done in two days is the sum of their individual contributions:

Work done in 2 days  Work done by A  Work done by B  1/12  1/18

Using a common denominator of 36, we get:

1/12  3/36  1/18  2/36

Therefore:

Work done in 2 days  3/36   2/36  5/36

Total Days to Complete the Task

Let ( n ) be the number of complete 2-day cycles needed to finish the task. After ( n ) complete cycles which take ( 2n ) days, the total work done is:

5/36 * n  1 (the whole task)

Solving for ( n ):

5/36 * n  1

( n 36/5 7.2 )

Since ( n ) must be a whole number, we take ( n 7 ) complete cycles which take ( 2 * 7 14 ) days. After 14 days, the remaining work is:

1 - 35/36  1/36

Final Day of Work

On the 15th day, A works again and can complete:

A can do 1/12 of the task in one day.

Since ( 1/12 > 1/36 ), A can finish the remaining work on the 15th day.

Therefore, A and B will finish the entire task in 15 days.

Keywords: work rate, alternative days, mathematical problem solving