Collaborative Efforts: Calculating Work Completion When Laborers Work Together

In this article, we will explore a classic collaborative labor problem and delve into the mathematical steps required to determine how much work can be completed when two laborers work together and then separately. We'll tackle a specific scenario where one laborer can complete a job in 60 days, and another in 40 days. By the end, you'll understand how to apply these calculations to real-world scenarios.

The Problem At Hand

Let's say we have two individuals, A and B, who are tasked with completing the same piece of work. A can complete the work in 60 days, while B can finish it in 40 days. They decide to work together for 12 days and then A leaves, leaving B to complete the remaining work alone. We need to determine how many additional days B will require to finish the work.

Step-by-Step Calculation

Step 1: Calculate Individual Work Rates

To begin, we calculate the work rate for each individual. A's work rate can be calculated as:

Work rate of A 1/60 of the work per day

Similarly, B's work rate is:

Work rate of B 1/40 of the work per day

Step 2: Determine Combined Work Rate

When A and B work together, their combined work rate is:

Combined work rate (1/60) (1/40)

To add these fractions, we need a common denominator of 120:

1/60 2/120, 1/40 3/120, thus combined work rate (2/120) (3/120) 5/120 1/24

This means together, A and B can complete 1/24 of the work in one day.

Step 3: Calculate Work Done in 12 Days

Now, we calculate the amount of work completed in the first 12 days:

Work done in 12 days 12 * (1/24) 1/2

Thus, half of the work is completed in the initial 12 days.

Step 4: Determine Remaining Work

With half the work completed, the remaining work is also 1/2.

Step 5: Calculate Days for B to Finish Remaining Work

B's individual work rate is 1/40 of the work per day. To determine how many additional days B needs to complete the remaining 1/2 work:

Days required by B (Remaining work) / (B's work rate) (1/2) / (1/40) 20 days

Therefore, B will need 20 more days to complete the remaining work.

Further Exploration

Let's look at an additional example where laborers A and B work on a combined task, but with a twist:

Suppose A can complete a work in 16 days and B in 44 days. The combined work rate is calculated as:

1/work rate of A - 1/work rate of B 0

1/a - 1/b 1/30

1 - (16/a) - (44/b) 0

We then simplify using the initial combined rate:

b - 8/15b - 16 44, solving for b we get: b 60 days

This deeper dive into work rate calculations not only reinforces the initial problem but also highlights the importance of systematic problem-solving in real-world applications.

Conclusion

The example provided above demonstrates the practical application of work rate calculations in the context of collaborative labor. Whether you are managing a construction project, organizing a community event, or just curious about problem-solving techniques, understanding work rates can be incredibly useful.

By breaking down the problem into manageable steps, we can ensure accuracy and efficiency in completing tasks, especially when working in teams.