A Collaborative Approach to Product Completion

Introduction

The efficiency of a team can be significantly enhanced by understanding and leveraging the individual and combined work rates of its members. This article explores a practical scenario where three individuals, A, B, and C, collaborate to complete 14 products, given their specific work rates. Understanding how to calculate and apply these rates is crucial for optimizing productivity and project timelines.

The Individual and Combined Work Rates

Let's delve into the problem step by step, starting with the individual work rates of A, B, and C.

Calculating A's Work Rate

First, we calculate A's work rate. A alone takes 10 hours to complete one product. Therefore, A's work rate can be calculated as follows:

Rate of A 1 product / 10 hours 0.1 products per hour

Calculating the Combined Work Rate of B and C

Next, we calculate the combined work rate of B and C. B and C working together can complete one product in 4 hours. The combined work rate of B and C is:

Rate of B and C 1 product / 4 hours 0.25 products per hour

Calculating the Combined Work Rate of A, B, and C

To find out the work rate when A, B, and C work together, we add the individual work rates of A and B and C:

Total Rate Rate of A Rate of B and C 0.1 0.25 0.35 products per hour

Calculating the Total Time Required to Complete 14 Products

Now that we know the total work rate, we can determine the time required to complete 14 products. Let ( T ) represent the time in hours required to complete 14 products:

Total Rate × T Total Work Done
0.35 products per hour × T 14 products
Solving for T:

T 14 products / 0.35 products per hour 40 hours

Verification Using Alternative Method

To verify the calculation, we can use an alternative method based on the combined work rates:

Using the Combined Work Rates to Find Time

Given that A, B, and C together can complete 14 products, we use the equation:

W/B W/C 1/4 (from the given data, 1/b 1/c 1/4)
Also, we have T/10 - 1/B - 1/C 14 (combined work of A, B, C)
Let's solve these step by step:

7T/20 14
Solving for T, we get:
T 14 * 20 / 7 40 hours

Conclusion

Through both methods, we have confirmed that A, B, and C will require 40 hours to complete 14 products when working together. This collaborative effort highlights the importance of understanding individual and combined work rates in optimizing productivity. Whether it's in a professional or academic setting, such calculations can significantly enhance project planning and execution.