Introduction

Understanding how different individuals can work together efficiently to complete a task is crucial in both professional and personal settings. This article explores a common problem: determining the time taken for two individuals to complete a job together, based on their individual work rates. We will use the concept of work rate and provide practical examples to help readers grasp the underlying concepts.

Understanding Work Rates

When two individuals work together to complete a task, their combined work rate is crucial. If one individual can complete a task in 3 hours and another in 2 hours, we can determine the time they will take to complete the task together using the concept of work rates.

Work Rate of Each Individual

- The first man can complete the work in 3 hours so his work rate is ( frac{1}{3} ) of the work per hour.

- The second man can complete the work in 2 hours so his work rate is ( frac{1}{2} ) of the work per hour.

Combined Work Rate

When both men work together, their combined work rate is:

[ text{Combined Rate} frac{1}{3} frac{1}{2} ]

To add these fractions, find a common denominator which is 6:

[ frac{1}{3} frac{2}{6} quad text{and} quad frac{1}{2} frac{3}{6} ]

So the combined work rate is:

[ text{Combined Rate} frac{2}{6} frac{3}{6} frac{5}{6} text{ of the work per hour} ]

Time to Complete the Work Together

If their combined work rate is ( frac{5}{6} ) of the work per hour, the time ( t ) taken to complete 1 whole work is:

[ t frac{1 text{ work}}{frac{5}{6} text{ work/hour}} frac{6}{5} text{ hours} ]

This simplifies to:

[ t 1.2 text{ hours} text{ or } 1 text{ hour and } 12 text{ minutes} ]

Thus, if both men work together, they will take 1.2 hours or 1 hour and 12 minutes to complete the work.

Additional Examples

Let's explore some additional examples to reinforce the concept:

Example 1

- First man completes the work in 5 hours so his work rate is ( frac{1}{5} ) of the work per hour.

- Second man completes the work in 4 hours so his work rate is ( frac{1}{4} ) of the work per hour.

Combined work rate is:

[ text{Combined Rate} frac{1}{5} frac{1}{4} frac{4}{20} frac{5}{20} frac{9}{20} text{ of the work per hour} ]

Time taken is the reciprocal of the combined work rate:

[ t frac{1 text{ work}}{frac{9}{20} text{ work/hour}} frac{20}{9} text{ hours} approx 2 text{ hours and } 13 text{ minutes} ]

Hence, they will take approximately 2 hours and 13 minutes to complete the work together.

Example 2

- One man can complete the work in 4 hours so his work rate is ( frac{1}{4} ) of the work per hour.

- The second man can complete the work in 5 hours so his work rate is ( frac{1}{5} ) of the work per hour.

Combined work rate is:

[ text{Combined Rate} frac{1}{4} frac{1}{5} frac{5}{20} frac{4}{20} frac{9}{20} text{ of the work per hour} ]

Time taken is again the reciprocal of the combined work rate:

[ t frac{1 text{ work}}{frac{9}{20} text{ work/hour}} frac{20}{9} text{ hours} approx 2 text{ hours and } 13 text{ minutes} ]

Both scenarios demonstrate that the combined work rate approach is consistent and helps in determining the time taken for multiple individuals working together.

Conclusion

The concept of work rate is fundamental in understanding how individuals can work together to complete tasks efficiently. By applying this concept, you can estimate the time required for collaboration, making it a valuable tool in various fields.