Collaboration Efficiency: How Long Will It Take Painter A and Painter B to Paint the Wall Together?

Imagine a scenario where Painter A can complete a painting job in 3 hours, and Painter B takes only 2 hours to finish the same job. The question arises: How long will it take them to complete the job if they work together? To answer this, we can use the concept of their combined work rates.

Understanding Work Rates and Combined Efforts

Let's start by calculating the work rates of each painter. A work rate is the fraction of the job each painter can complete in one hour.

Calculate the work rates: Painter A: 1 job / 3 hours 1/3 of the job per hour Painter B: 1 job / 2 hours 1/2 of the job per hour Add their work rates to find the combined rate: Combined rate 1/3 1/2

For addition, we need a common denominator. The least common multiple of 3 and 2 is 6.

1/3 2/6, 1/2 3/6

Combined rate 2/6 3/6 5/6 of the job per hour

Find the time to complete the job together:

If they work together at a rate of 5/6 of the job per hour, the time t to complete one whole job is the reciprocal of their combined rate:

t 1 / (5/6) 6/5 hours

.Convert to minutes if needed: 6/5 hours 1.2 hours 1 hour and 12 minutes.

Thus, working together, Painter A and Painter B can complete the job in 1 hour and 12 minutes.

Generalizing the Concept

In more complex scenarios, the calculation method remains similar but the numbers change. Here are a couple more examples to illustrate the concept:

Example 1: A 5 hours, B 8 hours

Using the least common multiple (LCM) approach:

Calculate the efficiency: LCM of 5 and 8 is 40 units. Efficiency of A: 40/5 8 units per hour. Efficiency of B: 40/8 5 units per hour. Combined efficiency: 8 5 13 units per hour. Total time taken: 40/13 3 hours and 1/13 hours.

Thus, working together, they can complete the job in 3 hours and 1/13 of an hour.

Example 2: A 5 hours, B 8 hours - Revisited

Using the fractions method:

Efficiency of A: 1/5 per hour. Efficiency of B: 1/8 per hour. Combined efficiency: 1/5 1/8 13/40 (common denominator method). Total time taken: 1 / (13/40) 40/13 hours.

Convert to hours and minutes: 40/13 hours is approximately 3 hours and 4 minutes and 36.923 seconds.

Thus, working together, they can complete the job in approximately 3 hours, 4 minutes, and 36.923 seconds.

Generalizing the Formula

For any two painters (Person A and Person B) with individual times A 5 hours and B 8 hours, we can derive the following steps:

Calculate individual work rates: Person A: 1/5 of the job per hour. Person B: 1/8 of the job per hour. Combined work rate: 1/5 1/8 13/40 of the job per hour. Time to complete the job together: 1 / (13/40) 40/13 3 hours 04 minutes 36.923 seconds.

Thus, working together, they can complete the job in 3 hours, 4 minutes, and 36.923 seconds.

Conclusion:

Whether you use the common denominator method or the LCM method, the key to solving these types of problems is to find the combined work rate and then use the reciprocal to determine the time needed. This approach can be applied to any similar problems where you need to find the time taken for collaborative efforts.