Calculating Total Time for Two Painters to Complete a Task Together

In this article, we will explore the concept of collaborative work and how to calculate the total time required for two painters to complete a task together, using both arithmetic and algebraic methods. We will determine the efficiency of each painter, find their combined work rate, and calculate the total time needed to complete the job.

Introduction to Painting Efficiency

Painting is a common activity where efficiency plays a crucial role. Different painters have different speeds at which they can complete a task. By understanding and calculating their individual work rates, we can determine how much work each can accomplish in a unit of time.

Understanding the Task and Painters' Efficiency

Let's consider two painters, A and B, working on a wall.

A can paint a wall in 5 hours.

B can paint a wall in 8 hours.

E(fficency) of A 40/5 8 units per hour

E(fficency) of B 40/8 5 units per hour

Combined efficiency of A and B 8 5 13 units per hour

Total time taken by A and B 40/13 3 hours and 1/13 hours

General Formula for Work Rate

Let's generalize the formula for work rate and apply it to the given scenario.

Efficiency of A (1/5) per hour

Efficiency of B (1/8) per hour

Combined efficiency of A and B (1/5) (1/8) 13/40

Total time taken by A and B 1 / (1/5 1/8) 40/13 hours 3 hours 4 minutes 37 seconds

Another method involves solving the equation:

1/t 1/5 1/8 (8 5)//40 13/40

t 40/13 3.076923 hours

Practical Application

Let's apply this knowledge to a real-world scenario:

Person A can paint a wall in 5 hours, and person B can paint the same wall in 8 hours.

Let's calculate the time it takes for both to paint the wall together:

Efficiency of A (1/5) per hour

Efficiency of B (1/8) per hour

Combined efficiency of A and B (1/5) (1/8) 13/40

Total time taken by A and B 1 / (1/5 1/8) 40/13 hours 3 hours 4 minutes 37 seconds

Further Analysis

It's important to note that the total time required for the painters to complete the job together depends on their combined efficiency. Here, the combined efficiency is 13 units per hour, hence the total time is calculated as follows:

40/13 hours 3.076923 hours ≈ 3 hours 4 minutes 37 seconds

Another example can be given to illustrate the complexity of such tasks:

Painter A can complete a 2-hour job in 1 hour, while painter B can complete a 1-job in 8 hours.

In one hour, painter A can complete 1/2 of the job, and painter B can complete 1/8 of the job.

Combined, they can complete (1/2 1/8) 5/8 of the job in one hour.

The reciprocal of 5/8 gives the total time required, which is 8/5 hours or 1 hour 36 minutes.

Therefore, painter A and B together can complete the job in approximately 3 hours 4 minutes 37 seconds when working in parallel.

Conclusion: Efficient time management for collaborative tasks like painting requires a clear understanding of the work rate of each individual and their combined efficiency. This method can be applied to various real-world scenarios, making it a valuable tool in project management and resource allocation.