Collaborative Work Rate: A Practical Example with Work, Rate, and Time Problems

Work, rate, and time problems are common in mathematics and primarily test a person's understanding of how different tasks can be accomplished by individual workers or teams, considering the rates at which they work. This article provides a detailed look into solving such problems, specifically focusing on collaborative work and how to determine the time required to complete a task when multiple workers are involved.

The Puzzle

The problem presented is: A can build a wall in 15 days, B can build it in 10 days, while C can completely demolish the wall in 12 days. If they start working at the same time, in how many days will the work be completed?

Understanding Work Rates

The first step in solving this problem is to understand the concept of work rate. Work rate essentially refers to the amount of work an individual can complete in a given period of time. It is usually expressed as a fraction or a fraction of the total work done in one unit of time (day, hour, etc.).

Determining Work Rates

We start by determining the work rates of A, B, and C:

A's work rate: A can build the wall in 15 days. So, A's work rate is 1/15 of the wall per day. B's work rate: B can build the wall in 10 days. So, B's work rate is 1/10 of the wall per day. C's work rate: C can completely demolish the wall in 12 days. So, C's work rate is negative and is -1/12 of the wall per day.

Combining Work Rates for Net Work Rate

To find the net work rate when A, B, and C work together, we need to add their individual work rates. The net work rate is calculated as:

Net Rate Rate of A Rate of B Rate of C

Substituting the values we have:

Net Rate 1/15 1/10 - 1/12

To combine these fractions, we need a common denominator. The least common multiple (LCM) of 15, 10, and 12 is 60. Converting each fraction to have this common denominator:

A's work rate: 1/15 4/60 B's work rate: 1/10 6/60 C's work rate: -1/12 -5/60

Adding these rates together:

Net Rate 4/60 6/60 - 5/60 (4 6 - 5)/60 5/60 1/12

This means that together, A, B, and C can complete 1/12 of the wall in one day.

Calculating the Total Time to Complete the Work

To find the time required to complete the wall, we take the reciprocal of the net work rate:

Time to complete the wall 1 / Net Rate 1 / (1/12) 12 days

Therefore, the work will be completed in 12 days.

Alternative Methods for Solving Work Rate Problems

Method 1: Using a Common Multiple

The first alternative method involves using a common multiple of the given time periods. For instance:

15 × 12 / (15 - 12) 180 / 3 60 days

Method 2: Adding Work Rates

Another method is to calculate the 1-day work for A and B together and then find the total time:

A's 1 day work 1/15 B's 1 day work 1/12 So 1 day work of AB 1/15 1/12 4/60 5/60 9/60 So they can do whole job in 1 / (9/60) 60/9 20/3 6.67 days

Method 3: Using LCM

The third method uses the least common multiple (LCM) of the time periods to simplify the calculation:

Let work LCM(15, 12) 60 units So eff of A to B 60/15 60/12 4 5 9 So they together can do job in 60/9 6.67 days

Conclusion

Understanding work rate, rate, and time problems, and using these methods can simplify solving complex work rate scenarios. Whether it's a simple addition of individual rates or using the least common multiple, these methods help in determining the total time required to complete a task.

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