Word Lesson: Working Together
In order to solve problems involving working together, it is necessary to

The goal of a “working together” problem is generally to figure how fast a job can be completed if two or more workers complete the job together.

Suppose that it takes Janet 6 hours to paint her room if she works alone and it takes Carol 4 hours to paint the same room if she works alone. How long will it take them to paint the room if they work together?

First, we will let x be the amount of time it takes to paint the room (in hours) if the two work together.

Janet would need 6 hours if she did the entire job by herself, so her working rate is of the job in an hour. Likewise, Carol’s rate is of the job in an hour.

In x hours, Janet paints of the room and Carol paints of the room. Since the two females will be working together, we will add the two parts together. The sum equals one complete job and gives us the following equation:

We are now ready to solve this equation to determine how long it will take the two females to paint the room if they work together.

 Multiply each term of the equation by the common denominator 12 Simplify Collect like terms hours Solve for x

Remember that x represents the amount of time it takes to paint the room (in hours) if the two work together. So, working together, the two females can paint the room in only hours or 2 hours and 24 minutes.

Example Group #1
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 One garden hose can fill an above-ground pool in 10 hours. A second hose can fill the pool twice as fast as the first one. If both hoses are used together to fill the pool, how many hours will it take? What is your answer?
 It takes Tom 4 hours to build a fence. If he hires Jack to help him, together they can do the job in just 3 hours. If Jack built the same fence alone, how long would it take him? What is your answer?

Example Group #2
No audio files were recorded for this set of examples.
 It takes a man one hour to mow his lawn. It takes his son 90 minutes to mow the same lawn. How long will it take if father and son work together to mow the lawn? 59.3 minutes 36 minutes 36 hours 24 minutes What is your answer?
 One pipe fills a pool in 4 hours. A second pipe, used to drain the water from the pool, can empty the pool in 8 hours. The owner of the pool mistakenly opened both pipes. How long will it take to fill the pool if one pipe is filling and the other is emptying? 2 hours and 40 minutes 15 minutes 5 minutes 8 hours What is your answer?

As you can see, this type of problem requires that you carefully establish what x will represent. You must then set up rates for each “worker,” put the rates together using addition or subtraction to write an equation, and carefully solve using a common denominator to multiply times every term of the equation. At the conclusion of the problem, you must check for the reasonableness of your solution.

D Saye

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