Thursday, April 05, 2012

Algebra: Rate Problems, Part 2

In this second part, we are going to solve rate of work word problems. If you missed the first part where we covered rate of travel word problems, click here to be directed there now.

A rate of work problem will be one in which two (or maybe more) objects or people perform a certain task at different rates. You are asked to find how long it will take them to do a job/task together, or maybe how long an invidual would take if working alone. I've devised a formula you can use to work these types of problems:

Example 1: Painter A can paint the Smith's house in 12 hours. Painter B could paint the same house in only 9 hours. How many hours would it take to paint the Smith's house if the two painters worked together?

First identify what elements are given and define a variable as the item you don't know. Here, the time it would take the painters to paint the Smith's house together is our variable x.
Set up the equation and plug in numbers. In order to add fractions on the left, we must first obtain a common denominator. The least common denominator here is 36 because both 9 and 12 are factors of 36. So, to get 36 for the first denominator, we multiply by 3/3 and for the second, multiply by 4/4. After that, we can add these fractions--remember to add the numerators and keep the same denominator. Then, we can cross multiply and solve a simple equation for x.
Let's solve another one.

Example 2: A computer can process some data in 3 hours. If it works together with another computer in the office, they can process the same data in only 1 hour. How long would the second computer take to process this data if it didn't work with the first computer?

What are we given? We are told that computer A takes 3 hours to do this task and that working together, computers A and B take only 1 hour. What are we trying to find? We want to know how long computer B would take if it worked alone. This information is boxed in green in the figure below.

In this situation, the common denominator we need to use on the left side is the product of the two existing denominators, which is 3x. So multiply the first fraction by x/x and the second by 3/3 to get the common denominator. Then add as usual. If you left the right side as 1/1, then cross multiply. If you reduced this to just 1, then simply begin solving for x by multiplying both sides by the denominator, 3x.

I hope you see now that these problems are not very difficult or time-consuming. Just pay attention to what you are given and list your variables accordingly. Let's try another.

Example 3: Sam has a school project to do. She can either work with two partners or work on her own. Her best friends, Jade and Amanda, would like to be partners with her. Sam knows she could do the project on her own in 12 hours. Jade could do the project on her own in 8 hours, but Amanda would need 16 to do the project by herself. Should Sam do the project on her own or partner with her two best friends?

This problem is a little different in that there are three acting people. We know that Sam could do the project on her own in 12 hours. In order to determine if she should work with her friends, we need to know how long it would take the three of them to complete the project. Let person A be Sam, person B be Jade, and person C be Amanda. Then their individual times are listed below.
In the formula, simply add in another fraction for Amanda's time. The work is below. Notice that now you need to get a common denominator for three numbers. I chose 48. You might find that adding two fractions, then adding that sum to the third fraction, would be easier than adding three fractions at the same time. You should get the same answer either way.
From this we can see that Sam should definitely work on the project with her best friends. Even though one of them is slower than Sam, their combined effort is worth it! In fact, even if both of her friends were a little slower than her, they would all benefit by working together.

Let's try one more with three acting parties.

Example 4: John can do a task in 3 hours. His coworker, Jerry, can do the task in 4 hours. Together, John, Jerry, and Joel can do the same task in 1 hour. How long would it take Joel to do the task on his own?

This one is set up like the last, but with the variable x in a different place. Here is how you can set it up:
Now, to add these fractions, I suggest adding the two with numbers first, then adding this sum to the one with a variable. The common denominator I would use is 12. Then, add this fraction 7/12 to the variable fraction 1/x. Next, your common denomintaor will be 12x. Once you've added these fractions, solve for x by multiplying both sides by the denominator 12x. We get an answer of 2.4 hours for Joel to do the task on his own.
I hope this post has helped you see that rate of work problems are not very difficult to do if you take your time and read the problem carefully. Look for words like "together" and "individual" or "alone" to signify which place in the equation a number needs to go.