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Don't get "worked" up: tips for solving work rate problems

Note: this article originally appeared as a blog post by Mike McGarry.

See if you can crack this problem in less than 2 minutes.

Problem:

Mike paints a fence in 9 hours.

Marty can paint the same fence in 5 hours.

 Column A Column B The time it takes Mike and Marty, working at a constant rate, to  paint the fence 3 hours

Work rates are one of my favorite problems to teach. Students usually have a formula in their heads that they vaguely remember. Even if they know the formula, they take awhile to put the numbers in the correct places. Assuming they don’t make a mistake, the problem can take them 2 minutes to finish.

What if I could show you a way to finish the problem in less than 15 seconds?

And that’s with no messy formulas.

Okay first things first let’s conceptually work through the problem.

Ask yourself, how much of the job does each person finish in one hour. With Mike, he finishes 1/9 of the job in one hour, because it takes him 9 hours to finish the entire job. With Marty he finishes 1/5 of the job in one hour. Add those two rates together, 1/9 + 1/5 = 14/45 and then Flip It! and you get 45/14. That is greater than 3, so the answer is (A).

Not bad. No cumbersome x and y, or Work Rate 1 and Work Rate 2, Total Work Rate, as many books on the market show you.

But imagine an even faster way. Ready?

All you have to do is multiple the hourly rate to find the numerator and add the rates to find the denominator.

Or more succinctly put, multiply the top; add the bottom.

9 x 5 = 45, 9 + 5 = 14. 45/14.

It’s that easy.

Let’s try two new numbers.

Mike = 15 hrs, John 5 hrs.

Now here’s all you have to do: multiply the top; add the bottom. In other words, multiply the time it takes Mike to do the job by the time it takes John to do the job. Then divide that by the sum of the time it takes Mike to do the job and the time it takes John to do the job.

(15 x 5)/(15 + 5) = 75/20 = 3 ¾ hrs.

Because it’s so easy try the next numbers:

7 hrs and 4hrs, Combined work rate: (Don’t look below till you’ve solved it)

Ans: 28/11 hrs

I told you—no need to get “worked” up!