Person A does a job in A hours, Person B does the same job in B hours. How long does it take them to do the job together?Teaching the work problem can turn into a debacle fast. Really fast. Most teachers boil these down to an algorithm they themselves might not understand.
"Here's the formula, kids. Learn it. Know it. Use it. Now let's move on to something else."
That's never set well and a couple of years ago, it dawned on me that a work problem is nothing more than a complex rate problem.
Person A can do 1/A of the job per hour and person person B can do 1/B of the job per hour. How much of the job do they complete per hour when working together?
This has been a game changer. Find the common denominator and add 'em up. Now we have a common rate. From there it's pretty easy to determine how long it takes to complete the job. We give no formulas and kids understand how to find an answer and why it works.
We had kids derive two different algorithms for work problems.
x = (AB)/(A+B)
1/A + 1/B = 1/X
Sure, kids. Use it. G'ahead.
But then we added the dreaded twist.
Person A can do the job in A hours. If person B helps, they can do the job in X hours. How long would it take person B to do the job when working alone?
I give you, Clinton's Theorem:
I kid you not, this kid came up with this all himself. By "integer portion of X" he means, everything to the left of the decimal. He's truncated the number to make things "simpler."
As he was presenting this to us before we left for break, I was baffled. It works, but why? So today, I sat down next to him and had him explain the formula to me again, I took notes and then simplified the expression. I won't deprive you the joy of doing so yourself.
Seriously, have at it.