Now Nathan is here to fix your introduction to combined work rate problems.

We’re all familiar with combined work rate problems:

Suppose one painter can paint a fence in twelve hours, and the second painter takes six hours to paint the same sized fence. How long would it take the two painters together to paint the fence?

OR

Jane can mow the lawn in 30 minutes and Frank can mow the lawn in 50 minutes. How long will it take for them to mow the lawn together?

Well, here’s Nathan’s business-oriented approach to combined work problems. Here’s a clip I edited from the second season.

I can’t lie. I howled with laughter when I saw this clip.

So, you tell me: how quickly can 40 maids complete the job?

It’s an obviously absurd concept and, as we find out later in the episode (spoiler alert!), the cleaning service comes in a bit over the mathematical answer. But it does make for a great final act of the problem. You can check out the full, unedited clip here. You’ll find the “solution” though the clip also demonstrates how the theoretical and the practical don’t always mesh well.

I say this task could make a good “intro” to combined work problems, because the real meat (and challenge) of these problems come when the subjects work at different rates. After working through this problem, you could reverse it: how many maids will it take to (theoretically) clean a house in 30 minutes? From there you can get into more complex examples, such as this task from Illustrative Mathematics: Harvesting the Fields.

PS. I realize this post is from another era. Not only because the clip is from an episode of a show that aired in 2014, but also because “bleeped math” was (and I’m going to say “still is”) a fun little tool in our teaching toolbelt. But some tools never go out of style. Unlike Nathan’s giant suit.