## What is Multiplication – part III

In the first installment of this series we looked at multiplication as what happens when you have a number of groups and each group consists of an equal number of members.  Our example was 3 rolls of Mentos, where each roll has the same number of Mentos candies, 14.

In the second installment, we looked at pictures being scaled up to bigger pictures of the same shape.

Now I’d like to take the investigation in a whole different direction, and look at man hours.  Acquaintances of ours run a landscaping business, and bill their clients based on man hours.  They send over some of their landscapers, and these landscapers work for some time on the yard or the garden, and then, later, the client gets billed for the number of man hours, at \$50 per man hour.  It got me thinking that here we have very practical people, with dirt on their pants, billing their clients for something so abstract that I realized that I didn’t know what it was.  What is a man hour?  It is neither a man nor an hour.  You can’t hold it, you can’t see it, but we know it costs you.  Unlike a mile, I can’t trace one out.  Unlike an hour, I can’t ring a bell at the beginning and at the end to mark its passing.
And yet we all know what a man hour is for.  If one of his men works for an extra hour, he gets to bill an extra man hour.  Of course, in practice, it doesn’t happen very often that one man works for an extra hour.  Almost always, the owner sends a crew, and they all come and leave in a single pickup truck.  He sends a crew of 3 for a 4-hour stint, and then bills for 12 man hours.  If one  day one crew member is sick, the owner would expect the same job to take 6 hours to do for his remaining two crew members, and he’d again bill 12 man hours.  The owner, and the client, may well think of the size of the job as being 12 man hours, and bill for that without closely consulting their stopwatches.

So here we have a multiplicative structure:

$\begin{matrix} 1 & man & for & 12 & hours: bill & 12 & man\ hours \\ 2 & men & for & 6 & hours: bill & 12 & man\ hours \\ 3 & men & for & 4 & hours: bill & 12 & man\ hours \\ 4 & men & for & 3 & hours: bill & 12 & man\ hours \\ 6 & men & for & 2 & hours: bill & 12 & man\ hours \\ 12 & men & for & 1 & hour: bill & 12 & man\ hours \end{matrix}$

It’s almost like a man hour is somehow a man multiplied by an hour, however we would think of such a thing.  It behaves that way.  It gets its meaning from the scenario of billing.  It rests on a rough assumption that the crew members he sends all handle the same work load.  If this assumption is too far out of whack with reality, he may bill his trainees at reduced rates, e.g. only half a man hour per hour worked.  If he and the client have built up trust, and the crew returns regularly to do the same kind of pruning and leaf raking etc, there may be no problem to send in his fastest guy and allow him to bill a man hour for 45 minutes of (fast) work.  In making these kinds of adjustments, the owner of the business is himself using multiplicative thinking to account for different work speeds.