# The Penny Problem

Normally, I don’t think there should be penalties for paying with pennies – it is legal tender, after all – but in this case the man obviously created quite the disruption. He was acting like a douche, no doubt. But a \$140 fine… seriously? Maybe they got angry that the man demanded that they count the pennies. Alas, they didn’t have to. Because “counting” the pennies would have taken less than five minutes (if you have the right equipment in hand). Here’s how.

First, weigh the pennies. This can be done by weighing a rigid container, adding the pennies into the container, and then weighing it again – the difference between the original and final weights is the contribution from the pennies. I’m assuming they have weights since this is a clinic, after all. Let’s call the mass of the pennies M.

Here are some facts about pennies. Pre-1982 pennies, in general, weigh 3.11 grams each. Post-1982 pennies, in general, weigh 2.5 grams each. (Source) Finally, the guy claims he has given 2500 pennies. Let the number of pre-1982 pennies be p1. Let the number of post-1982 pennies be p2. Realizing that the total mass of the pennies and the total amount have to be consistent, one can set up the following system of equations:

One can solve this system using an online calculator such as the one here. As it turns out, this method is pretty reliable, as the set of possible masses that would yield sensible solutions to p1 and p2 is relatively small:

It is unlikely that there would be more pre-1982 than post-1982 pennies from a statistical perspective (after all, we have a pretty large sampling of pennies), and it is even more unlikely that we could have a negative amount of pennies. Thus I deduce that the mass of 2500 pennies should be around a little less than 14 to a little less than 15.5 lb. Any less, and the guy’s bs’ing you. Any more, you can take the extra money. ğŸ˜‰

The best part is, you can print out a fake copy of a results sheet from what you claim is a spiffy “penny-counting machine”. Say the “machine” is in a restricted area, and if he tries to dispute your “machine”‘s results; he can go ahead and count them – the hard way.

P.S./Edit: According to this link, the heavier pennies consists of around 20% of all pennies in circulation right now. As a result, the pennies should most likely weigh around 14.5 lb (a ratio of around 500 : 2000 p1 to p2).