the density of money

I have a jar of assorted coins. I sort out the quarters separately (because I need them for laundry) but otherwise every so often I throw my change in here. It's getting kind of heavy; one of these days I'll cash it in for real money.

As I was lifting it this morning, I began to wonder -- if I knew how much it weighed, could I tell from that approximately how much money it contained?

Then I went to buy breakfast -- which cost me $5.25, and I paid with a $20 bill. I got $14.75 in change -- a ten, four ones, seven dimes and a nickel, because there were no quarters. The woman working the cash register said she was sorry they were out of quarters; I replied that dimes were okay because at least they're small.

So, here's the question: what's the density of money in, say, dollars per kilogram?

A U.S. dime weighs 2.268 grams; that's $44.09 per kilogram.

A U.S. quarter weighs 5.670 grams; that's also $44.09 per kilogram.

A U. S. nickel weighs 5.000 grams; that's $10.00 per kilogram. Wikipedia says that"nickels have always had a value of one cent per gram", which is interesting if true in part because nickels have been minted since 1866. Clearly the designers of the nickel were thinking in metric.

A U.S. penny weighs 2.5 grams; that's $4.00 per kilogram.

I presume the fact that the quarter is exactly two and one half times the weight of the dime has something to do with the fact that they're both made out of the same alloy, an 11:1 copper/nickel mixture; the nickel is three-fourths copper and one-fourth nickel; the penny is 97.5% zinc and 2.5% copper.

Right now, metalprices.com reports that copper sells for $7.895/kilogram; zinc, $3.436/kilogram; nickel, $36.20/kilogram. Thus dimes and quarters, if you melt them down, could sell for $10.25/kg; nickels, $14.97/kg; pennies, $3.54/kg. I'm surprised to learn that nickels are worth less than the metal underlying them, because you hear this more often about pennies, even though it's not actually true. It does, however, cost more to make a penny than that coin is worth, and the U. S. Mint has passed regulations about the melting down of pennies and nickels.

But what's the density of "money"? That's a bit trickier. Let's assume that on any given transaction, I pay with a whole number of dollars; furthermore assume that the "fractional part" of my change is equally likely to be 0, 1, 2, ..., 99 cents, and that it's given back to me with the smallest number of coins possible. The easiest way to do the computation is to assume that I make 100 transactions, in which I get 0, 1, 2, ..., 99 cents back. Now I have $49.50. How much does it weigh?

Well, twenty times I got 0 pennies; twenty times I got 1 penny; and so on up to 4 pennies. So I have 20*(0+1+2+3+4) = 200 pennies.

I'll never get more than one nickel. I get a nickel if the fractional part of my change is 5-9, 15-19, 30-34, 40-44, 55-59, 65-69, 80-84, or 90-94 cents; there are 40 numbers there. So I get 40 nickels.

The rest of what I get is dimes and quarters; since dimes and quarters have the same "money density" I won't distinguish between them. I get $45.50 worth of dimes and quarters. (In fact, I get 150 quarters and 80 dimes.)

Together, all these coins weigh 1732 grams; thus the density of money appears to be $28.58 per kilogram.

But in reality, it won't be nearly this much. I try to get rid of change when I'm carrying it, and a lot of businesses now set their prices so that they don't have to deal with nickels. (At one of my favorite coffee shops, all the prices are multiples of 25 cents. The problem with this is that people will bitch and moan when that inevitable day comes when they raise the price of a large coffee from $1.75 to $2; if they were willing to deal with nickels they'd only have to raise it to $1.80.) And I'm more likely to spend quarters than any other coin, because they work the laundry machine (They buy newspapers, too; the Inquirer costs 50 cents. The machines take nickels, dimes, or quarters, but usually I use two quarters.)

I can't weigh my jar of money to tell you what its actual density is -- I don't have a scale. But when I cash it in I'll let you know how many of each kind of coin the coin-counting machine says it had.