The tax rate inside the Tucson city limits is 8.6 percent — that's a combination of 5.6 percent for the state of Arizona, 0.5 percent for Pima County and 2.5 percent for the city of Tucson. If you buy one item at one of the $1 stores — it costs $1 — the tax is 8.6 cents ($0.086), but the merchant rounds up, as is legal, and you'll pay 9 cents ($0.09).
If you are fortunate enough to find something somewhere that costs 5 cents ($0.05), first: that's amazing; second: the sales tax is 43/100 of a cent ($0.0043), which is less than ½ of a cent. Chances are, you won't be charged a sales tax because the amount is rounded down.
We might assume that the governments charging that tax are essentially OK with that because you can't really buy anything for a nickel anyway and they got to round up when you bought something for a dollar.
However, back in 1937, there were a lot more things one could buy for a nickel.
The sales tax in Arizona was 2 percent on most retail items, which means one would pay 2 cents ($0.02) sales tax on an item that costs $1. That would also mean that for items costing less than 25 cents ($0.25) if merchants rounded down, you would not pay a sales tax. This is assuming you only purchased the one item because, of course, the sales tax is calculated on the total purchase.
Fortunately for consumers and unfortunately for the tax commission and local governments, there were quite a few items costing 25 cents or less back then, and Arizona wanted to charge tax on them. Without half-cent coins, and with the law stating only the federal government can make legal tender including coins, the city had to find a way to collect tax on these purchases.
They began minting tax tokens that represented one-tenth and five tenths of a cent. They were called one mill and five mill tokens. The name comes from the fact that a one mill token represents $0.001 or one one-thousandth of a dollar. "Mill" refers to thousand in Latin.
This allowed consumers to pay a 1 mill tax on an item that cost 5 cents ($0.05). A consumer would rarely escape a sales tax this way.
Once the tokens were available, there was plenty of confusion. (The Morgue Lady assumes her explanation hasn't confused you already.)
From the Arizona Daily Star, Thursday, Sept. 2, 1937:
Tomorrow, we'll tell you of a problem the tokens caused for slot machine operators.