The end of give a penny take a penny #hashtag

Let's say that you own a store in Michigan you don't like pennies. You want to price the items in your store so that no matter what people buy, the total price will not require the use of pennies. One idea might be take one cent off for every dollar. So, if an item was priced at $0.99, the total after tax would be $1.05. An item priced at $1.98 would give a total of $2.10 after tax. It’s really easy to do this for any single item. By adding or subtracting a few pennies, it’s easy to get an after-tax total that would not require the use of pennies.

The problem is with combinations and rounding. Following this pattern, consider two items priced at $4.95. Buying the individual items would result in a purchase price of $5.247 which rounds to a nice $5.25. The use of pennies has been avoided. However, if someone put two of these items on the same bill, the price would double and one would end up with a total of $10.494 which rounds down to $10.49. The plan has been foiled.

So, our first plan has failed. Rather than guess and check, let’s look at the algebra. Let P be the price in pennies in the pre-tax price and N to be the number of nickels after tax. What we are looking for is a P and N that are whole numbers and satisfy the equation.

Having reduced the fraction, the only way that N can be a whole number is if P is a multiple of 250 (otherwise, the fraction on the right hand side of the equation will not simplify to a whole number). So, there is a way to avoid the use of pennies. If every item in the store is priced at $2.50 or whole number multiple of $2.50, then post-tax total will always work out to a number that does not require the use of pennies. So, there you go, problem solved. 

* What is wrong with pennies? For one reason, they are economically inefficient. You can read about it here or just search for one of the posts from my friend Mankiw. Another reason might be historic. Consider South Carolina (which has a similar six percent sales taxes to Michigan) or Georgia (the math would be similar with their four percent sales tax). There are probably some folks out there still bitter about the "War of Northern Aggression."

One might argue that getting rid of the penny would make more copper available; copper is an important item in all sorts of electronics. This isn't a very good argument. While modern pennies look like they are made out of copper, there is only the tiniest layer of copper covering a zinc disc. Only 2.5% of a freshly-minted penny is actually copper. On the other hand, nickels, dimes and quarters are at least 75% copper. So, as the old adage goes, you cannot judge a coin by the plating that covers it. Also, all that shimmers is not necessarily copper.