# Mathematics of change money

Do you know any results or articles about change money? Something like the statistics of different value notes in a cash box. Or answers to questions which distribution of notes values is best for starting a day in a shop. I mean obviously you need more small value notes than large ones. After one day of selling you probably have more large notes as they don't go away easily.

-
But what is change money? –  Hui Yu Jul 16 '12 at 19:29

Regarding the comment (since I have insufficient reputation to comment), "change money" is the cash kept in a drawer at the start of business in a shop. You will want to have some money on hand, so that if someone comes in with, say, a \$20 bill, you can give them appropriate change for a \$17 purchase.

To answer the question, you can fairly easily derive appropriate statistics. Without loss of generality, assume that all transactions are whole-dollar transactions.

First, you need to know what the available denominations are. Then, you must estimate the different "tender" amounts -- i.e. the cash a customer gives you before you give them change. You don't need to estimate the distribution of their denominations: \$45 given for a \$41 purchase is identical whether it is given as 2 \$20s and 1 \$5, or 4 \$10s and 1 \$5.

For a reasonable range of these values, estimate what your returned change would be for purchase values up to that amount minus \$1. For example: If the customer gives you \$20, for the following purchases you need the following bills:

\$1 purchase: 1 \$10, 1 \$5, 4 \$1

\$2 purchase: 1 \$10, 1 \$5, 3 \$1 ...

Some patterns will obviously repeat here.

Because it is straightforward to compute this (this is just doing mixed-base arithmetic), the only interesting part comes from estimating the histogram of the purchase amount. This will vary based on your business; a cafe will have different values than a bookstore, for instance.

Therefore, you have one random variable, the purchase amount, which can be easily associate with different change amounts for different "tender" values. The "tender" value is not really a random variable, because the customer almost always gives the smallest denomination that fits a purchase.

Alternatively, you could consider the delta -- the value in change -- as the random variable, which will have a deterministic "change value" associated with it. Again, this is just converting some random number with mixed-base arithmetic.

-
"revenue" value is probably the wrong term. I am unable to recall a proper term for the money that a customer hands to a clerk at a store. –  Arkamis Jul 16 '12 at 19:45
The word you are looking for is probably "tender" –  M Turgeon Jul 16 '12 at 20:03
For the ongoing problem, \$45 tendered as 2 \$20's and 1 \$5 doesn't leave you as well positioned as if you get 4 \$10s and 1 \$5. The cash you take in can be used for change later in the day. – Ross Millikan Jul 16 '12 at 21:55 Also, when you get a \$20 bill for \$1, you not only have the option to give 1 \$10 bill, 1 \$5 bill and 4 \$1bills, but you also could give e.g. 3 \$5 bills and 4 \$1 bills. Of course you would only do this if you don't have and \$10 bills left, but there can be more interesting situations: Assume there are also \$2 bills, and you get a \$10 bill for \$4. Then you have the option to give either 1 \$5 bill and 1 \$1 bill, or 3 \\$2 bills. So which one is better? –  celtschk Jul 16 '12 at 22:05
Ah yes, tender. Thank you. I have edited to reflect that update. –  Arkamis Jul 17 '12 at 2:49

Coin production figures by the US Mint are available here. I presume coin figures in circulation and in change trays would be in similar proportions. No doubt figures for bills and for other countries are also available to those willing to type a few cleverly chosen words into the web.

-