## Sunday, May 10, 2015

### An Open-Ended Problem, Part 2

Study the numbers in this photo until a question occurs to you. Then answer your question. Repeat as desired.

Reader Stacy W pursued the implications of treating the word "Due" as the visible portion of "Change Due." In that case, assuming 2.59 and 0.57 were the only charges, then her question was, "How much money did the customer give the cashier, in order that the change due would be 7.00?"

Answer: 10.16, likely in the form of a ten-dollar bill and sixteen cents in change. Good question!

Reader Marni pursued the implications of treating the word "Due" as the visible portion of "Amount Due." In that case, what was the rate of tax paid?

Answer: 0.57 on a pretax amount of (7.00 - 0.57) comes to (0.57)/(6.43) = 8.864967(...)%. Good question!

Some of the questions that occurred to me, along with my answers to them:

1. How much in total were the item(s) not shown?

I took "Due" to be the visible portion of "Amount Due." But this raises a puzzle: if 2.59 is the price of a retail item (probably a drink, because of the "oz"), while 0.57 is the amount of tax levied on the sale, then as reader Marni noted, some additional item(s) must have been purchased, because 2.59 and 0.57 don't add up to 7.00. It appears the cash register display was simply too small to show all of the items purchased. So how much in total were the item(s) not shown?

Answer:  (7.00 - 0.57) - 2.59 = 3.84.

Pursuing some more questions along these lines:

2. What was the pretax subtotal?

The pretax subtotal was 7.00 - 0.57 = 6.43.

3. Were all of the purchased items subject to tax?

Certain retail items are exempt from sales tax, so it's possible that the 0.57 amount was levied on some but not all of the purchased items. Note, however, that if any of the items purchased here were exempt, then the local sales tax rate would have to be at least 14.8% (0.57/3.84), which is higher than in any U.S. city. The transaction could conceivably be in a VAT country, where rates can be high. But as discussed further below, we are going to take the transaction to be in the U.S., in which case it follows that the full pre-tax total of 6.43 was subject to sales tax.

4. What can we say about the local sales tax rate?

While 8.864967(...)% gives the rate of tax that was actually paid, this was probably not the sales tax rate prescribed by law in the location where the photo was taken. Legislators aren't going to set the tax rate at a funky, non-terminating decimal value like that. Rather, issues of precision and/or rounding are probably involved here, which makes things tricky since rounding is a non-invertible function. But let's see what we can do.

The register in the given photo is a Micros point-of-sale system. The rounding scheme is presumably configurable in the proprietary Configurator module; however, freely available documentation doesn't seem to specify how many digits of internal precision the cash register uses for its tax computations, nor the logic by which it rounds fractions of a cent. I tried to submit a helpdesk ticket about this, but I couldn't do that because I'm not a Micros customer. I did establish that the Casio PCR-T465 cash register provides several user-configurable options: multiply and round in the usual way (based on 0.5); multiply and round up; multiply and round down; or select a state-specific tax table to compute the tax via lookup. It wouldn't surprise me if the Micros system gave managers the same options.

(And now I have to digress, because suddenly I remember ringing up customers at my parents' diner when I was a kid. The cash register at Rip's Drive-In was a clicking, clanging, chugging machine. An NCR model like the one in the photo, electric-powered but not electronic. We figured the tax by remembering the amount for common orders, or computing mentally, or consulting a laminated table. Computing sales tax was easy for eight-year-old me, but I always panicked inside while figuring the customer's change, because nobody had ever explained to me the method of counting up. So to figure the change for a ten-dollar bill on a forty-nine-cent tab, I stood there for what seemed like endless seconds while the customer waited behind me, my eyes staring into space as I crossed out zeros and borrowed ten on the chalkboard in my mind. Cashiering would have been easier if I'd been taught mental computation strategies in addition to the standard right-to-left subtraction algorithm. Things could have been worse, too: a friend tells of his sister's sidekick from teenaged years who worked a register in those days and never realized that there was any specific quantitative relationship holding between the change, the amount of the tab, and the amount of the dollar bills tendered. Her method of figuring change was to paw through the coin tray with brisk confidence until she had collected what seemed to her like a convincing enough collection of coins to present to the customer.)

For purposes of analysis, I will model the Micros register as follows:
(i) the register multiplies the amount of the sale (a decimal to hundredths) by a finite-decimal tax rate, working to full precision (exact multiplication);
(ii) the register then rounds the result to the nearest cent by rounding fractions less than 0.5 to 0 and rounding fractions 0.5 or greater to 1.
Given this model, the programmed sales tax rate must lie in the range from A to B, where
A = 0.565/6.43
B = 0.575/6.43.
To see this, let R be the programmed sales tax rate and let T be the internal result of the tax computation, noting that (by (i)) T = 6.43R. Now by (ii), T must satisfy 0.565 <= T < 0.575. Dividing through by 6.43, we have A <= T/6.43 < B. But T/6.43 = R, so the programmed rate lies in the range from A to B, as claimed,

Carrying out the division, we have
A = 8.78693(...)%
B = 8.94245(...)%.
According to the model, and assuming that the tax rate is programmed correctly, the local sales tax rate lies somewhere between these two percentages.

6. Where might the photo have been taken?

I downloaded the image file and examined its EXIF data to look for latitude and longitude. (If you don't know how to do this, you can upload the image to whereisthepicture.com.) Unfortunately, it turns out there's no geotagging in the EXIF data, so we can't locate the transaction that way. An image search on the phrase "join us in the" plus the word childhood did not yield a match to the baby in the photo. (I'd been hoping to identify a corporate campaign.)

Dollar signs are used to denote currency amounts in the United States and many other places. The only language in the photo is English, which rules out some of these countries; the unit of measure "oz" probably rules out many others. In any case, let's assume the transaction took place in the United States.

This website shows minimum and maximum sales tax data for every state. Examining the data, one finds 14 states with minimum and maximum rates that are consistent with A and B:
Alabama, Arizona, Arkansas, California, Colorado, Illinois, Kansas, Louisiana, Missouri, New York, Oklahoma, South Carolina, Tennessee, Washington.
Thus, even though we don't learn the exact location of the transaction, we still rule out out a great many potential locations for the photo (dozens of states and foreign countries). One could repeat the analysis with other reasonable models of the cash register. I'm guessing the results won't change very much, but feel free to check that.

Sometimes a sleuth has to go with a hunch. I noticed from the sales tax data that there is only one state in the U.S. where the maximum sales tax rate falls in the range from A to B. That's New York, where the maximum sales tax rate is 8.875%. Digging a little further, one discovers that this statewide maximum is attained in only one city, New York City. So if I had to guess, I'd guess that the local sales tax rate was 8.875% and that the photo was taken in NYC.