There’s a restaurant in downtown San Francisco called ‘wichcraft. They make sandwiches.

This post is about their menu. Specifically, if you saw their menu could you work out the sales tax rate in effect?

It’s a silly question, at first. But consider their breakfast options. They cost, in descending order of price, $7.35, $6.90, $5.12, $4.68, $4.45, $4.01, $3.34, $2.45. (These are *not* the prices you’ll see on the menu at their web site, because their menu has their New York City prices.) These prices look a bit strange, but we might guess that they’re round numbers after tax. And in a US context, if you’re expecting cash transactions, “round numbers” means multiples of 25 cents.

Take the *differences* between these; they’re 45, 178, 44, 23, 44, 67, and 89 cents. Just looking at this sequence I can see some quantization; I guess that that 23-cent difference becomes a quarter after tax, the 44- and 45-cent differences become two quarters, and so on. So those differences are, in post-tax quarters, 2, 8, 2, 1, 2, 3, and 4, which add up to 22. In particular $2.45 becomes n quarters after tax, for some n, and $7.35 becomes 3n.

Then, $7.35 is three times $2.45; thus n is 11, and $2.45 pre-tax becomes $2.75 post-tax. The pre-tax prices, and the corresponding post-tax prices, are:

$7.35 | $6.90 | $5.12 | $4.68 | $4.45 | $4.01 | $3.34 | $2.45 |

$8.25 | $7.75 | $5.75 | $5.25 | $5.00 | $4.50 | $3.75 | $2.75 |

So what’s the tax rate? Each one of these prices gives us some small interval which contains the tax rate. Let the tax rate be x percent; then we must have, for example, that 7.35(1+x/100) is between 8.245 and 8.255, from which x must be between 12.177 and 12.313. We can do a similar computation for each price. The highest lower bound that we get is (4.995/4.45)-1 = 12.247 percent; the lowest upper bound is (5.255/4.68)-1 = 12.286 percent.

One last assumption – the tax rate is a round number. So it must be 12.25 percent.

But the California State Board of Equalization says the sales tax in San Francisco is 8.5 percent! And this is in conjunction with a couple places I saw on Clement Street a few weeks ago that charged $4.38 for a dim sum special, which are what inspired this post; add the tax and it’s $4.75 — I held this post back because from that single data point it’s hard to show much.