### Some basic math for waiters

October 8th, 2007When a group of several people eat together at a restaurant (or bar, coffee shop, etc…) there are common ways to split the bill:

- One person pays everything.
- Split evenly.
- Each pays for their own portion.

The exact values are of course a bit fluid on the last two options, since the numbers may be rounded. Currency is discrete rather than continuous, after all. Not only that, but it’s often simpler to divide up to the main coin and not the sub-coins^{[1]}.

The payment can be done by cash. In that case the people would usually just collect enough, pay with it, and divide the change between themselves when the change comes back. The work on properly dividing the charge is on the customers in these cases.

Sometimes, though, people pay with credit cards. Which means that many times the waiters will just receive a bunch^{[2]} of cards, with simple instructions on how to divide the charge between them.

The common one is of course “Split it evenly”. And these are the cases where money is often rounded to higher coins, since apparently most waiters have a problem with fractions. I can recall maybe 1-2 cases, ever, where the individual charges weren’t rounded with one person paying the extra.

When things are not split evenly, well, that’s when the fun begins. And by “fun” I mean an all too common tragic comedy of errors.

The simple case is when the customers still calculate the amounts in advance. In this case the waiter receives exact instructions in the style of “Put 100 on this card, and 150 on that card”. Simple. Easy.

And they still sometimes manage to get it wrong:

- The bill comes back split evenly.
- The amounts are charged correctly, but on the wrong cards. In this example, the first card is charged 150, and the second 100.
- All of the cards are charged the same amount, which is one of the sub-amounts. So, for example, for this 250 bill either both cards will be charged 100, or both will be charged 150.
- Some of the cards may be charged correctly, and some will be charged an unrelated amount. This is because the complexity of the task got the waiter confused and he/she charged an amount due for another customer entirely.

I had all of these happen to me, as a customer in restaurants.

One time I had two of them happen in a series. The waitress made a mistake (#3 above), I alerted her, and she came back with a “correction” that included another type of mistake (#4 above). When there’s a charge, and a cancellation, as a customer you’re requested to sign on both. If you simply don’t sign on the charge, it creates all sorts of complications. So I ended up having to sign five times for my bill that day. What did I tell you? Fun!

It also happens, though, that the job of dividing the charge is placed on the waiter. Sometimes the customers know the difference between what they’re supposed to be billed for, but not the final amount.

In which cases someone has to do the calculation. It’s a simple enough calculation, you know the total, and you know the differences.

And the natural tendency would be to let the waiter do it. People just had a meal, are finishing up, and they need to pay the bill. Why would they want to do the work, as easy as it is, when there’s a waiter that will have to process the charges anyway and is being paid for it?

Makes sense.

Except it doesn’t. Because many waiters seem a bit deficient in the math department.

The latest time this happened to me was a couple of weeks ago. I was finishing a meal with a friend. We basically shared the dishes, so almost everything was supposed to be split evenly. The only difference was that I had an extra glass of some medium-pricey alcohol.

The waitress arrived, and saw the two credit cards on the tray with the bill. The dialog between me and the waitress went something like that:

Waitress: Should I split this up?

Me: Yes, but it’s 70 more on this card.

Waitress: Right. 70 on this card, and the rest on the other card.

Me: No. Split it between the cards, so that this card is charged by 70 more than the other card.

Waitress: Eh…

Waitress: Hmm….

Waiterss: I’m…. err… not….

Me: It’s simple. Just split evenly, add 35 to this card, and reduce the other 35 from the other card.

Waitress: Ah. Yes. OK, sure.

And this is the math lesson for today. If you want to divide a sum *X* between *N* people so that everyone pays the same except for one who pays an extra *Y*, this is what you do:

- Divide
*X*by*N*. Let’s call that*A*for average. You already know how to do that. This*A*would be what you’d charge each card if you had to split evenly. - Divide
*Y*by*N*. Let’s call that*B*. This value is like the average of the differences. Mathematically it’s the exact same process as the previous step, so if you knew how to do it, you know how to do that. - Everyone, except the person who has to pay more, pays
*A-B*. You know how to do subtraction already. It’s the same thing you’d do if someone paid part by cash and part by credit card, and you’d have had to reduce the cash amount from the total to get the credit card charge. - The person who has to pay more pays
*A + [(N-1)*B]*. Basically all the B’s you reduced from the bills of the other people, you add to this one’s bill. You already know how to do addition too. It’s just like what you’d do if someone asked you to charge the tip on the card as well, telling you how much is the charge and how much is the tip. You already know how to do multiplication as well, it’s what you’d do if you got everyone else’s cards and they all told you they have to pay B.

That’s it. Easy. Simple steps. And these are all things that waiters are supposed to know how to do already.

Except sometimes they don’t.

In this case, for example, I was indeed charged 35 more. The other card? Charged exactly the amount of an even split.

Wait, wait, I know what you’re thinking. In this case it would mean that the total would come to 35 more than the real total, right? So the waitress, or at least the cash register computer, should notice something is off, right?

Right.

But they had a simple solution for that. You see, the final bill came back printed with three items:

- Credit card charge : A
- Credit card charge : A+35
- Refund : -35

So the total was absolutely correct, making the waitress feel perfectly happy about it. No problem if it all adds up, after all.

Except that, of course, we didn’t get that refund. The bill did not come back with 35 cash, nor did one of the credit cards get a refund (which would have kind of defeated the whole purpose, but at least would have meant the amount of money passed from us to the restaurant would have been correct).

Our poor waitress didn’t quite see the problem. It all adds up after all, and the total is right. Luckily another waitress/supervisor did see the light immediately after a very brief explanation.

Waiters should learn a little basic math. Me, I should learn not to trust waiters to do even the most basic math. I think I learned my lesson. Now it’s their turn.

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