# Determining the maximum value

I received at my work some meal tickets valued at 8.80 euros a piece, and my wife who works at another firm, receives meal tickets at 7.50 euros a ticket. The neighborhood that we work at accepts these for food, but doesn't return change. When we go out together, we want to maximize the value of the tickets by arranging some combination of the meal tickets.

So my question is: what is the algorithm to determine the best value?

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Is it feasible to pay for most of a meal with the tickets, and then pay the rest with real cash? If so, you wouldn't have to waste any money. Otherwise: unless there are hundreds of items on the menu, just looking at all of the appetizing combinations one-by-one shouldn't take too long. – Lopsy Jan 11 '13 at 18:58
Here are the assumptions: you don't have another form of payment, so you will almost always have to pay more than the meal price. Second, although it is nice to have enough of a selection to get exactly the right combination of tickets, in practice this doesn't really work out. – Mark McWhirter Jan 11 '13 at 19:02
Convince the restaurant to let you have a standing tab that is deducted against the tickets. (of course, this is not a 'math' solution) – Calvin Lin Jan 11 '13 at 19:07
By the way, how much does the "average" meal cost? Or does it vary wildly depending on where you go? – Lopsy Jan 11 '13 at 19:07
maybe you can handle it this way xkcd.com/287 ;) – k1next Jan 11 '13 at 19:49

The prices between 10 and 20 euros that you can exactly hit using combinations of 8.80-euro and 7.50-euro tickets are $15.00, 16.30,$ and $17.60$. Between 20 and 30 euros, there's $22.50, 23.80, 25.10,$ and $26.40$. So heuristically, if a meal is anywhere between 13.70 and 17.60 euros, or anywhere between 21.20 and 26.40 euros, then you won't be wasting more than 1.30 no matter what.
But in general, I'm afraid there's nothing much better than the approach you were probably already using, adding up various appetizing meals and seeing how many tickets they use. Remembering the target values - $15.00, 16.30,$ and $17.60$ on one end, $22.50, 23.80, 25.10,$ and $26.40$ on the other - should speed up the process, however.
Finally, and this is important, don't get caught in the tempting logical trap of thinking you're "wasting money" when you're not. If there's one meal that costs $15.01$ and a another meal that costs $16.30$ exactly, then sure it feels like the latter "wastes less money". But in the end, they both use one 7.50 ticket and one 8.80 ticket, so it doesn't matter - you should just pick whichever one is tastier.