## The Diner Check Problem

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

### The Diner Check Problem

I heard this one from my father when I was about five or six years old.

Three people enter a restaurant and order meals for which they are charged a total of \$30. Each of them chips in \$10. (When my father first put this to me, the bill was 30/-; and if you understand what 30/- means, you must be nearly as old as I am!).

A few minutes later, the waiter returns to their table to apologise for overcharging. The total bill should have been \$25; and the waiter lays five \$1 coins on the table.

The three diners each take back \$1, and give the waiter the remaining \$2 as a reward for honesty.

So in the outcome, each diner has paid \$9 for the meal, amounting to \$27. The \$2 for the waiter brings the amount spent to \$29. What happened to the last dollar?
Alan Masterman
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### Re: The Diner Check Problem

They paid 8.3333 each, so they paid 25 between them not 27. There is no "last dollar."

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### Re: The Diner Check Problem

But each put in \$10, and each took back \$1, so the \$9 payout is irreducible...?
Alan Masterman
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### Re: The Diner Check Problem

Alan Masterman » February 25th, 2018, 12:42 pm wrote:So in the outcome, each diner has paid \$9 for the meal, amounting to \$27. The \$2 for the waiter brings the amount spent to \$29. What happened to the last dollar?

Each pays \$10, takes back a dollar, and tips \$2. So they paid \$11 each, not \$9. The extra \$1 each came out of their pockets, or whatever. This sort of thing happens at every group meal I go to, and always causes interminable arguments and confusion. Absolute nightmare. I used to be a waiter, and don't get me started.

Lomax

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### Re: The Diner Check Problem

They paid \$25 for the meal and \$2 as a tip for a total of \$9 each (\$10 - \$1 returned). Cheap tip.

Forest_Dump
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### Re: The Diner Check Problem

It states quite clearly at the beginning that they paid \$30 for the meal. They only take back \$1 each, which means \$27 have been collectively paid - and that's before they each contribute a \$2 tip. That makes \$11 each. Please, you're giving me flashbacks to merry social occasions that have ended in quarrels. Think of what the tips to the therapist will cost me.

Lomax

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### Re: The Diner Check Problem

Oh, I see. They only tipped the waiter \$2 altogether, not each. Mea culpa. I hear ya, Forest. I'll revise my previous answer: collectively they chipped in \$27, not \$29, because the \$2 tip is included in the \$27. The last three dollars (not one) are what went back into the pockets, one each.

Lomax

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### Re: The Diner Check Problem

I'm always amazed how when someone sticks a \$ in front of a number, I can no longer add and subtract right.

Now if this was an accounting message board (instead of science and philosophy) no one would have a problem figuring out what's going on. But just stick a \$ in front of a number and suddenly, anyone with a science background can suddenly no longer add and subtract numbers correctly.

But yea, there's no missing dollar. 3 people paid \$10 each. The restaurant got \$30 and gave \$5 back so the restaurant now has \$25. 3 people each took \$1 each and collectively paid the waiter an additional \$2 (each person paid \$9.66666667 each). There's no missing dollar, just an expectation that somehow, there's another dollar that's gone missing.

My question is, how the hell did each person pay a fraction of a penny?

Dave_C
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### Re: The Diner Check Problem

Small point but \$9.6666667 is \$29.0000001?

Watson
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### Re: The Diner Check Problem

Alan Masterman » February 25th, 2018, 9:16 pm wrote:But each put in \$10, and each took back \$1, so the \$9 payout is irreducible...?

No, wrong. They each thought they'd paid \$10 each but they hadn't. They had actually paid \$25 (divided by 3 means they'd all paid \$8.33) not \$30.

I am just puzzled why this is puzzling to anyone? I guess if you get caught up in the numbers without looking at the obviousness of the situation it can get confusing? (Pinker points out this in the difficulty we have with dealing with logical problems in completely abstract contexts - did I post that one here a couple of months ago??)

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### Re: The Diner Check Problem

It is just a numerical slight of hand, as in don't look at the \$8.33. Look at the 10, less a dollar, which is the 9. The 9 is also the \$8.33 rounded up. So in either case, the \$9.00 each is \$27.00 and \$1.00 back to each. The \$2.00 tip is the rounding amount \$0.66 x 3 = \$1.98, so the tip is the distraction.

Watson
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### Re: The Diner Check Problem

In the US, \$2 is a terrible tip. Since 20% is now standard, they should have just let the waiter keep the \$5. Fractions are, yes, a red herring when each person doesn't have to separately provide their own exact share. Difficulty would only arise if the waiter had to return the money individually to each diner (and each suffered from OCD).

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### Re: The Diner Check Problem

Dave_C » February 25th, 2018, 4:46 pm wrote:3 people each took \$1 each and collectively paid the waiter an additional \$2 (each person paid \$9.66666667 each). There's no missing dollar, just an expectation that somehow, there's another dollar that's gone missing.

My question is, how the hell did each person pay a fraction of a penny?

Just £9 each. Remember they each put in 10 and each got 1 back. There's 25 for the food and 2 for the tip, which makes 27 - or for long, 3 x 9.

Lomax

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### Re: The Diner Check Problem

I like Lomax's last succinct answer.

Dave_C's question "how the hell did each person pay a fraction of a penny?", possibly warrants an explanation.

The question of fractions of a shilling never became an issue, because the diners took a whole shilling each back and gave 2 whole shillings to the waiter.

IF the diners had decided not to tip the waiter, THEN, and only THEN, fractions of a shilling would have become an issue. They would have each been entitled to one third of 5 shillings refund, which would have been 1.6667 shillings, and in that case only, they would have paid 8.3333 shillings each. (This, by the way is the mathematical share of what they each paid to the restaurant). Tipping 0.6667 shillings each to the waiter kept all of the transactions in whole dollars, so that fractions never became an issue.

doogles
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### Re: The Diner Check Problem

I woke up this morning with my subconscious working further on the diner problem. It wasn't relevant to the explanation of the diner 'missing shilling' problem per se, but a recall of a piece of history trivia associated with the scenario.

If the diners had decided to split the 5 shillings three ways and not give a tip, they could have done so without any fuss because there are 12 pence in each shilling. They would have taken 1 shilling and 8 pence each. The restaurant would had enough change to give each of them the 8 pence in the form of a sixpence coin and 2 pennies.

They could then have given the waiter eight pence each and he would have had a handful of tips.

I believe this 'subconscious imagery' is common in all of us by the way. I note that Sponge has raised the matter in the CTD thread. If I get the time today, I'll start a new thread on it.

doogles
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### Re: The Diner Check Problem

This is not a math problem, this is a psychological trick:
Alan Masterman » February 25th, 2018, 12:42 pm wrote:So in the outcome, each diner has paid \$9 for the meal, amounting to \$27. The \$2 for the waiter brings the amount spent to \$29. What happened to the last dollar?

The \$2 for the waiter is part of such \$27 (from which it must be subtracted, not added)

neuro
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### Re: The Diner Check Problem

If the total bill was \$25, then they each owed \$8.33 each, but since they each paid \$10.00 each, they each were due \$1.67 each in change (\$5 total). So of the 5 coins, they each took a \$1.00 back and donated the .67 each (\$2 total) as the tip.

...and Neuro is correct too! \$25 was the total bill. They paid \$27 total (\$9 x 3), leaving \$2 left for a tip.

RJG
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### Re: The Diner Check Problem

There are a bunch of variations on this puzzle. They can all be resolved by adding assets and subtracting liabilities. The same procedure can be repeated for every participant involved. You can always recover the original amount stated in the problem by adding liabilities to assets.

hyksos
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### Re: The Diner Check Problem

Dave_C » February 25th, 2018, 7:46 pm wrote:There's no missing dollar, just an expectation that somehow, there's another dollar that's gone missing.

All variations on this puzzle work the same way. Rattle off a bunch of money amounts at the end of the story and ask why they don't sum to \$X by a squirrely question that presumes the total should come out to \$X. There is no reason in the universe the numbers rattled off should sum to 30.

hyksos
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