Three guests pay 30 dollars for a hotel room that should have cost 25. Of the 5 dollars sent back to them, the bellhop keeps 2 and returns 3. Each guest has now paid 9 dollars, 27 in all, and 27 plus the bellhop’s 2 makes only 29, so where did the thirtieth dollar go? This page follows every dollar to its final owner and shows that the riddle’s sum adds two amounts that overlap.
Key terms
Missing dollar riddle
Definition The missing dollar riddle tells this story. Three guests share a hotel room and each pays 10 dollars, so the hotel receives 30 dollars. The manager then sees that the room costs only 25 dollars and gives the bellhop 5 dollars to return to the guests. The bellhop keeps 2 dollars and gives 1 dollar back to each guest. Each guest has now paid dollars, so the guests have paid 27 dollars in all, and the riddle adds the bellhop’s 2 dollars to get 29. It then asks where the thirtieth dollar went. The resolution is that no dollar is missing. The sum is correct arithmetic, but its two amounts are not separate parts of the 30 dollars, because the 27 dollars the guests paid are the 25 dollars at the hotel plus the 2 dollars with the bellhop. The correct accounts are
The first says the 30 dollars end up as 25 at the hotel, 2 with the bellhop and 3 back with the guests. The second adds the guests’ 3-dollar refund to the 27 dollars they paid in the end, which gives back the 30 they first handed over. The third takes the bellhop’s share out of the 27, which leaves the price of the room.
In plain words The riddle adds a part to an amount that already contains it. The 2 dollars the bellhop kept are part of what the guests paid, so adding them to the 27 counts them a second time, while the 3 dollars in the guests’ pockets are left out.
Example A ledger that lists who holds the 30 dollars after each step shows that nothing disappears.
| Step | Guests | Hotel | Bellhop | Total |
|---|---|---|---|---|
| Before paying | 30 | 0 | 0 | 30 |
| After paying | 0 | 30 | 0 | 30 |
| Manager gives 5 dollars to the bellhop | 0 | 25 | 5 | 30 |
| Bellhop returns 3 dollars | 3 | 25 | 2 | 30 |
Every transfer takes an amount from one holder and adds the same amount to another, so the total stays 30. The final row splits the money into three groups that do not overlap and together hold all 30 dollars, so their sum must be 30.
Double counting
Definition Double counting is the error of including the same item more than once when adding up a total. Adding the sizes of several groups gives the size of everything they contain only when no two of the groups overlap. To get back an original total, the groups must also cover all of it. For two groups and that may overlap,
where is the number of items in , the union is everything in at least one of the groups and the intersection is what they share.
In plain words Before adding numbers, check that each one counts different things and that together they count everything. If one group sits inside another, adding both counts the inner group twice.
Example A class has 12 students in the chess club and 8 in the music club, and 3 students belong to both. The number of students in at least one club is , not 20, because adding 12 and 8 counts the 3 students in both clubs twice. In the riddle, treat each dollar as an item. The 27 dollars the guests paid in the end contain all 2 of the bellhop’s dollars, so the overlap is 2 dollars and the two groups together hold only dollars. The 3 refunded dollars lie in neither group. Compared with the 30 dollars, the riddle’s sum is
where the is the bellhop’s money counted a second time and the is the refund left out. The apparent shortfall of 1 dollar is the difference .
Net payment
Definition The net payment of a payer is the total amount paid minus the total amount refunded,
It is the amount that has left the payer for good. When money moves only between the payer and the receivers, it equals the total that the receivers keep in the end. The original payment and the net payment describe different moments, before and after the refund, and they differ by exactly the refund.
In plain words The net payment says how much the payers are out of pocket once all refunds are in. That money may be split among several receivers, but their shares are parts of the net payment, not extra amounts to add to it.
Example Each guest paid 10 dollars and got 1 back, a net payment of 9 dollars, and the three guests together have a net payment of dollars. The receivers keep dollars, the hotel’s 25 and the bellhop’s 2, which is the same amount. Now change the story so that the bellhop keeps nothing and returns all 5 dollars to the guests. Their net payment becomes dollars, all of it at the hotel, and the riddle’s method gives . Nobody would ask where 5 dollars went, because the 5-dollar refund is plainly back with the guests, and the full account is .
Check yourself
- Change the numbers. The room really costs 20 dollars, the manager sends 10 dollars back, and the bellhop keeps 4 dollars and returns 2 to each guest. What is the guests’ net payment, what total does the riddle’s method give, and where are the 30 dollars?
Show the answer
Each guest has paid dollars, so the guests’ net payment is 24 dollars, which is the hotel’s 20 plus the bellhop’s 4. The riddle’s method adds the bellhop’s share again and gets , so it seems to lose 2 dollars. The 30 dollars are 20 at the hotel, 4 with the bellhop and 6 back with the guests, and . The flawed sum counts the 4 dollars twice and leaves out the 6, so it falls short by .
- The room costs 24 dollars, the manager sends 6 dollars back, and the bellhop keeps 3 dollars and returns 1 to each guest. Now the riddle’s method gives . Does that mean the method is right in this case?
Show the answer
It does not. The guests’ net payment is dollars, which is the hotel’s 24 plus the bellhop’s 3. Adding the bellhop’s 3 dollars again still counts those dollars twice and still leaves out the 3 refunded dollars. The total comes out right only because the double-counted amount and the omitted amount happen to be equal. A correct account uses groups that do not overlap,
where the first 3 dollars are with the bellhop and the second 3 dollars are back with the guests.
- Show that, whatever the numbers, the riddle’s sum differs from the original payment by the bellhop’s share minus the refund. Then use this for a room that costs 21 dollars, where the manager sends 9 dollars back and the bellhop keeps 6 and returns 1 to each guest.
Show the answer
Let be the original payment, the hotel’s final amount, the bellhop’s share and the refund that reaches the guests. Every dollar ends with exactly one holder, so . The net payment is , and the riddle’s sum is
So the riddle’s sum equals the original payment exactly when , falls short when the refund is larger than the bellhop’s share, and overshoots when the refund is smaller. In the original story and , which gives . In the new case , , and . The net payment is , the riddle’s sum is , three dollars too many, and the correct account is .