Three guests check into a hotel room that costs $30. They each pay $10. Later the clerk realizes the room is only $25, so he gives the bellhop $5 to return.
The bellhop pockets $2 and gives $1 back to each guest.
Now each guest has paid $9, for $27 total. The bellhop has $2. That is $29. Where did the last dollar go?
Answer
There is no missing dollar. The misleading sum is $27 + $2 = $29 instead of $25 + $2 + $3 = $30.
The guests paid $27 total after the refund. Of that, $25 stayed with the hotel and $2 stayed with the bellhop. Adding the bellhop's $2 back onto the $27 double-counts and invents a fake $29.
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Where the fake dollar comes from
The framing trick here has been written up for a long time, and it is worth seeing somebody else lay out the same bad addition:
