The initial payment of $30 is accounted for as the owner takes $25, the waiter takes $2, and the diners get a $3 refund. It adds up. After the refund has been applied, we only have to account for a payment of $27. Again, the owner keeps $25 and the waiter gets $2. This also adds up.
There is no reason to add the $2 and $27 – the $2 is contained within the $27 already. Thus the addition is meaningless. Instead the $2 should be subtracted from the $27 to get the revised bill of $25.
This becomes clearer when the initial and net payments are written as simple equations. The first equation shows what happened to the initial payment of $30:
- $30 (initial payment) = $25 (to owner) + $2 (to waiter) + $3 (refund)
The second equation shows the net payment after the refund is applied (subtracted from both sides):
- $27 (net payment) = $25 (to owner) + $2 (to waiter)
Both equations make sense, with equal totals on either side of the equal sign. The correct way to get the waiter's $2 and the guests $27 on the same side of the equal sign ("The waiter has $2, and the guests paid $27, how does that add up?") is to subtract, not add:
- $27 (final payment) - $2 (to waiter) = $25 (to owner)
