Who owes whom after a group bill
Four people, nine expenses, three different cards. Everybody agrees what was spent and nobody can say who should pay whom. The arithmetic is genuinely simple, but it has to be done in the right order — and the step almost everybody skips is the one that turns nine payments into two.
Two numbers per person, not a web of debts
The instinct is to track debts between people: Ann owes Bob for the taxi, Bob owes Cara for lunch, Cara owes Ann for the tickets. That is the hardest possible representation, and it grows quadratically — six people can owe each other thirty different ways.
Work in two numbers per person instead. What they paid, and what they consumed. Everything else is derived from those, and neither of them depends on anybody else's arithmetic being right.
- Paid: the total of every expense that person put on their own card.
- Consumed: their share of everything — their own items, plus their portion of anything shared.
- Balance: paid minus consumed. Positive and the group owes them; negative and they owe the group.
The balances must add to zero
Add every balance together and the result is zero, always. Money that somebody paid is money somebody else consumed, so the positives and the negatives are the same total seen from two sides.
This is the single most useful check in the whole process. If your balances do not sum to zero, you have not found a disagreement about fairness — you have an expense that nobody claimed, or one counted twice. Look for the missing line rather than arguing about the split.
Then settle between the two lists
Now there are only two groups: people who are owed, and people who owe. Nobody in the first group ever pays anybody, and nobody in the second is ever paid. That alone removes most of the transfers people would otherwise make.
Take the largest debtor and the largest creditor, move the smaller of the two amounts between them, and repeat. Somebody drops off the list every time, so a group of N people is always settled in at most N minus one payments — four people, three payments, whatever tangle you started with.
Why fewer payments is worth the trouble
The same money moves either way, so this can look like tidiness for its own sake. It is not. Every transfer is a chance to mistype an amount, a fee if anybody is crossing a currency, and one more thing for somebody to forget on Monday.
It also changes the social texture of settling up. Two payments feels like closing a trip; eleven feels like admin, and admin is what people quietly stop doing halfway through.
Rounding, and the cent nobody owes
Split thirty by four and you get seven fifty each. Split ten by three and you do not — somebody has to absorb a cent, and if the arithmetic is done twice by two people it may be a different somebody each time.
Fix it by deciding once, at the point of the split, rather than at the point of payment: give the remainder to a fixed person in a fixed order, and record that. The amount is trivial and the argument about it is not, because it is never really about the cent.
How platy24 does it
Balances are recalculated from the items and the claims every time anything changes; they are never stored, so they cannot drift from what is on the tab. The settle-up screen runs exactly the reduction above and shows only the payments that are actually left.
Anything nobody has claimed is reported separately rather than being quietly spread across everybody — an unclaimed dish is a question for the table, not a rounding error to hide.