Who owes who: how the calculation actually works
Five people, seventeen expenses, four different payers. The question is always the same, and it has exactly one right answer: how much does each person owe, and to whom.
5 min read
You get back from a weekend away. One person paid for the house, another for the food, a third for fuel and tolls, and two people paid for nothing at all. In the group chat, someone types "ok so", and the conversation rarely ends well.
The maths is simple. What is hard is doing it out loud, in the right order, without asking the wrong question first.
The only formula that matters
For each person:
Balance = what they paid out − what they owe + what they have repaid − what they have received
A positive balance means the group owes them money. A negative one means they owe the group.
And one property worth checking every single time:
The balances of a group always add up to exactly zero.
That is not a nice turn of phrase, it is a test. Every euro someone paid out is charged to someone, including to the person who paid it. If your balances do not sum to zero, you have missed an expense, missed a participant, or counted a repayment on the wrong side. Do that check before arguing about anything else.
The mistake that breaks everything: mixing up "paid" and "owed"
By far the most common one, and it sounds perfectly sensible.
Most people reason expense by expense: "Alice paid 300, so we each owe her 100." Then for the food: "Bruno paid 90, so we each owe him 30." The result is four transfers, two of which cross each other, and a conversation about who should pay first.
Reasoning by balance gets the same economic outcome in two transfers. That difference is not cosmetic: every transfer is one more chance to forget, to send the wrong amount, or to keep someone waiting.
The shortest way to settle up
Once you have the balances, you have to decide who pays whom. The rule: line up the debtors and the creditors, and clear the largest debt against the largest credit until nothing is left.
In the example above, Chloe owes 130, Bruno owes 40, Alice is owed 170. Chloe sends €130 to Alice, Bruno sends €40 to Alice. Two transfers, and everyone is at zero.
The bound worth remembering: for n people you never need more than n − 1 transfers. With five people, four transfers always suffice, and two or three usually do. If your tally suggests more, it is still thinking expense by expense.
The cent that will not divide
Three people, a €10 expense. Each share is €3.333… In cents you can only write 3.33 or 3.34, and 3.33 × 3 = 9.99. One cent is missing.
Dropping it is the worst option: your total no longer equals the expense, and the zero-sum property breaks. Over a hundred expenses the gap becomes visible, and a tally that does not add up loses all its authority.
The correct method is called largest remainder: give everyone their share rounded down, then hand out the leftover cents to whoever had the largest fractional part. On €1 split three ways: 0.34, 0.33, 0.33. The sum is exactly €1.
Two requirements are worth stating:
- the shares always add up to the expense, to the cent;
- the tie-break is deterministic: running the same numbers twice gives the same answer. A cent that wanders between two screens destroys trust faster than an outright error.
Four ways to split one expense
Not every expense is split equally, and a tally that only knows how to do that forces you to cheat.
| Mode | When | Example |
|---|---|---|
| Equal shares | The usual case | Groceries, a tank of fuel |
| By shares | Someone counts double | A couple in a group of friends, a room for two |
| By percentage | A fixed key, agreed in advance | 40 / 60, in proportion to income |
| Exact amounts | The bill itemises who had what | A restaurant, an invoice line by line |
Expenses that do not concern everyone
This is the most common case after the first evening, and the one that breaks the most spreadsheets.
Four people out of six go to a restaurant. The €92 bill is not split six ways, it is split four ways. Everyone's balance has to reflect that immediately, without opening a second tally for "Tuesday's subgroup".
That is why a useful tally reasons in terms of the participants in an expense, not the members of the group. The group says who can see and enter things; the expense says who pays.
What Kotisso does about it
Kotisso applies exactly what is above: every expense is split using one of the four modes, balances are recalculated live for everyone, the zero-sum property is checked, and the settlement plan suggests the shortest route.
Two things are worth calling out:
- the exchange rate is frozen when you enter the expense. Something paid in baht is converted once, at the rate you type in, and never moves again. Fetching today's rate would change everyone's balance every morning, three months after the trip ended;
- the tally exports, to a spreadsheet or a PDF. The file has one column per person, every row sums to zero, and each person's column sums to their final balance. The zero-sum property becomes checkable in Excel, which ends the discussion faster than any explanation.
The whole tally is free: all four split modes, the export, the receipts and the settlement plan.
Common questions
Should we settle after every expense? No, and it is counterproductive: you multiply transfers for an identical result. Let the balances run and settle at the end, or once a month for a flatshare.
What if someone paid for a person outside the group? Do not invent a ghost member. Record the expense at the amount actually shared, and treat the outsider's share as a repayment when it arrives, or as a separate expense.
Can you add someone halfway through? Yes, and their balance starts at zero: earlier expenses do not concern them unless you explicitly edit those expenses to include them. That is the correct behaviour, and also the one that surprises people most.
What if a balance is never settled? It stays in the tally, visible to everyone, until a repayment is recorded. That is also why you cannot leave a group while you still owe money: money would disappear from everyone else's books.