Splitting $10 three ways: where the extra cent goes

The short version

$10 between three people is $3.34, $3.33 and $3.33. Somebody pays a cent more, because ten does not divide by three and the remainder has to land somewhere.

What you must not do is round each share and hope. Give everyone $3.33 and you have collected $9.99, so the bill is a cent short. That failure is well known. The one nobody mentions is that rounding can go the other way and collect more than the bill, which is worse.

Rounding does not just lose money

Everyone has seen the missing cent. Almost nobody has noticed that the same shortcut can overshoot, and that when it does you are collecting money that belongs to no one. There is no correct person to give a surplus back to, which is precisely why it is the worse of the two failures.

That last row is the one to sit with. Five cents between three people, rounded, produces six cents. A 20 percent error, from an operation that looks like being careful.

The rule that always adds up

It is called largest-remainder allocation, and it is three steps you could do on paper.

• Work out the exact share. $10 across three is 333.333 cents each.
• Give everyone the floor. 333 cents each, which is $9.99 handed out and one cent still in your hand. The leftover is never more than one cent per person, so this always terminates quickly.
• Distribute the remainder, biggest loser first. Everyone lost 0.333 of a cent here, so it is a tie, and the tie-break decides. One person gets $3.34.

The property that makes this correct rather than merely tidy: the parts sum to the total by construction. You never distribute more cents than you held back, and you never stop while holding one. There is no amount and no number of people for which it drifts.

Who pays the extra cent, and why it is not arbitrary

When everyone lost the same fraction to rounding, something has to break the tie. The obvious choice is the first person in the list, and it is the wrong one, for a reason that has nothing to do with fairness.

Sort by position and the answer changes when the order changes. The same three people and the same $10 give a different result depending on how the list happened to be built, which means two devices looking at the same expense can compute two different splits and both be following the rules.

So the tie-break is on something stable. Halvy breaks ties by user id, and the effect is checkable: split $10 across Ana, Ben and Cleo, then split it again with the list reversed to Cleo, Ben, Ana. The extra cent lands on Ana both times.

Why that property is worth having

Because your phone and the server split the same expense separately, and they have to agree without conferring. A deterministic tie-break means a balance computed offline on a train matches the one computed on a server an hour later, to the cent. Sort by position instead and you get a class of bug that looks like a sync failure and is actually two correct answers disagreeing.

Try it on your own numbers

Put in any amount and any number of people. The left is the allocator the app uses; the right is rounding each share, which is what a calculator gives you.

A cent does not matter. Until it does.

On one dinner, this is pedantry. Nobody is chasing a penny, and a friend who did would be a worse problem than the penny.

It matters when a record runs for months. Every expense that does not divide evenly adds another cent of drift, in whichever direction the rounding happened to fall, and the errors do not cancel. After a few dozen expenses the balances are out by an amount too large to ignore and too small to explain.

That is the real cost, and it is not the money. It is that nobody can tell whether the discrepancy is rounding or a genuine mistake, so the whole record stops being evidence. A shared ledger people no longer trust is worse than no ledger, because they have stopped keeping receipts too.

Which is why this is worth getting right once, in the thing doing the arithmetic, rather than adjudicating later. The fewest payments that settle everyone up depends on it completely: netting people off only works if the parts reconcile, and how to split shared costs fairly covers choosing the method before any of this arithmetic applies.

Common questions

Every figure here came from running Halvy’s allocator against a naive round-each-share for the same inputs, including the cases where rounding overshoots. The determinism claim was checked by splitting the same amount across the same three people twice, with the list reversed the second time.