Why the cents go missing, and how they come back
Divide $847.20 by weights of 3, 2 and 2 and the exact answers are $363.0857…, $242.0571… and $242.0571…. Round each to the cent independently and you get $363.09, $242.06 and $242.06, which sums to $847.21 — a cent invented out of nothing. Round each one down instead and you get $847.18, two cents short. Neither is a settlement you can send to three people.
The fix is to stop rounding independently. Every amount on this page is held as an integer number of cents. Each share is floored to the chosen unit, the shortfall is measured as a whole number of cents, and that shortfall is handed back out one unit at a time. Because the leftover is exactly the sum of what the flooring removed, the parts always add back to the original total. The check block at the bottom of every result prints that sum next to the total, and it is there so you can confirm it rather than take it on trust.
The four leftover rules
| Rule | What it does | When it fits |
|---|---|---|
| Largest remainder first | Sorts by how much each share lost to flooring and gives one unit each, starting with the biggest loss, until the leftover is gone | Paying money out, where the fairest distribution matters and everyone can see the working |
| All to the last row | The final row absorbs the whole leftover | Collecting money in, where the organiser takes the odd amount |
| All to the first row | The first row absorbs the whole leftover | Same, when the responsible party is listed first |
| Leave it undistributed | Nothing absorbs it; the amount is reported | When the remainder goes somewhere outside the list — a float, a tip jar, a rounding account |
Largest remainder is the default. It is the same allocation method used to apportion seats to parties from vote counts, and its property is that no share is ever more than one unit away from its exact value. It never gives one row several extra units while another goes short. When the fractions tie exactly — three equal shares of $10.00, say — ties break in the order the rows were written, so the first row named takes the extra penny.
Rounding to something larger than a cent
Setting the unit to a dollar, or to five dollars, makes each share a round figure and pushes correspondingly more into the leftover. Split $100.00 three ways at the dollar and the exact shares are $33.33 each; floored to whole dollars they are $33.00 each, leaving $1.00, which goes to one row. The result is $34.00, $33.00, $33.00. That is a bigger gap between people than a penny, so a coarse unit is worth choosing only when round numbers matter more than evenness — reimbursements handed over in cash, for instance.
There is one wrinkle. If the total itself is not a whole number of units — $100.05 split at the dollar — then the five cents cannot be part of any whole-dollar allocation. It rides along with the leftover, so the row that takes the remainder ends up at $34.05 while everyone else is on a round dollar. A warning appears when this happens so it does not look like a bug.
A worked example you can check by hand
Total $847.20, shares Alex 3, Blake 2, Casey 2, rounded to the cent, largest remainder. In cents the total is 84,720 and the weights sum to 7. Alex's exact share is 84,720 × 3 ÷ 7 = 36,308.571; Blake and Casey are each 84,720 × 2 ÷ 7 = 24,205.714. Floor all three: 36,308 + 24,205 + 24,205 = 84,718, leaving 2 cents. The two largest fractions are Blake's and Casey's at .714, so each takes one cent: 36,308, 24,206, 24,206. That is $363.08, $242.06 and $242.06, and the three add to $847.20 exactly. Alex is the one who loses out, by just over half a cent, because Alex's fractional part was the smallest — which is precisely what the rule is supposed to do.
Questions people ask
Can I write the shares as percentages?
Yes. Write Alex 30% and the percent sign is ignored, leaving the weight 30. They do not need to add to 100, because only the ratio between the numbers is used. Three people written as 20, 30 and 40 add to 90, and they are normalised to 22.22, 33.33 and 44.44 percent and given the entire total between them. If you genuinely want to hold 10 percent back, add it as its own row with a name like "reserve" so it appears in the split and in the check.
Which leftover rule should I use?
There is no correct answer, only an agreed one. Paying money out to several people, largest remainder distributes the difference most evenly and is easy to justify because it is the same method used for apportioning seats. Collecting money in, sending the odd amount to a single row is simpler to explain and usually lands on whoever is running the collection. What matters more than the choice is making it before you circulate the numbers, because changing it afterwards means every figure moves.
What happens to a share with a weight of zero?
It receives nothing, which is correct. The one case to watch is the leftover rule: if you send the remainder to the first or last row and that row has a weight of zero, it receives the leftover and nothing else, which looks strange in a settlement. A warning fires when a zero weight is present. Negative weights are read as zero, because a negative share inverts the split into something that is arithmetically defined but has no sensible reading.
Can I split something that is not money?
The tool is built around money and formats everything as dollars, with the arithmetic in integer cents. For hours, units or points you can read the amounts as plain numbers and set the rounding unit accordingly — the dollar for whole units, the cent for two decimal places — and the exactness guarantee still holds, since it is a property of the integer arithmetic rather than of the currency. What will not work well is anything with a non-decimal subdivision, like hours and minutes.