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The cent that will not divide is placed by rule, not by taste. Ten dollars in cents is 1,000, and 1,000 does not divide by 3: every share is exactly 3.3333…, so the three shares floor to 3.33 and one cent is left over. This page shows where that cent goes, on three bills, and why rounding each share on its own cannot place it.
What one $10.00 bill becomes, split 3 ways
Every share is exactly $3.3333…, so the 3 shares floor to $3.33 each and 1 cent is left. All 3 fractional parts tie, so the first-listed person takes the cent.
Type any amount and any number of people. The split below is the same integer-cent arithmetic every calculator on this site runs: every share is floored to the cent, then the leftover cents go to the largest fractional parts.
Every share is floored to the cent first, then the leftover cents go to the largest fractional parts — so the shares always add back up to the bill exactly.
Three views of the same rule: the worked cases side by side, the division behind the $10.00 case, and what one cent is worth at three different bill sizes.
Money has a smallest unit: the cent. Ten dollars is 1,000 cents, and 1,000 does not divide by 3, so a third of the bill is 333.3333… cents — a number that does not exist as money. Something has to give, and the cents where the division stops are the leftover. This is not a rounding error and not a mistake in the arithmetic: the bill simply cannot be divided into three equal amounts that are all whole cents.
Some bills avoid the step entirely. Six dollars is 600 cents and 600 divides by 3 exactly, so three shares of 2.00 add back to 6.00 with nothing left to place. The rule below is still applied to those bills — it just finds nothing to do, and that is the correct answer rather than a rounding that happens to balance.
Floor every share to the cent first, then hand the leftover cents to the shares with the largest fractional parts. The parts always add back up to the bill exactly, and which person takes the extra cent is decided by arithmetic rather than by taste.
Two things are worth noticing about it. First, the flooring happens before anything else: the floored shares deliberately add to less than the bill, and the shortfall is measured rather than hidden. Second, the leftover is handed out by the size of each share's fractional part — the share that came closest to a cent gets first refusal on the missing cents. Ties happen whenever the split is even, because then every fractional part is identical; the tie is broken by order, so the first-listed people take one cent each until the leftover runs out.
The tempting move is to round every share to the nearest cent and let the total land where it lands. On the $10.00 case each share is 333.3333… cents, which rounds down to 333; three shares of 333 are 999 cents, so the three payments come to $9.99 and the bill is a cent short. The same move on the $0.05 case goes wrong in the other direction: each share is exactly 2.5 cents, rounding half-up takes both shares to 3, and the two payments come to $0.06 — a cent more than the bill.
Rounding each share on its own can leave the total short or push it over, and which of the two happens depends on the shape of the numbers. The largest-remainder rule can do neither. It only ever moves cents that the flooring step already set aside — exactly as many as were set aside — so the total is exact by construction rather than by luck.
| Bill | Split between | Exact share | Floored shares | Leftover | Final shares | Adds back to |
|---|---|---|---|---|---|---|
| $10.00 | 3 people | $3.333333…1,000 ÷ 3 cents | $3.33 × 3= $9.99 | 1 cent | $3.34 / $3.33 / $3.33 | $10.00 ✓ |
| $100.00 | 3 people | $33.333333…10,000 ÷ 3 cents | $33.33 × 3= $99.99 | 1 cent | $33.34 / $33.33 / $33.33 | $100.00 ✓ |
| $0.05 | 2 people | $0.0255 ÷ 2 cents | $0.02 × 2= $0.04 | 1 cent | $0.03 / $0.02 | $0.05 ✓ |
| $0.20 | 3 people | $0.066666…20 ÷ 3 cents | $0.06 × 3= $0.18 | 2 cents | $0.07 / $0.07 / $0.06 | $0.20 ✓ |
| $6.00 | 3 people | $2.00600 ÷ 3 cents | $2.00 × 3= $6.00 | none | $2.00 each | $6.00 ✓ |
On $0.05 split two ways the leftover cent is 20% of the bill, and the larger share is 150% of the smaller one — one person genuinely pays half again as much. On $10.00 the leftover cent is 0.1% of the bill and the larger share is 100.3% of the smaller; on $100.00 those figures are 0.01% and 100.03%. Same arithmetic, three meanings: on a five-cent bill the rule is deciding something, and on a hundred-dollar bill it is housekeeping nobody can see.
Never more than one fewer than the number of people, because each share loses less than one cent to the floor. Three people can leave one or two cents — $10.00 ÷ 3 leaves one, and $0.20 ÷ 3 leaves two, which become $0.07, $0.07 and $0.06 — and ten people can leave up to nine. The leftover is always a handful of cents at most, and the rule places every one of them.
The leftover cent appears in every split, not just the even ones: per item, by percentage, by income, by room size. Which basis is chosen is a different question from this one — the split by percentage page decides who owes a share of the bill, and this page decides what happens to the cents that cannot be divided. The basis changes the numbers; the leftover rule keeps them exact.
Every calculator here runs the same integer-cent arithmetic, so a figure can be checked on one page against the same figure on another. The split bill calculator applies the rule to a whole bill with tax, tip and items; the split by income and split utilities pages apply it to weighted totals. Everything is computed in the browser and nothing is uploaded — the privacy page says what that means, and the about page says who builds these pages.
Because 1,000 cents does not divide by 3. Each share is exactly 333.3333… cents, so each one is floored to 333, which gives 999 cents in total and leaves one. The leftover cent goes to the largest fractional part; all three fractions are identical, so the tie is broken by order and the first-listed person takes it. That is why the three figures read 3.34, 3.33 and 3.33 and still add back to $10.00 exactly.
Because rounding each share on its own breaks the total. 333.3333… rounds to 333, so three rounded shares come to $9.99 — a cent short of the bill. On the $0.05 case the exact share is 2.5 cents, half-up rounding takes both shares to 3, and they come to $0.06 — a cent over. Whether the total ends up short or over depends on the numbers; flooring first and placing the leftover does neither, because it only moves cents the floor already set aside.
The first-listed people, one cent each. An even split gives every share an identical fractional part, so there is no largest fraction to prefer and the order of the list breaks the tie. If that stings, rotate who is listed first from one bill to the next — the arithmetic does not care, and the totals stay exact either way.
Fewer than the number of people, because each share loses less than one cent to the floor. Three people can leave one or two cents: $10.00 ÷ 3 leaves one, and $0.20 ÷ 3 leaves two, which become $0.07, $0.07 and $0.06. Ten people can leave up to nine. The leftover is never more than a handful of cents, and the rule places every one of them.
Because a share of a cent is not a cent. Two people and five cents is 2.5 cents each, which no one can pay or receive. The two shares floor to 2 cents each, one cent is left, and it goes to the first-listed person — so one person pays 3 cents and the other pays 2, and the two still add to the 5 cents that were on the table.
By at most the number of leftover cents. The exact share is unchanged — the rule only decides who carries the cents the division could not place. Nobody can end up paying a materially different amount because of it, and the largest move it can make between two people is the whole leftover, which on any bill of a few dollars or more is a cent or two.
On small bills. On $0.05 split two ways the leftover cent is a fifth of the bill and the two shares differ by half; on $100.00 split three ways it is a hundredth of a percent and nobody can see it. The rule is the same in both cases — what changes is what the cent is worth relative to the money it came from.
Yes, always. The flooring step decides how many cents are left over, and the placement step hands out exactly that many cents — not one more, not one fewer. That is why every table on this page ends in an exact total, and why the split bill calculator never shows a total that disagrees with its parts.
Yes. Every calculator here computes in whole cents and places remainders largest-remainder-first, so the answer is the same whether the split is even, per item, by percentage or by income. This page shows that one step on its own, where it is easiest to see, rather than burying it inside a larger calculation.
After they are applied. The bill that gets split is the final figure, with tax, tip and any discount already carried through it, and the cent-splitting rule runs once on that number. The split by percentage page and the split by income page apply the same rule to their own totals.