Rounded numbers do not add up

A column of correctly rounded numbers frequently does not add to the rounded total. Six percentages that each round to the nearest whole can sum to 99 or 101, and neither the parts nor the total is wrong: rounding is not additive, so doing it before you add and doing it after you add are different operations with different answers. The only question is which of the two you meant.

This is not a floating-point problem and it does not go away with more decimal places. It is arithmetic, it is guaranteed to appear in any table long enough, and the fix is a decision about where the leftover goes rather than a better calculator.

Why does the column not match the total?

Because each row is moved independently, and the moves do not cancel. Rounding a value up borrows from nothing and rounding it down gives to nothing, so the sum of the errors is whatever it happens to be.

Every row there is rounded correctly and the column is two out. Six values each gaining around a third of a point is not an error that cancels, it is an error that accumulates, and a longer table drifts further.

What is the largest remainder method?

The standard way of forcing a set of rounded shares to hit their total. Round everything down first, count how many units are left over, and hand them out to the rows with the largest fractional parts until the total is reached.

Applied to the table above, every row floors to 16, which leaves four units to distribute across a target of 100. They go to the rows with the biggest fractions — F at 0.8, then D and E at 0.7, then one of the three rows sitting at 0.6. Five rows read 17, one reads 16, and the column adds to 100.

It is the same method used to turn vote shares into seats, where it goes by other names. The property that matters is that it is a rule: it produces the same answer every time, and it can be explained to whoever asks why one row is a unit lighter than its neighbours.

Where does this actually cost something?

Anywhere a total is a commitment rather than a summary. Splitting a bill of 100 between three people gives 33.33 each and loses a penny; allocating a discount across order lines has to reconcile with the discount actually granted; apportioning a shared cost between departments has to sum to what was spent.

Tax is the strict case, because both the line amounts and the total are declared figures that somebody may check. Rules vary on whether tax is computed per line and summed or computed once on the total, and the two give different answers — which is exactly why the rules specify one rather than leaving it to the implementation.

Payroll has the same shape spread over time. A monthly deduction rounded to the cent twelve times does not match the annual figure rounded once, so a system has to decide whether the year or the month is the number it is willing to defend.

Does rounding the total separately fix it?

It fixes the total and breaks the relationship between the total and the rows, which is usually the worse trade. A reader who adds the column and gets a different number assumes a mistake, and on a printed invoice they are entitled to.

The workable approach is to nominate an absorber: one row, chosen by a rule, that takes the remainder so that the parts and the total agree. The largest-remainder method is one such rule, and a fixed choice — the largest line, or the last — is another, provided it is applied consistently rather than per document.

The one thing that does not work is rounding twice on the way, which compounds the problem instead of resolving it. That failure has its own article, and the short version is to carry full precision until the final presentation.

What should be visible to the reader?

The fact that an adjustment happened, where it is material. A footnote saying figures may not sum due to rounding is not an apology, it is the honest statement that the table is showing rounded values and the total is the real one.

Percentages deserve particular care because a reader is more likely to add them. A set of shares presented as whole numbers is expected to reach 100, and when it reaches 101 without explanation the table loses credibility it did not need to lose.

Questions people ask

Does banker’s rounding solve it? It removes the systematic upward bias from ties, which helps over many values, but it does not force a set of shares to reach their total. The two problems are separate and need separate fixes.

Should I round at all before totalling? Not for the arithmetic. Carry the unrounded values through, total them, and round only what is displayed — unless a rule requires rounding at each step, as some tax regimes do.

What about more decimal places? They shrink the discrepancy and never remove it. A column of values to two places still fails to sum to a two-place total; it just fails by a smaller amount, which is arguably worse because it is harder to spot.

Is the largest remainder method fair? It is deterministic and explainable, which in practice is what is being asked for. Every apportionment rule has a scenario that looks unfair, and the literature on seat allocation is largely a catalogue of them.

Rounding is a presentation decision and it belongs at the end. The rounding calculator shows what each rule does to a value, and the percentage calculator gives the exact shares to apportion from before anything is rounded at all.