Rounding, and the invoice that is a penny out

Every line on the invoice is correct. The total is correct. They differ by a penny, and the customer wants to know which one is the mistake. Neither is. This is rounding behaving exactly as it must.

The basic rule, and the digit that decides

To round to a number of decimal places, look at the digit immediately after the cut. Below 5, cut and stop. Above 5, cut and add one to the last digit you kept. Exactly 5 with nothing after it is a tie, and the tie is the whole argument.

Half-up against half-even

Half-up is the rule taught in school: ties go up. 2.5 becomes 3; 0.125 at two decimals becomes 0.13.

Half-even, sometimes called banker's rounding, sends a tie to whichever candidate ends in an even digit. 2.5 becomes 2, 3.5 becomes 4, and 0.125 at two decimals becomes 0.12.

The reason half-even exists is that half-up pushes every tie in the same direction, so a column of them drifts. Round 0.5, 1.5, 2.5 and 3.5 to whole numbers:

• True sum: 0.5 + 1.5 + 2.5 + 3.5 = 8
• Half-up: 1 + 2 + 3 + 4 = 10
• Half-even: 0 + 2 + 2 + 4 = 8

Half-up is two whole units high across four numbers. Half-even lands on the true total because it sent two ties up and two down.

On messy real data exact ties are uncommon, so the drift rarely matters. It matters enormously where values land on exactly half a penny by construction — which is what happens the moment you take a percentage of a price.

Where the missing penny comes from

Three identical lines at $3.30, with a 15% service charge.

Round each line, then total:

• Per line: 3.30 × 0.15 = 0.495, which rounds half-up to $0.50.
• Three lines: 0.50 × 3 = $1.50.

Total first, then round:

• Subtotal: 3.30 × 3 = $9.90.
• Charge: 9.90 × 0.15 = 1.485, which rounds half-up to $1.49.

$1.50 against $1.49. Same prices, same rate, one penny apart. Under half-even the gap widens: 0.495 still goes to 0.50 because 0.50 ends in an even digit and 0.49 does not, but 1.485 goes down to $1.48, so the two methods land two pennies apart.

Round per line and every line matches its own arithmetic. Round the total and the invoice matches the stated rate. You cannot have both.

So pick one and say so

Choose a rule, apply it everywhere, and show the reconciliation where the customer can see it. If you are matching another system — a payment processor, an accounting package, a tax filing — match its rule rather than the one you prefer.

Whether sales tax is computed per line or per invoice is set by the rules of the place you are trading in, not by arithmetic, and it does differ. That is a question for the relevant tax authority's own guidance.

The other way to be a penny out

Splitting. $10.00 three ways is 10.00 ÷ 3 = 3.3333..., so $3.33 each, and 3.33 × 3 = $9.99. A penny is left unallocated, and no rounding rule will place it. Somebody has to be handed it deliberately.

Round once, at the end

Never round a number that has already been rounded. Take 2.45: round it to one decimal and you get 2.5, then round that to a whole number and you get 3. Round 2.45 straight to a whole number and you get 2. The intermediate step invented a tie that was not there.

Carry full precision through every intermediate calculation and round once, for display, at the end.

Significant figures answer a different question

Decimal places count positions after the point. Significant figures count meaningful digits from the first non-zero one.

• 0.00123456 to two decimals is 0.00 — every digit destroyed. To three significant figures it is 0.00123.
• 12,345 to two decimals is 12,345.00 — nothing removed. To three significant figures it is 12,300.

Significant figures track how much you actually know. Decimal places track how you are displaying it.

One thing to know about calculators

Computers store decimals in binary, and most decimal fractions have no exact binary form. 2.675 is held as a value a hair below 2.675, so asking software to round it to two decimals returns 2.67, not the 2.68 you expected. The rounding is not broken; the tie you thought you had was never quite a tie.

The same thing happens to the 0.495 from the invoice above: typed into a program it is stored fractionally under half a penny, so it rounds to 0.49 rather than the 0.50 the paper rules give. Where exact decimal behaviour is required — money, always — work in whole pennies as integers and never let a fraction of a penny exist in the first place.

The free Rounding tool rounds a value to as many decimal places as you ask for, up to ten, and also shows it at three significant figures, with trailing zeros trimmed off. It is the display step. The decision about where to round, and how often, is the part above.