Rounding decimals

7.2649 rounded to the nearest hundredth is 7.26, because the digit after the hundredths place is 4.

Split 10,000 won three ways and one won goes missing

A made-up puzzle first. Three friends share a 10,000-won taxi fare. The calculator says 3,333.333... won each. Nobody pays a third of a won, so each share is rounded to 3,333 won. Add the three shares and you get 9,999 won. Where did the last won go? Hold on to your answer. It comes back in the part about percents that do not add to 100.

Only the digit after the kept place counts

A calculator says a share of a bill is 17.3333333 dollars, but nobody pays a third of a cent. Rounding keeps the digits that matter and drops the rest. The OpenStax rule has four steps: find the place you are rounding to, look at the digit immediately to its right, add 1 to the place if that digit is 5 or more and leave it alone if it is less than 5, then remove every digit to the right. So 17.3333 to the nearest cent is 17.33.

Take 7.2649. To the nearest hundredth, the hundredths digit is 6 and the next digit is 4, so the 6 stays: 7.26. To the nearest tenth, the tenths digit is 2 and the next digit is 6, so the 2 becomes 3: 7.3. To the nearest whole number, look at the 2 in the tenths place: 7. Only the one digit right after the chosen place counts; the 9 at the end does not ripple forward.

Carrying can roll over several places. 3.996 to the nearest hundredth: the 9 in the hundredths place gets 1 added, which makes it 10, so it becomes 0 and carries into the tenths, which also becomes 10 and carries into the ones. The answer is 4.00. Keep the two zeros if you were asked for hundredths; they show how precise the number is.

Round at the end, not in the middle

Rounding too early lets small errors pile up. Three items cost 2.346 dollars each at a wholesale price. If you round each to 2.35 first, three cost 7.05. If you multiply first, 3 × 2.346 = 7.038, which rounds to 7.04. A cent is small, but across a thousand items the same habit adds 4 dollars: 1,000 × 2.35 = 2,350 against 1,000 × 2.346 = 2,346.

Rounded percents may not add to 100

Split something three equal ways and each part is 1/3, about 33.3%. Three of those add to 99.9%, not 100%. Charts and surveys often show totals of 99% or 101% for this reason alone. Nothing is missing; each rounded number lost or gained a little. The taxi fare's lost won is the same thing. Each share dropped a third of a won, three times.

2.5 in Python, 1.005 in JavaScript

The rule above sends an exact half upward: 2.5 rounds to 3. Not every tool agrees. In the Python programming language, round(2.5) gives 2, because Python rounds halves to the nearest even digit, a method that keeps long sums from drifting upward. JavaScript has a different surprise: (1.005).toFixed(2) gives 1.00, because the computer stores 1.005 as a binary number a hair smaller than 1.005. For homework and shop prices, use the 5-or-more rule.