An estimate you can produce in one second is the cheapest error check in arithmetic, and it catches the only mistakes that really matter: the ones out by a factor of ten. Getting an answer slightly wrong is annoying. Getting it wrong by a decimal place is the mistake that puts a decimal point in the wrong place on an invoice. Estimating first is what stops that, and it takes less time than checking the calculation.
Look only at the digit immediately to the right of the place you are rounding to. If it is 5 or more, round up. If it is 4 or less, leave the digit as it is. Everything further right is irrelevant.
That last part matters. Rounding 3.4499 to one decimal place gives 3.4, not 3.5 — the 9s further along do not cascade back. Rounding in stages, first to two places and then to one, is how people talk themselves into the wrong answer.
Watch the carry. Rounding 3,961 to the nearest hundred gives 4,000, not 3,900, because rounding the 9 up moves into the thousands column. Any digit that rounds up through a 9 carries just as it would in addition.
Drill it on Rounding Race, which shows which digit decided it whenever you miss one.
Round both numbers to one significant figure, do the easy sum, and you have a size to compare against.
47 × 23 is roughly 50 × 20 = 1,000, so an answer near 1,081 is right and 108 is not.
618 ÷ 29 is roughly 600 ÷ 30 = 20, so an answer of 21.3 is believable.
£8.99 × 7 is roughly £9 × 7 = £63, so anything near £63 is fine and £6.30 is not.
19% of 240 is roughly 20% of 240 = 48, so the real answer should be a little under.
None of those takes more than a second, and each one rules out the class of error that actually costs something.
Estimate first, calculate second, compare. If the two disagree by more than a little, one of them is wrong — and it is worth knowing which before you act on it.
An estimate is more useful when you know its direction. If you rounded both numbers up, the estimate is too big. If you rounded both down, it is too small. If you rounded one each way, the errors partly cancel and the estimate is close.
47 × 23 estimated as 50 × 20: one number went up by 3, the other down by 3. Those pull in opposite directions, and 1,000 against a true 1,081 is a 7% miss — good enough.
618 ÷ 29 estimated as 600 ÷ 30: the top went down and the bottom went up, so both push the estimate down. The true answer must be more than 20, and it is 21.3.
This is what makes estimating a skill rather than a shrug. Knowing the direction turns 'about twenty' into 'a bit over twenty'.
Rounding to two decimal places fixes how far right you stop. Rounding to two significant figures fixes how many digits of information you keep, counting from the first non-zero one.
0.004829 to two decimal places is 0.00 — which throws the number away entirely. To two significant figures it is 0.0048, which keeps what matters. For small numbers, significant figures is almost always the right choice.
For money, decimal places usually wins, because the unit is fixed: two decimal places is pence, and that is what the answer has to be expressed in whatever its size.
Rounding too early
Round at the end, not in the middle. Rounding a figure and then multiplying it by a hundred multiplies the rounding error by a hundred too. Carry the full precision through the arithmetic and round only the answer.
Rounded numbers that no longer add up
Three figures rounded to whole numbers may not sum to the rounded total — 1.4, 1.4 and 1.4 each round to 1, and the three 1s add to 3 while the real total, 4.2, rounds to 4. This is normal, it is not an error, and it is why financial tables sometimes carry a note about it.
Assuming 5 always rounds up
In the convention used here and in schools, it does. Some statistical and financial work rounds half to even instead — 2.5 to 2, 3.5 to 4 — to stop a systematic upward drift across many roundings. It is worth knowing that the other convention exists if you ever compare totals with someone and find a penny missing.
Look at the digit immediately to the right of the place you are rounding to. If it is 5 or more, round up; if it is 4 or less, leave the digit alone. Digits further right make no difference.
Round each number to one significant figure and do the easy calculation. 47 × 23 becomes 50 × 20 = 1,000, which is close enough to check a real answer of 1,081.
Decimal places count from the decimal point; significant figures count from the first non-zero digit. For small numbers like 0.004829, significant figures keep the information and decimal places throw it away.
At the end. Rounding partway through and then multiplying scales the rounding error up with everything else.
Because each figure is rounded independently. Three values of 1.4 each round to 1, totalling 3, while the true total of 4.2 rounds to 4. It is expected, not a mistake.
In the usual school convention, yes. Some statistical work rounds half to even instead, to avoid a systematic upward bias when many numbers are rounded together.
One second of estimating catches the errors worth catching. Get the rounding automatic on [Rounding Race](/games/rounding-race), then use it on every calculation you do — the estimate before the answer, every time.