Rounding and estimation

Not every calculation has to be exact. When you are checking whether the money will stretch, how many packs of flooring to buy, or whether a figure on an invoice makes any sense at all, an approximate answer is enough — and it is often better, because you can work it out in your head. Two skills do that job: rounding a single number and estimating the result of a whole calculation.

Rounding to a place

Rounding a number to tens means replacing it with the nearest multiple of ten. The number line shows it best.

Instead of measuring that distance every time, it is enough to look at the deciding digit — the first digit to the right of the place you are rounding to:

• the deciding digit is $0, 1, 2, 3$ or $4$ — round down (leave it),
• the deciding digit is $5, 6, 7, 8$ or $9$ — round up (add one).

Every digit to the right of the rounded place becomes a zero. The result is written with ≈, read as approximately equal to: $63 \approx 60$.

The same number $2847$ gives three different approximations, because each time it is rounded to a different place: to tens, to hundreds and to thousands.

Watch out for one trap: only one digit decides. The number $2849$ rounded to hundreds is $2800$, even though it ends in a nine — because the direction is set by the tens digit, which is $4$. Rounding it in stages, first to tens and then to hundreds, would give the wrong $2900$.

When the deciding digit is 5

The number $25$ sits exactly halfway between $20$ and $30$ — it is no nearer to either. The distance rule settles nothing here, so an agreement is needed. School uses the simplest one: a 5 rounds up.

It is worth knowing that this is a convention, not a theorem. Always rounding a 5 upwards nudges results high over the long run, which is why statistics and accountancy sometimes agree on something else.

Estimating a result

Estimation is rounding put to work: simplify the numbers first, then do the arithmetic in your head.

The exact answers are $703$ and $798$ — the estimates are within a few per cent of them, and we got there without paper. That is enough to answer the questions that come up most often in practice: will it be enough and does this figure make sense.

Estimation as a check on exact arithmetic

The most valuable use of estimation is checking a result worked out exactly — on a calculator, in columns or in a spreadsheet. An estimate will not tell you whether an answer is precise, but it catches an error of an order of magnitude instantly: a lost decimal point, an extra zero, a misplaced separator.

If the estimate says $800$ and the calculator says $79.8$ or $7980$, the problem is not that the estimate was rough — the problem is a mistake in the calculation.

Just remember that rounding both numbers the same way shifts the result systematically: since $40 > 38$ and $20 < 21$, some of the error cancelled out here. Round both factors up and your estimate is certainly on the high side — which is a virtue when you are working out how much material to order.

Common mistakes

• Rounding in stages — the direction is set by one deciding digit, not by a chain of successive roundings: $2849 \approx 2800$, not $2900$.
• An equals sign instead of ≈ — a rounded or estimated value is an approximation, so it takes the $\approx$ sign.
• Confusing the places — to hundreds means zeros in the tens and units places, not keeping three digits.
• Estimating where precision is required — an estimate will not do for a payment, a tax return or a dimension fed to a cutting machine; it is there for checking and for deciding, not for the final figure.