Reasonableness and Estimation
Estimating by rounding
Estimating starts with rounding: replacing numbers with nearby easy ones, usually to the nearest ten, hundred or whole dollar. To estimate 312 + 489, round to 300 + 500, which is 800, close to the exact answer of 801. Rounding both numbers the same way keeps the estimate close, and the easier the rounded numbers, the quicker the estimate. An estimate is not the exact answer but a sensible guide to about how big it should be.
Estimating products and quotients
The same rounding works for multiplication and division. To estimate 39 × 21, round to 40 × 20, which is 800, near the exact answer. To estimate 203 ÷ 4, round to 200 ÷ 4, which is 50. Choosing numbers that are easy to multiply or divide, sometimes called compatible numbers, makes mental estimation possible even for large calculations. The estimate gives a ballpark figure to expect before, or instead of, working out the exact result.
Is the answer reasonable?
An estimate is most useful for checking whether an answer is reasonable. After working out, or being told, a result, a quick estimate shows whether it is in the right range. If 412 + 389 is given as 1801, an estimate of 800 shows the answer is far too large, so a mistake has been made. If 48 × 21 is given as about 1000, that matches an estimate of 50 × 20, so it is reasonable. Checking against an estimate catches slips like a misplaced digit or a wrong operation.
Estimating with money
Estimation is especially handy with money. Rounding each price to the nearest dollar gives a quick total: four dollars ninety-five, three dollars ten and six dollars twenty round to five, three and six dollars, about fourteen dollars altogether. This is enough to know whether there is enough to pay, without adding the cents exactly. Because prices are often just under a whole dollar, rounding up slightly tends to give an estimate a little above the true total, which is a safe way to check there is enough money.
Checking a money answer
The same idea checks money answers and change. For three items at $19.95 each, rounding to $20 gives about $60, so a total near $60 is reasonable and a total of $200 is not. To estimate change from $50 for a bill of $38.40, round the bill to $40 and subtract: about $10 to $12 change. Estimating first means an exact answer, or a till receipt, can be trusted or questioned at a glance.
Checking with confidence
Checking reasonableness with confidence means estimating on purpose: round to easy or compatible numbers, work out a quick estimate, and compare it with the exact answer or the situation. In money problems, rounding to whole dollars gives a fast, safe check on totals and change. An estimate will not match exactly, and it is not meant to, but an exact answer far from a sensible estimate is a signal to look again. This habit of checking turns careful estimation into a reliable guard against mistakes in every kind of problem.