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 three hundred and twelve plus four hundred and eighty-nine, round to three hundred plus five hundred, which is eight hundred, close to the exact answer of eight hundred and one. 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 thirty-nine times twenty-one, round to forty times twenty, which is eight hundred, near the exact answer. To estimate two hundred and three divided by four, round to two hundred divided by four, which is fifty. 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 four hundred and twelve plus three hundred and eighty-nine is given as one thousand eight hundred and one, an estimate of eight hundred shows the answer is far too large, so a mistake has been made. If forty-eight times twenty-one is given as about one thousand, that matches an estimate of fifty times twenty, 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 nineteen dollars ninety-five each, rounding to twenty dollars gives about sixty dollars, so a total near sixty is reasonable and a total of two hundred is not. To estimate change from fifty dollars for a bill of thirty-eight dollars forty, round the bill to forty dollars and subtract: about ten to twelve dollars 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.