Estimating a calculation by rounding

A field measures thirty-eight point seven metres by twenty-one point four metres. Someone keys its area into a calculator with a decimal point in the wrong place, and the display reads eight thousand two hundred and eighty-one point eight. You can catch that slip in your head in a few seconds, with no calculator at all. The method that does it is the one a question means when it says estimate.

This is video two of six in Rounding, Estimation and Bounds. If rounding to one significant figure feels shaky, Rounding to significant figures and decimal places is the video that sets it up.

To estimate means one method, in two steps. First, round every number. Then calculate with the rounded numbers. Unless the question names another accuracy, each number is rounded to one significant figure. An estimate is not exact, so it never gets an equals sign. It gets the approximately-equal sign, two wavy lines, which is read as is approximately equal to. Back to the field. Thirty-eight point seven, to one significant figure: the first significant figure is the three, and the next digit is an eight. Eight rounds the three up, which gives forty. Now round twenty-one point four to one significant figure. Twenty. The first significant figure is the two, and the one behind it leaves the two alone. Next comes the calculation, with the rounded numbers only. Forty times twenty. Four times two is eight, and the two zeros go back on the end, which makes eight hundred. Thirty-eight point seven times twenty-one point four is approximately equal to eight hundred. The field is roughly eight hundred square metres. Now, about how many times too big is eight thousand two hundred and eighty-one point eight? About ten times. Eight thousand is ten times eight hundred, so the decimal point has slipped one place. The true area is eight hundred and twenty-eight point one eight square metres. One examiner's report on a Foundation paper records what happens when the rounding step is skipped. Not using any estimation meant they were unable to achieve more than two marks. Those students worked with the exact numbers, and a close answer still did not count as an estimate. The fix is to write the rounded numbers down first, where the examiner can see them.

Division follows the same two steps. Take six hundred and twelve point three divided by nineteen point eight, to be estimated. Three students start it differently. Line A is six hundred divided by twenty. Line B is six hundred and twelve divided by twenty. Line C is six hundred divided by twenty-five. Which line, A, B or C, is the right first line of the estimate? Line A. Both of its numbers are genuine roundings to one significant figure. Line B stops short of one figure, and line C swaps in a number nobody rounded to. Twenty-five might feel friendlier to divide by, but it is not nineteen point eight rounded. A number of your own choosing can pull the estimate anywhere. A rounded number keeps the first figure, and that keeps the estimate close. Line A, then. Six hundred divided by twenty. Divide both numbers by ten first, which gives sixty divided by two. Sixty divided by two is thirty. Six hundred and twelve point three divided by nineteen point eight is approximately equal to thirty.

A square root gets estimated a different way. Rounding forty-eight to fifty does not help, because fifty has no neat square root. Instead, you look for the nearest square number you know. Six squared is thirty-six, and seven squared is forty-nine. Forty-eight sits between them, and it is much closer to forty-nine. That makes the square root of forty-eight approximately equal to seven, a little under it, because forty-eight is a little under forty-nine. Here's a different root: estimate the square root of eighty-three. About nine. Nine squared is eighty-one, the nearest square number, so the square root of eighty-three is approximately nine, a little over it.

Before any rounding, read the accuracy the question names. One significant figure is the default, used only when the question names nothing else. Some questions name their own accuracy, and then that accuracy replaces one significant figure. Estimate twenty-three point six times four point one, each to the nearest whole number. Ninety-six. Twenty-three point six rounds to twenty-four, and four point one rounds to four. Twenty-four times four is ninety-six. Rounding to one significant figure instead gives twenty times four, which is eighty. Eighty is a fair estimate, but not the one this question asked for. One examiner's report on a Higher paper says this plainly. Students should refrain from rounding numbers to one significant figure, simply because they see estimate. A second report puts the habit that fixes it in one line. Candidates need to carefully read the instructions given within questions.

Before this video closes, the method once more, then two estimates to do in your head. What are the two steps of an estimate, in the right order? Round every number first. Then calculate, and write the approximately-equal sign. Try this: estimate seventy-nine point two divided by four point one. It comes to twenty. Seventy-nine point two rounds to eighty, four point one rounds to four, and eighty divided by four is twenty. Now a root: estimate the square root of sixty-two. About eight. Eight squared is sixty-four, the nearest square number, and sixty-two sits a little below it. And the field from the start: forty times twenty says roughly eight hundred square metres, which is how the slipped decimal point gets caught.

When rounding first and calculating second feels automatic, press the thumbs-up on this one. In a list of thumbed videos, it shows that topic is in the bag. If it still feels wobbly, keep it saved for later with no thumb, and it stays visible as one to revisit. Estimating comes with the questions, so a few more and it settles.

Next in this chapter: Upper and lower bounds of a rounded value, which asks what a rounded number could have been before it was rounded.

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