Upper and lower bounds of a rounded value

A news report says a concert crowd was four thousand three hundred people, correct to the nearest hundred. Nobody counted exactly four thousand three hundred. The real crowd could have been smaller, or bigger. There is a smallest crowd that would be reported that way, and a largest. Both edges come from one halving step.

This is video three of six in Rounding, Estimation and Bounds. Rounding to the nearest hundred or to one decimal place is taught in Rounding to significant figures and decimal places, if that needs a refresh first.

A rounded number stands for a whole stretch of true values. The bottom edge of that stretch is the lower bound. The top edge is the upper bound. The edges sit half a unit either side of the rounded value. The unit is the accuracy it was rounded to. The crowd was rounded to the nearest hundred, which makes the unit one hundred. Half of one hundred is fifty. Write that halving step down every time. It is half the unit, never half the number, and half of four thousand three hundred would be nowhere near. The reason is halfway. Halfway between four thousand two hundred and four thousand three hundred is four thousand two hundred and fifty, and from there upward, counts round to four thousand three hundred. The lower bound is four thousand three hundred take away fifty. That is four thousand two hundred and fifty people. Now find the upper bound of four thousand three hundred, to the nearest hundred. Four thousand three hundred and fifty. Four thousand three hundred add fifty, because halfway up to the next hundred is where counts start rounding to four thousand four hundred. A crowd comes in whole people, so the biggest real count is one less, four thousand three hundred and forty-nine. The bound is still four thousand three hundred and fifty, the edge where the rounding switches.

The same halving step works on decimals. A length is seven point two centimetres, rounded to one decimal place. One decimal place means the unit is nought point one. Half of nought point one is nought point nought five. Using nought point nought five, find both bounds of seven point two centimetres. Seven point one five and seven point two five centimetres. Seven point two take away nought point nought five gives the lower bound, and add nought point nought five gives the upper.

Truncation is a different instruction. To truncate means to cut digits off and never round. Seven point two nine truncated to one decimal place is seven point two, because the nine is thrown away unused. A machine truncates to one decimal place, and its reading is seven point two centimetres. Pair A is seven point one five and seven point two five. Pair B is seven point two and seven point three. Pair C is seven point one and seven point two. Which pair, A, B or C, gives the bounds of the true length? Pair B. Truncation only cuts digits off, so the true value never sits below what is shown. It starts there and runs up towards the next value that could be shown. Now put the two seven point twos side by side. Rounded, the bounds are seven point one five and seven point two five, half a unit either side. Truncated, they are seven point two and seven point three, a whole unit, all of it above. Take a student who is told that one point two was truncated, and who gives its bounds as one point one five and one point two five. That answer is marked wrong. Why are one point one five and one point two five wrong here? Those are rounding edges, half a unit either side. Truncation only cuts off, so the stretch runs from one point two up to one point three. One examiner's report on a Higher paper records real students doing this. Most candidates were unable to appreciate that the question focused on truncation, rather than rounding and treated the question as if they had been told one point two was the result of a rounding process. As a result, they gave the bounds as one point one five and one point two five. These bounds could not be given any credit. The habit that fixes it is to find the word rounded or truncated in the question before you find a single bound.

Three bound questions to round this off, two of them on numbers this video has not used. For a rounding bound, what exactly gets halved? The unit of accuracy, never the number itself. Nearest hundred halves to fifty. Next: sixty kilograms, to the nearest ten kilograms. What are its bounds? Fifty-five and sixty-five kilograms. Half of ten is five, taken away and added. Now a truncated one: a timer cuts to whole seconds and shows nine seconds. Bounds? Nine and ten seconds. Truncation cuts off, so the true time starts at nine and runs up towards ten. And the concert crowd reported as four thousand three hundred sits between four thousand two hundred and fifty and four thousand three hundred and fifty.

If you can now give both bounds and spot a truncation, leave a thumbs-up on this video. It marks the topic as one you're solid on, so you can skip it when you revise. If it is not quite there yet, save the video, keep it un-thumbed, then give it a day and look again with fresh eyes.

Next in the chapter comes Bounds of a calculated result, a Higher topic, where two rounded measurements go into one calculation.

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