Decimals and standard form

A school is ordering a hundred and eighty drawstring PE bags, and two people price the cord and the cord stoppers on the same calculator. One gets one hundred and five pounds eighty-four. The other gets ten thousand five hundred and eighty-four. Only one of those numbers can go on the purchase order, and the calculator was right both times.

This video belongs to Maths: number, units and data, a set you can take in any order. If converting units feels shaky, the video on units, symbols and metric prefixes covers it.

Decimals carry money, sizes and times, and where a digit sits sets its value. The first place after the point is tenths, the next is hundredths, and the next is thousandths. Is fourteen pence one pound forty, nought point one four, or nought point nought one four? Nought point one four pounds. There are a hundred pence in a pound, so you divide fourteen by a hundred. That is the move for every money sum: pence become pounds when you divide by a hundred. Twenty-three pence is nought point two three pounds, and eleven pence is nought point one one pounds. Now the order. Each bag uses one point six metres of cord at twenty-three pence a metre, and two cord stoppers at eleven pence each. The question wants the total for all the bags, in pounds. Price everything in pounds from the start. Cord for one bag is one point six times nought point two three pounds, which is nought point three six eight pounds. Next, the stoppers. Two times nought point one one pounds is nought point two two pounds. Add them for one bag: nought point three six eight plus nought point two two is nought point five eight eight pounds. That stays unrounded. Roughly how many pounds should a hundred and eighty bags cost? About a hundred. Nearly sixty pence each, times a hundred and eighty, is a little over a hundred pounds. Now the exact sum. A hundred and eighty times nought point five eight eight pounds is one hundred and five pounds eighty-four, close to the estimate. The other number came from pricing in pence. Fifty-eight point eight pence a bag, times a hundred and eighty, is ten thousand five hundred and eighty-four. That is pence, and divided by a hundred it is the same one hundred and five pounds eighty-four. So turn pence into pounds before you multiply, and say a rough check out loud. A check of about a hundred pounds catches ten thousand straight away.

Next, rounding. A question tells you how precise its answer must be, and the last line of your working rounds to that, and only that. A laser cutter follows an outline at eighteen millimetres per second, and one coaster's outline is three hundred and seventy-seven millimetres long. The question wants the time for one coaster to the nearest whole second. Time is distance divided by speed. Three hundred and seventy-seven divided by eighteen is twenty point nine four four, and the digits carry on. What is twenty point nine four four seconds to the nearest whole second? Twenty-one seconds. The first digit after the point is a nine, so twenty rounds up to twenty-one. Now a batch of a hundred and fifty coasters. This time the question wants minutes, to one decimal place. For the batch, do you multiply by a hundred and fifty, or divide by it? Multiply. A hundred and fifty coasters take longer than one, so the number has to grow. One examiner's report, on a stitching question that scaled a time up by thirty, says: A single method mark was available for recognising that their answer from question twenty-four point one had to be multiplied and not divided by thirty. The habit is one question to yourself: does the batch take longer than one item? If it does, the answer must be bigger, so multiply. There are two routes to the batch time. One keeps the full value: twenty point nine four four and on, times a hundred and fifty, is three thousand one hundred and forty-one point six seven seconds. Divided by sixty, that is fifty-two point three six minutes, so fifty-two point four. The other route rounds first: twenty-one times a hundred and fifty is three thousand one hundred and fifty seconds, which is fifty-two point five minutes. Fifty-two point four or fifty-two point five: which route is right, and why? Fifty-two point four. The other route rounded the single coaster to twenty-one seconds, and that small error was then multiplied a hundred and fifty times. So keep the full value in the calculator through every step, and round once, at the end. Write one line per step with its unit, so method marks can still be given even if the last line slips. The same report adds this about finishing the sum: Where students did work through to an answer, many were rounded to the nearest second and gained maximum marks.

Money gets its own rounding rule. Money has two decimal places, the pennies, so the nearest penny means two decimal places of a pound. A supplier's total comes to forty-three point two five seven five pounds. One student writes forty-three pounds twenty-six. Another writes forty-three pounds thirty. Which of the two is rounded to the nearest penny? Forty-three pounds twenty-six. Forty-three pounds thirty is rounded to the nearest ten pence, one decimal place too few. On a question where a money answer had to be rounded, one report records: A common misunderstanding with this question was rounding up. Several students did not round up to the nearest one penny, but the nearest ten pence arriving incorrectly at an answer of five hundred and ninety pounds eighty. The way to avoid it is to underline the rounding instruction in the question before you start, then end on a round line that names it: to the nearest penny, two decimal places.

Now very big and very small numbers. Standard form writes a number as a value from one up to, but not including, ten, multiplied by a power of ten. A factory moulds two hundred and forty thousand bottle caps a week. Two students write that in standard form. One writes twenty-four times ten to the four. The other writes two point four times ten to the five. Which of the two is written in standard form, and why? Two point four times ten to the five. Twenty-four is bigger than ten, so the other one is the right number in the wrong form. To get there, move the point until one digit sits in front of it. From two hundred and forty thousand to two point four, the point moves five places, so the power is five. Each cap has a mass of two point five grams. On the calculator, type two point four, press the times ten to the x key, type five, then times two point five, and press equals. The display reads six hundred thousand, so the caps have a mass of six times ten to the five grams. As an ordinary number in kilograms, divide by a thousand: six hundred thousand grams is six hundred kilograms of plastic a week. Next, a very small size. One layer of graphene is about nought point nought nought nought nought nought nought three four millimetres thick. To reach three point four, the point moves seven places to the right. Is the power of ten for that graphene layer positive or negative? Negative, because the number is smaller than one. So the graphene layer is three point four times ten to the minus seven millimetres thick. How many layers stack up to one millimetre? Type one, divide, three point four, times ten to the x, minus seven, equals. If the display shows a fraction, press the S to D key. It reads two million, nine hundred and forty-one thousand, one hundred and seventy-six point four seven one. In standard form to one decimal place, that is two point nine times ten to the six. Almost three million layers make one millimetre.

That covers money, rounding and standard form. Four short questions now, and the numbers in them are ones you have not met in this video. A clip costs forty-five pence. What do two hundred clips cost, in pounds? Ninety pounds. Forty-five pence is nought point four five pounds, and two hundred of them is ninety pounds. Next, what is twelve point four three seven five pounds to the nearest penny? Twelve pounds forty-four. Two decimal places, and the seven after the three rounds it up. Now write nought point nought nought five in standard form. Five times ten to the minus three. It is smaller than one, and the point moves three places. One more: in a sum with several steps, when do you round? Once, at the end, to the precision the question states. And the PE bags: the purchase order says one hundred and five pounds eighty-four, because the pence were turned into pounds before anything was multiplied.

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Coming up next in this chapter: Ratio and scale drawings.

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