Fractions and percentages: profit, waste and survey results

A forty-pound hoodie is thirty per cent off, and at the till another twenty per cent comes off. Thirty plus twenty is fifty, so it should cost twenty pounds. The receipt says twenty-two pounds forty, and the till is not broken. The receipt comes down to one idea, and it starts with a vote on a pencil case.

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A percentage is a number of hundredths. Fifteen per cent means fifteen in every hundred, which as a decimal is nought point one five. It is always taken of some amount, so name that amount before you calculate. Fifteen per cent of sixty pounds is sixty times nought point one five, which is nine pounds. A fraction of an amount works the same way. A twelfth of something means sharing it into twelve, so you divide by twelve, and the answer must be smaller than the amount. Multiplying by twelve would make it bigger, so it cannot be a twelfth. Now a survey. Eighty Year Ten students voted for the material of a new pencil case. Recycled PET felt, a felt made from old plastic bottles, got twenty-eight votes. Cotton canvas got twenty, and PU leather-look fabric, a polyurethane coating on cloth, got twelve. Cork fabric's votes were never written down, only its share of the vote. Cork took twenty-five per cent of the eighty votes. How many votes is that? Twenty. Nought point two five times eighty is twenty. Check the total: twenty-eight plus twenty plus twelve plus twenty is eighty, so everyone is counted. Now the other way: one amount as a percentage of another. Divide the part by the whole, then multiply by a hundred. For PET felt that is twenty-eight over eighty, times a hundred, which is thirty-five per cent. Twelve votes out of eighty for PU fabric: what percentage is that? Fifteen per cent. Twelve eightieths is nought point one five, which is fifteen hundredths. Cotton canvas, twenty out of eighty, is twenty-five per cent, the same as cork. The four shares add up to a hundred per cent, and that is the check. As a fraction, cotton canvas is twenty eightieths. Divide top and bottom by twenty, and it is one quarter. For the designer, felt leads, but only thirty-five per cent chose it. It is the most popular choice, not the choice of most students, so a second material is still worth testing.

Percentage change is how a fall in output or a profit is measured. Every change is measured against the original, the amount you started with. Percentage change equals the change, divided by the original, times a hundred. The change is the difference between the two amounts. The original is the starting amount. A workshop laser-cuts three thousand six hundred acrylic keyrings in a normal week. In a week when the machine is serviced, it makes two thousand three hundred and fifty. The change is three thousand six hundred minus two thousand three hundred and fifty, which is one thousand two hundred and fifty. Which goes on the bottom, the normal week or the service week, and why? The normal week. It is the original, the amount you started from, so the change is measured against it. One thousand two hundred and fifty divided by three thousand six hundred, times a hundred, is thirty-four point seven two per cent. The question asks for a whole per cent, so the last line rounds. Thirty-four point seven two rounds up to thirty-five, and output fell by thirty-five per cent. Two other answers look believable. Dividing by the service week gives fifty-three per cent, measured from the wrong start. And sixty-five per cent is the output that was kept, not the fall. One examiner's report, on a question about production rates, notes: Students in most cases were well prepared for this question. In many instances, there was clear evidence of working out, sometimes with an incorrect final answer given. This allowed method marks for correct working to be awarded. The same report adds: In limited number of scripts, students did not correctly round up to eighty per cent meaning the final mark could not be awarded. Both are true of one question: the method was usually there, and the mark that got away was the rounding. So the fix is a round line of its own, naming the precision the question asks for, here a whole per cent. The same formula gives a profit. Each keyring costs forty-two pence to make and sells for one pound five. The profit is one pound five take away forty-two pence, which is sixty-three pence. As a percentage of the cost, what is that sixty-three pence profit? One hundred and fifty per cent. The cost is the original, so it is sixty-three over forty-two, times a hundred. Percentages can go over a hundred. A five hundred per cent mark-up means the price is the cost plus five times the cost, so you multiply the cost by six. It is not a fifty per cent rise, and nothing is taken away. A percentage change can be done in one step with a multiplier, one number that applies the whole change. Ten per cent off leaves ninety per cent, so you multiply by nought point nine. A thirty per cent increase makes a hundred and thirty per cent, so you multiply by one point three. Read what the question asks for. A batch of stools costs two hundred pounds, the price is thirty per cent more, and the question wants the extra amount, not the new price. For only the extra, multiply by one point three or by nought point three? Nought point three. One point three gives the whole new total, not the part that was added. Two hundred times nought point three is sixty pounds of extra. Two hundred times one point three is two hundred and sixty pounds, the new price. A report on a question about a thirty per cent increase records: Many responses were seen whereby candidates had simply multiplied the one hundred and fifty thousand pounds by one point three to get an answer of one hundred and ninety-five thousand pounds. Had they then taken away the original sum of money they would have had a correct answer of forty-five thousand pounds for the full two marks. The habit is to underline whether the question wants the increase or the new total, before you choose the multiplier. Now a discount. The workshop's acrylic supplier takes twelve and a half per cent off a large order. Eight hundred and twenty-nine pounds fifty, twelve and a half per cent off. Price paid? Seven hundred and twenty-five pounds eighty-one. Twelve and a half off leaves eighty-seven and a half per cent, so the multiplier is nought point eight seven five. Eight hundred and twenty-nine pounds fifty times nought point eight seven five is seven hundred and twenty-five point eight one two five pounds. To the nearest penny, that is seven hundred and twenty-five pounds eighty-one. Now the hoodie. Thirty per cent off forty pounds leaves seventy per cent, so forty times nought point seven is twenty-eight pounds. The next twenty per cent comes off twenty-eight pounds. What is the price now? Twenty-two pounds forty, the price on the receipt. Twenty-eight times nought point eight is twenty-two point four. The second twenty per cent was of twenty-eight pounds, not of forty, and that is why the two discounts never add up to fifty.

Last, waste. A school laser-cuts earrings from an acrylic sheet of a hundred and eighty thousand square millimetres. The earrings have a total area of a hundred and fifty thousand three hundred and fifty square millimetres. The question asks what percentage of the sheet is wasted, to two decimal places. One student divides the earrings by the sheet, multiplies by a hundred, and writes eighty-three point five three per cent. Eighty-three point five three per cent wasted: what has gone wrong? That is the percentage used. The earrings are the part that was cut out and kept, not the leftover. So find the waste first. The sheet take away the earrings is a hundred and eighty thousand minus a hundred and fifty thousand three hundred and fifty, which is twenty-nine thousand six hundred and fifty square millimetres. Twenty-nine thousand six hundred and fifty divided by a hundred and eighty thousand, times a hundred, is sixteen point four seven two two. To two decimal places, sixteen point four seven per cent is wasted. Check: sixteen point four seven plus eighty-three point five three is a hundred. On a timber question, one report says: The most commonly observed incorrect response was a calculation to show how much timber had been used, that is, nine tenths. The way to stop it is to label used and wasted in words before calculating, and to check the two percentages add to a hundred. Better nesting, which means fitting the parts closer together on the sheet, cuts the waste to nine point five per cent. How many square millimetres does better nesting save on each sheet? Twelve thousand five hundred and fifty square millimetres. The new waste is nought point nought nine five times a hundred and eighty thousand, which is seventeen thousand one hundred. Twenty-nine thousand six hundred and fifty take away seventeen thousand one hundred leaves twelve thousand five hundred and fifty.

Before you go, four questions on numbers you have not seen here, each one testing what the percentage is of. Thirty-six of a hundred and twenty students chose bamboo. What percentage is that? Thirty per cent. Thirty-six over a hundred and twenty is nought point three. Now output rises from four hundred a day to five hundred. What is the percentage increase? Twenty-five per cent. The change is a hundred, divided by the original four hundred. Here's a discount: fifteen per cent off a two hundred pound order. What multiplier, and what price? Nought point eight five, and a price of a hundred and seventy pounds. A different kind of question: in percentage change, which amount goes on the bottom? The original, the amount you started with. And the hoodie: thirty per cent of forty pounds, then twenty per cent of twenty-eight. Each percentage was of its own amount, which is why the receipt says twenty-two pounds forty.

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Up next in this chapter: Charts: frequency tables, bar charts, pie charts and histograms.

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