Percentages, percentage change and averages

What do fifty per cent off and a further twenty per cent off add up to? Picture a trainer shop window with that sign on it. The trainers were eighty pounds. Your friend says that is seventy per cent off, so they will cost twenty-four pounds. Another friend says they will cost more than that. Before you walk in with twenty-four pounds, who is right? The sums settle the argument, one percentage at a time.

A percentage is a number out of a hundred, so fifty per cent means fifty out of every hundred. Every calculation here is set out the same way. The formula goes first, written in words, because your paper may not print it for you. Then the numbers go in, with the pound sign or the unit on every line. Rounding waits until the last line, and one final sentence says what the answer means for the business. The first form is a percentage of an amount. Its formula in words is: percentage of an amount equals the percentage over a hundred, times the amount. Take the first part of the sign, fifty per cent off eighty pounds. Fifty over a hundred, times eighty pounds, is forty pounds off. Eighty pounds take away forty pounds leaves a sale price of forty pounds. Now the second part of the sign, a further twenty per cent off. Which price does that twenty per cent come off, the eighty pounds or the forty? It comes off forty pounds, because the sign says off the sale price, and the sale price is the price after the first discount. The first friend took both percentages from eighty pounds. Now your turn with the numbers. What is twenty per cent of forty pounds, and what do the trainers cost? Twenty over a hundred, times forty pounds, is eight pounds off. Forty pounds take away eight pounds leaves thirty-two pounds. The trainers cost thirty-two pounds, not twenty-four, so the second friend was right. The two percentages come off two different prices, which is why you cannot add them. Turning thirty-two pounds into one total discount needs the percentage change formula, so that sum waits until the formula is taught. Writing every line has a reason behind it. One examiner's report puts it like this: "Centres are encouraged to ensure that students show workings clearly for examiners to allow them to follow and award OFR." O F R stands for own figure rule. If one number slips early on, the steps you then work correctly from it can still be credited, as long as the examiner can see them. The habit is to write every line, even when a step feels obvious.

The second form asks what percentage one figure is of a total. Its formula in words is: percentage of a total equals the part over the total, times a hundred. Hilltop Holiday Lets earned a hundred and fifty thousand pounds in revenue last year, revenue being the money that comes in from sales. Ninety-six thousand pounds came from its cottages, and fifty-four thousand pounds came from its glamping pods. What percentage of Hilltop's hundred and fifty thousand pounds of revenue came from the fifty-four thousand pounds of glamping pods? Have a go at this one before I show you. Part over total gives fifty-four thousand pounds over a hundred and fifty thousand pounds, which is nought point three six. Times a hundred, that is thirty-six per cent. Thirty-six per cent of Hilltop's revenue came from the pods, which is more than a third of everything it earned. Sometimes a question gives only the parts and no total. Then the first line adds the parts together before anything is divided. For Hilltop, ninety-six thousand pounds plus fifty-four thousand pounds makes a hundred and fifty thousand pounds. Now back to the first form, in a new setting. The booking website takes three per cent commission on all of Hilltop's pod revenue, commission being a share of sales paid to whoever brought them in. You already know the formula from the trainers: the percentage over a hundred, times the amount. What commission does Hilltop pay on its fifty-four thousand pounds of pod revenue? Check the rate in your fraction against the question before you multiply. Three over a hundred, times fifty-four thousand pounds, is one thousand six hundred and twenty pounds. Write thirty over a hundred by mistake and you get sixteen thousand two hundred pounds, ten times too much. That is why the rate gets checked against the question before you multiply. The pods bring in over a third of Hilltop's revenue, so the commission the website takes is a cost to weigh against finding those guests another way. Rounding is the other place a money answer slips. One examiner's report on a commission question says, "Many students did not round correctly which led them to miscalculate the daily commission; the decimals were important for accurate workings and the final answer due to the question relating to currency." In plain words, rounding too early threw the money answer out. The fix is to keep every decimal until the last line, then round once, and for money that means to the nearest penny.

The third form measures how much a figure has changed, as a percentage of where it started. Its formula in words is: percentage change equals the new figure take away the original figure, over the original figure, times a hundred. The original figure is the earlier one, where you start from. You always divide by the original, because a change only means something compared with where the business began. Ashgrove Florist's total costs were thirty-two thousand pounds in twenty twenty-three, and thirty-six thousand eight hundred pounds in twenty twenty-four. What is the percentage increase in Ashgrove's total costs, from thirty-two thousand pounds to thirty-six thousand eight hundred pounds? Pause it there and work it out for yourself. Step one is the change. Thirty-six thousand eight hundred pounds take away thirty-two thousand pounds is four thousand eight hundred pounds. Step two divides by the original figure, the costs in twenty twenty-three. Four thousand eight hundred pounds over thirty-two thousand pounds is nought point one five. Step three multiplies by a hundred, which gives fifteen per cent. Ashgrove's total costs rose by fifteen per cent. Unless Ashgrove's prices or sales rose by at least as much, its profit will have been squeezed. Divide by the new figure instead, four thousand eight hundred over thirty-six thousand eight hundred, and you get thirteen point nought four per cent. That answer is wrong, because it measures the change against where Ashgrove ended up. One examiner's report on a question like Ashgrove's says, "However, there are still a significant minority that get confused and divide the change in the figures with the destination figure rather than the original number." The destination figure is the new one. The fix is to write the word original under the line of the fraction before any numbers go in. That same report also says, "This type of question has appeared on previous papers and, as a result, most candidates are well-versed in how to answer such questions." Both lines hold at once: most students know this question, and the slip that remains is the division. Divide by the original every time, and you are clear of the slip that minority still makes. Now a different case, where the figure is itself a percentage. A shop's market share, its slice of all the sales in its market, falls from twenty per cent to fifteen per cent. Is that fall from twenty per cent to fifteen per cent a five per cent fall? Decide, and have your reason ready. No. It is a fall of five percentage points, because percentage points are the plain gap between two percentages, and twenty take away fifteen is five. As a percentage change, it is five over the original twenty, times a hundred, which is a twenty-five per cent fall. The shop has lost a quarter of its share. Now back to the shop window. The trainers fell from eighty pounds to thirty-two pounds, so the first step, the change, is already done: forty-eight pounds off. As a percentage of the original eighty pounds, how much is forty-eight pounds off? Forty-eight over eighty is nought point six, and times a hundred that is sixty per cent off. The sign's two percentages never added up to seventy.

An average in a business exam means the mean. The mean is the total of the values divided by how many values there are. Its formula in words is: mean equals the total of the values over the number of values. Velo Bay hires out electric bikes at the seaside. Its hires were two hundred and ten in April, three hundred and forty in May, and three hundred and ninety-five in June. Then they were five hundred and sixty in July, six hundred and ten in August, and three hundred in September. What is Velo Bay's mean number of monthly hires for June to August? Pick the months first, then work it through. Step one picks only the three months the question asks for, and leaves the other three out. Step two is the total. Three hundred and ninety-five, plus five hundred and sixty, plus six hundred and ten, is one thousand five hundred and sixty-five hires. Step three divides by the number of months, which is three. One thousand five hundred and sixty-five over three is five hundred and twenty-one point six six six, with the sixes running on. Step four rounds once, at the end, to a whole hire. That gives five hundred and twenty-two hires a month. In its peak months, Velo Bay needs enough bikes for around five hundred and twenty-two hires a month, far more than in spring. Average all six months instead, and you get two thousand four hundred and fifteen over six, which is four hundred and two point five. That answers a question nobody asked. Take August's six hundred and ten minus June's three hundred and ninety-five instead, and you get two hundred and fifteen. That is a difference, not an average. One examiner's report on an average set over stated years says half the students scored nothing on it, and that most of those "did not use the correct time frame". Another common mistake, the same report says, was to calculate the difference between two years rather than the average. The fix is to circle the period in the question before you touch the numbers, then pick only those values.

Back through it now, and you answer each one first.

In a percentage change, which figure always goes under the line? The original figure goes under the line, because the change is measured against where you started.

Now a new case. A cafe sold four hundred coffees last week and five hundred this week. What is the percentage increase? The change is a hundred coffees. A hundred over the original four hundred, times a hundred, is a twenty-five per cent increase.

Now one with percentages already in it. A market share rises from ten per cent to twelve per cent. How many percentage points is that, and what is the percentage increase? It is two percentage points. As a percentage change, two over the original ten, times a hundred, is a twenty per cent increase.

Now the last of them. Before you add anything up for a mean, what do you do first? Circle the period the question gives, and pick only the values inside it.

And the shop window. The trainers cost thirty-two pounds, which is sixty per cent off, because the second twenty per cent came off the forty-pound sale price.

If you can say this one's done, give it a thumbs-up and move on. If it has not settled, save it, and say divide by the original out loud once, because a rule said out loud tends to stick.

Next in the chapter is Qualitative and quantitative data: reading graphs, charts and tables. Hopefully that's next in your feed, but if not, search for ScholaFly B U oh two, oh three.

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