Computing a percentage in your head by splitting it

0:00Here is a grid with four percentages down the side and seven numbers across the top. Twenty-five, ten, twenty and five per cent. And a hundred, two hundred, fifty, eighty, ten, thirty-five, and two hundred and eighty-seven. Twenty-eight cells, and exactly one of them is filled in. Twenty-five per cent of a hundred is twenty-five. The rest are yours, and you are not allowed a pen. There is a single fact underneath every cell, and once you have it the grid is quicker to do than to write down.

0:39Start with the easiest cell you were not given. Twenty-five per cent of forty. Most people take a quarter, then wonder whether they were allowed to. They were, and the reason is short. Twenty-five out of a hundred is a quarter. So twenty-five hundredths of forty and a quarter of forty are not two calculations that agree. They are one multiplication written twice. Fifty out of a hundred is a half. Ten out of a hundred is a tenth.

1:11Which means you never have to divide by a hundred if a friendlier fraction is sitting right there. Ten per cent you already know: it is moving the decimal point. Ten per cent of the seven numbers is ten, twenty, five, eight, one, three point five, and twenty-eight point seven. So what is twenty per cent of each? Double each of those. Twenty parts in every hundred is exactly twice ten parts in every hundred.

1:44Twenty, forty, ten, sixteen, two, seven, and fifty-seven point four. Forty per cent is double again. And notice what has happened: the numbers across the top never moved. Only the percentage changed, and the answers changed with it in exactly the same proportion. So write down what is going on, once. Y per cent of a value is y over a hundred, times that value. The value is sitting still, and y is out in front of it as a plain multiplier.

2:20Everything else in this video is that one line being reused. Double the y and the answer doubles, because the value over a hundred never moved. Add two y's and the answers add, for the same reason. Twenty hundredths plus five hundredths is twenty-five hundredths, and the value over a hundred is the common factor sitting outside the bracket. That is the distributive law, and it is the whole of mental percentage arithmetic.

2:54So here is the toolkit. Fifty, twenty-five, twenty, ten, five, and one. A half, a quarter, a fifth, a tenth, half a tenth, and a hundredth. Those six reach every whole percentage from one to a hundred, which is worth checking rather than believing. Fifteen per cent is ten plus five, and five is half of ten, so fifteen per cent of two hundred and eighty-seven is twenty-eight point seven plus fourteen point three five.

3:27Forty-three point zero five, in your head, from a number nobody would choose. Seventy-five is a half and a quarter. Fifty-five is a half and a five. Seventy is a half and a twenty. Ninety takes four pieces, or one subtraction: everything, less a tenth. Now the part usually left out, because it never seems to matter. Percentages add. But percentages of what? Take twenty per cent of eighty and five per cent of fifty, and add them.

4:02Sixteen plus two and a half is eighteen and a half. Twenty-five per cent of eighty is twenty. Those are not the same number. Put seven pairs through it and adding the percentages regardless gives the right total on four of them. And that is not luck. It is right on exactly the four where the two quantities matched, and wrong on the three where they did not. The shared value is the common factor. Take it away and there is nothing to factor out, and nothing left to add.

4:38One more thing falls out of that line, nearly free. What is five per cent of forty? And what is forty per cent of five? Both are two, and that is not a coincidence about these numbers. X per cent of y is x times y, over a hundred. Y per cent of x is y times x, over a hundred. Multiplication does not care which way round you write it, so the two are equal for every pair you could pick.

5:09So when a percentage is awkward and the number beside it is friendly, swap them and do the easy one. Porridge. Millet to water, two to seven. The first mistake is to call the millet two sevenths. It is not. Two and seven are both parts, so the whole is nine, and the millet is two ninths of the pot. Before working it out, bracket it. Half of nine is four and a half, and two is well under that, so the millet is under fifty per cent.

5:42Half of four and a half is two and a quarter, and two is still under that, so it is under twenty-five per cent. Ten per cent of nine is nought point nine, and twenty per cent is one point eight, and two is above that. So the answer is between twenty and twenty-five per cent, and we have not touched the division yet. It is twenty-two point two two per cent, and the water is seventy-seven point seven eight - exactly where the brackets said.

6:15Now one that looks like more of the same and is not. Five hundred millilitres of that porridge. How much millet? A hundred millilitres holds twenty-two point two two, so five hundred holds five times that, which is a hundred and eleven point one. Except it does not. The true answer is a hundred and eleven point one one. The gap opened because twenty-two point two two was already rounded, and multiplying by five multiplied the rounding too.

6:50The exact millet share is two hundred ninths of a per cent, and five hundred millilitres of it is a thousand ninths of a millilitre. So round at the end, never in the middle. An estimate is for checking, not for feeding back in. A test is out of seventy-five and you need eighty per cent. What is the lowest mark that does it? Three ways, worth seeing side by side.

7:19As a fraction: eighty over a hundred is four fifths, and four fifths of seventy-five is sixty. As a decimal: nought point eight times seventy-five is sixty. Out of a hundred you would need eighty, so out of seventy-five you need seventy-five times eighty hundredths. Sixty. Same number three times, because all three are the same multiplication in different handwriting. Pick whichever your head is fastest at. There is no correct one.

7:53Now run the whole thing backwards. A cyclist has done forty per cent of a ride, and that forty per cent is ninety-two kilometres. How much further? The tempting move is to multiply ninety-two by forty, and that gives three thousand six hundred and eighty, which is not a bicycle ride. Here is the one you can do while walking. Forty per cent is ninety-two, so twenty per cent is forty-six.

8:24Sixty per cent is forty plus twenty, so it is ninety-two plus forty-six, which is a hundred and thirty-eight kilometres. No equation, no unknown, just the splitting rule pointed the other way. You could also find the whole ride first, and it is two hundred and thirty kilometres. But then you must remember to subtract, because the question asked what is left, not how long the ride is. That last step is where the marks go.

8:57One last habit, the one that outlasts the arithmetic. Take eight pairs of amounts and rank each pair by its percentage alone, ignoring what the percentage is of. That gets five of the eight right, often enough that nobody notices. It fails on fifty per cent of five hundred and ten against fifty per cent of five hundred and fifteen: equal percentages, unequal answers. And it fails on five per cent of forty against forty per cent of five, where it picks a winner and those two are the same number.

9:36So try this one. Ten per cent of a day, or one per cent of a week? The week is seven times longer, so a hundredth of it ought to be the bigger stretch. It is not. Ten per cent of a day is two point four hours and one per cent of a week is one point six eight hours. A percentage is not a number yet. It is a number waiting to be told what it is a percentage of, and until you say, it cannot be compared with anything.