The FDP trio: fraction, decimal and percentage as one object

0:00Write fifty per cent. Now write fifty over a hundred. Now write one half. Now write nought point five. Four different pieces of writing, and the claim of this whole video is that they are not four numbers. They are one number in four costumes, and nothing about it changes when it changes clothes. That is a slogan until you say what it has to mean. It has to mean that anything you do with one of them, you can do with any of the others and land on the same answer.

0:34Not nearly the same. The same. And if that holds, moving between them is not a calculation. It is a change of handwriting. Test it on a number. Take twenty-four. Fifty per cent of twenty-four. Half of twenty-four. Nought point five times twenty-four. Twelve. Twelve. Twelve. One example proves nothing, so look at why rather than at that. Fifty per cent means fifty hundredths. Fifty hundredths reduces to one half. One half written as a decimal is nought point five.

1:14At no point in that chain did the number move. Every step only rewrote it. So there was never a conversion and then a multiplication. There was one multiplication, written three ways. And if that is the pattern, ten per cent should have a decimal doing the same job. It does. Nought point one. Here is a table with three rows. Per cent. Fraction. Decimal. Eight columns across the top: fifty, a hundred, twenty-five, seventy-five, ten, one, five and forty-three.

1:52The top row is filled in the whole way across. Underneath, exactly two cells are filled. Fifty over a hundred, and nought point five. Fourteen cells are blank, and every one is a number you already have. You are not being asked to work anything out. You are being asked to write a number you already know in a costume it is not wearing. That distinction is the whole topic.

2:21So why is it a re-labelling and not a calculation? Because of how our digits are built. Take thirty-one per cent. That is thirty-one hundredths. Now look at where hundredths live in a decimal. First place after the point, tenths. Second place, hundredths. So thirty-one hundredths is nought point three one. The same digits, in the same order, in the place that already means hundredths. A percentage counts hundredths, and the second decimal place counts hundredths.

2:56They were never two ideas that happen to agree. They are one idea with two spellings. Run it across every whole percentage from one to ninety-nine and it is the same number every single time. Now fill the table, and put two of those columns next to each other on purpose. Ten per cent is ten hundredths, which is one tenth, which is nought point one. One per cent is one hundredth, which is nought point nought one.

3:29Those two look alike and are not alike, and writing them side by side is the cheapest protection there is. A hundred per cent is one whole thing. No hundredths are left over, so there are no places after the point. And forty-three per cent is forty-three hundredths, nought point four three, and a fraction that refuses to get tidier. Forty-three shares no factor with a hundred. If you went hunting for a neater form and could not find one, that was not a mistake. There is not one.

4:07Now a shortcut that almost works, and almost is the dangerous part. Since a percentage is hundredths, people read a decimal by covering the point and taking the digits. Nought point four three. Cover the point. Forty-three per cent. Correct. Nought point two five, twenty-five. Nought point seven five, seventy-five. Nought point nought five, five. Nought point nought one, one. That is five of the eight columns, all correct. Now try nought point five. Cover the point, and it answers five per cent. It is fifty.

4:47It is right exactly when the decimal has two places written out, and never otherwise. Nought point five has a second place. It is a nought, and nobody writes it. Which brings us to two things that look identical and differ by a factor of a hundred. Nought point five is fifty per cent. Nought point five per cent is nought point nought nought five. Put those side by side once and you never confuse them again.

5:19And the digits stop being the percentage entirely once you go past the whole. One point two is a hundred and twenty per cent, and the digit after the point is a two. Two and a half is two hundred and fifty per cent, and the digit after the point is a five. Percentages above a hundred are not errors. They are simply more than the whole of the thing.

5:46There is one place the smooth story gets rough, and it is better met head on. One third. Write it as a decimal and the threes never stop. So its percentage is thirty-three point three three, then more threes, forever, and every written version has been cut short somewhere. Which fractions do this? A decimal stops when the denominator is built only out of twos and fives. Twos and fives, because those are the factors of ten, and ten is what every decimal place is worth.

6:22Four, five, eight, twenty, twenty-five, fifty. Those stop. Three, seven, nine, eleven, thirty. Those do not. So a percentage ending in a rounded tail came from the second kind, and the rounding is a decision, not an error. But there is a trap inside that test, and it is an easy one to fall into. The test has to be run on the fraction after it is reduced, not on it as it was handed to you.

6:54Seventy-two over a hundred and fifty. That hundred and fifty carries a three, so you expect trouble. It reduces to twelve over twenty-five, and twenty-five is clean. Nought point four eight. It stops. Ninety-nine over a hundred and fifty reduces to thirty-three over fifty. Nought point six six. It stops as well. A hundred and thirty over three hundred looks like the same situation and is not. It reduces to thirteen over thirty, and the three survives.

7:31Test the denominator as written and you are right most of the time, and wrong on exactly the fractions where reducing throws the awkward factor away. Now a picture that carries all three at once. A bar, cut into equal cells. Five cells. The whole bar is a hundred per cent, so one cell is twenty. Shade four and that is four fifths, which is eighty per cent. Write seventy-five at the far end and those four cells are sixty.

8:06Change nothing except the label at the end, and the quantity changes while the percentage does not. Ten cells. One cell is ten per cent, six cells is sixty, and against ninety at the end, six cells is fifty-four. Four cells. One cell is twenty-five per cent, three cells is seventy-five, and against a hundred and forty, three cells is a hundred and five. The cell count fixes the fraction. The end label fixes the quantity. Neither one on its own tells you anything.

8:44Attach a quantity and the three costumes start earning their keep. Sixty millilitres of orange paint, three quarters of it red. How much red? Three quarters of sixty. Seventy-five per cent of sixty. Nought point seven five times sixty. Forty-five millilitres, three times over, and you choose which of the three you actually do in your head. That is the part that gets missed. They are interchangeable, so pick whichever is easy.

9:17Nought point eight times seventy-five beats eighty over a hundred times seventy-five. But a quarter of forty beats nought point two five times forty. Same number, different costume, and one costume makes the arithmetic disappear. One last thing, and it is the nicest thing here. An animal is offered two measures of food and eats one. The next day, three, and eats two. The day after, four, and eats three.

9:51As fractions: one half, two thirds, three quarters, four fifths. As percentages: fifty, sixty-six point six seven, seventy-five, eighty. Keep going. On the ninety-ninth day it is offered a hundred and eats ninety-nine. Ninety-nine per cent. Every day is a bigger share than the day before, and not one of them ever reaches a hundred. Because exactly one measure is left behind every day, and one measure never changes. What changes is the pile it is left out of. One out of a growing pile is a shrinking share that never quite becomes nothing.