Percentage increase and decrease, and choosing the right base

0:00Every percentage change is a fraction, and the whole difficulty of this topic is the number underneath it. Change means change from something. Name that something before you do anything else. It is always the value you started from. Never the value you ended at, and never the two averaged. That sounds like a convention you could take or leave. It is not. It has a consequence almost everybody finds surprising, and you will be able to say exactly why.

0:34Start with the easy direction. A kilogram of tomatoes cost thirty, three years ago. It costs forty-two now. The increase is twelve. Twelve out of what? Out of thirty, because thirty is where it started. Twelve over thirty is two fifths, which is forty per cent. So the price rose by forty per cent over the three years. Notice the shape of that fraction, because the next one has exactly the same shape.

1:08A theatre's average attendance was a hundred and sixty before. It is a hundred now. The decrease is sixty. Sixty out of what? Out of a hundred and sixty, because that is where it started. Sixty over a hundred and sixty is three eighths. Thirty-seven and a half per cent. Same fraction, same rule, opposite direction. And here the value it started from is the larger of the two, while for the tomatoes it was the smaller. So dividing by the bigger number is not the rule, though it happened to work once.

1:47Now the surprise. Suppose attendance climbs back. A hundred returns to a hundred and sixty. The gap is sixty again. The very same sixty. But this time it started at a hundred, and sixty out of a hundred is sixty per cent. So the fall was thirty-seven and a half per cent, and the climb back is sixty. One gap, two percentages. Nothing is inconsistent. The fractions simply have different numbers underneath.

2:20Percentage changes are not reversible, and that is the single most useful thing here. So how big does the fall have to be to undo a rise? A rise lands you above a hundred, and to get back you must lose that much out of the bigger number you are standing on now. Which is always a smaller share than the rise was, because the thing underneath has grown. Do the numbers. A rise of a quarter is undone by a fifth: twenty-five per cent up, twenty per cent down.

2:56A rise of a half takes a third: fifty up, thirty-three point three three down. A doubling takes a fall of a half: a hundred up, fifty down. And the theatre's own pair falls straight out of it. Undoing a rise of sixty takes a fall of thirty-seven and a half. But for a small change the two are almost the same. A rise of one per cent is undone by a fall short of one per cent by a hundredth of a point.

3:27That near-miss is exactly why the false belief survives. Here is what a wrong base actually looks like. Fuel cost sixty ten years ago and costs a hundred now. By what percentage did it rise? Forty out of sixty. That one does not stop: it is two hundred thirds, sixty-six point six seven to two places. Six answers were offered, and two of the wrong ones have a traceable source.

3:58Forty per cent is the change measured against the price it ended at, instead of the one it started from. Sixty per cent is the old price read as a share of the new one, which answers a completely different question. The other three come from nowhere, and it is better to say so than to invent a pattern that is not there. One more thing worth catching: the right answer is offered written sixty-six point six six. That is the number cut short, not rounded. Rounded, it is sixty-six point six seven.

4:37Now the identity that makes all of this portable. Two statements. A population is a hundred and sixty-five per cent of what it was. And: the population increased by sixty-five per cent. Do those say the same thing? Call the old value p. The first says the new value is a hundred and sixty-five hundredths of p. One point six five p. The second says the new value is p, plus sixty-five hundredths of p. Which is also one point six five p.

5:15Yes. One statement in two dresses. And that is the general fact underneath everything else here: a change of anything per cent is a multiplication by one plus that many hundredths. A herd growing five per cent is multiplied by one point nought five. Not by nought point nought five, and not by one point five. Now a third phrasing, and it is not the same statement at all. Increased by a hundred and sixty-five per cent.

5:50That means p, plus one point six five p, which is two point six five p. Put the three side by side. A hundred and sixty-five per cent of p is one point six five p. Increased by sixty-five per cent is one point six five p. Increased by a hundred and sixty-five per cent is two point six five p. On a starting value of a thousand, that is one thousand six hundred and fifty, one thousand six hundred and fifty, and two thousand six hundred and fifty.

6:29The first two are the same place. The third is a different place, and the two readings differ by exactly the whole original amount. One word deserves a mention, because it sounds like a separate calculation and is not. Inflation is the percentage increase in prices. That is the whole of it. Rice at thirty-eight one year, forty-two the next. The increase is four, out of thirty-eight. Four over thirty-eight is two hundred nineteenths. Ten point five three per cent, to two places.

7:07The same fraction as the tomatoes, with an economic name attached to it. Now run the whole thing backwards. A number increased by twenty per cent becomes ninety. What was the number? In multiplier form this is one step. Something times one point two is ninety, so the something is ninety divided by one point two. Seventy-five. Check it. Seventy-five, plus twenty per cent of seventy-five, is ninety. Here is the wrong route, and it is tempting. Take twenty per cent off the ninety instead. That gives seventy-two.

7:49But seventy-two grown by twenty per cent is eighty-six point four, not ninety. It lands three point six short. Twenty per cent of seventy-five and twenty per cent of ninety are different amounts, which is the entire reason for naming the base out loud. Sometimes the question hides the base, and you have to go and find it. Someone pays four hundred and forty thousand for a car after the dealer takes fifteen per cent off.

8:22What was the price before the discount? The fifteen per cent came off a price he did not pay, so what he paid is eighty-five per cent of it. Divide, do not multiply. Four hundred and forty thousand over nought point eight five, which is about five hundred and seventeen thousand six hundred and forty-seven. Adding fifteen per cent to what he paid gives five hundred and six thousand, and taking fifteen per cent back off that lands at four hundred and thirty thousand one hundred. Nine thousand nine hundred short of where it has to land.

9:02And notice the correct answer is not a round number. It does not even stop. An awkward answer is not evidence of a mistake. One last one, and it is this whole video in a single picture. A rectangle's length grows by ten per cent, and its area has to stay exactly what it was. By what percentage does the breadth fall? Not by ten. Multiply by one point one and then by nought point nine and you get nought point nine nine. The area has shrunk.

9:39The breadth has to be multiplied by the reciprocal of one point one, which is ten elevenths. That is a fall of one eleventh. Nine point nought nine nought nine per cent, going on for ever. And it is exactly the fraction the theatre gave us: undoing a rise of ten takes ten elevenths of ten. Take a rectangle twenty by eleven, area two hundred and twenty. Stretch the length to twenty-two and the breadth must drop to ten. Two hundred and twenty again.

10:14A rise of ten per cent is undone by a fall of less than ten, every single time, because change means change from something, and the something moved.