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Two times more, or twice as much?

This is something that has been bugging me a long time, so I just had to make a write-up to clarify the subject a bit.

A very common mistake, or just something people don't really pay attention to or care about, is to say "two times more" (or any number of times, two is just an example here) when they actually mean "twice as much".

"But they're the same, what's the difference?" I hear people already murmuring inside their heads. Wrong, they're not the same, and the difference can be (in a very concrete, real-life way) huge. I'll give you a couple of examples to clarify the difference.

If you have one dollar and I promise to give you one time more (meaning 100% more) it's very easy, grade-school math to calculate that 1 + 1 * 1 = 2. When people say "it costs two times more" they often mean that something, which used to cost 1 dollar now costs 2 dollars. But, how could that be when it was already shown that just "one time more" already makes that 1-dollar item cost 2 dollars; how could "two times more" be as much as "one time more"? The answer is simple: it isn't - "two times more" would make that 1-dollar item cost 3 dollars (1 + 2 * 1 = 3)! "X times more" means addition on top of what you already have!

"Nonsense", I hear people saying, but it isn't. "One time more" means the same as "one hundred percent addition", or "twice as much", or "double the amount". "Two times more" means the same as "two hundred percent addition", or "three times as much", or "triple the amount".

People also very often make the same mistake in the opposite direction, only this time the difference between what's being said and what's actually being meant is far bigger than in the case of addition.

"His Toyota moves three times slower than my Porsche" is how you might hear someone bragging about his new car. What a lot of people just don't realize is that while they mean, for example, that the Toyota is moving at 100km/h (60mph) and the Porsche at 300km/h (180mph), they're actually saying that the Toyota is moving 600km/h (360mph) on reverse - quite a feat for any car, I'd say! Again, an example to clarify what I'm saying:

John has one dollar which he gives to charity, leaving him with no cash on him. This means he gave away 100% of - or one time - his capital stock.

A week later, John has ten dollars in cash, which he loans to his uncle Bob. Once again, John is left with nothing on him. Thus, once again, he gave away one time the amount he had - all 100% of it.

So, if one time less means you're left with zero, regardless of how much you originally had, wouldn't it clearly imply that two times less means you subtract the original amount twice, which would not only take away everything you had, but it would also drag you one time the original amount on the negative scale: 1 - 2 * 1 = -1. Once again, very simple grade-school math. So, if the Porsche guy wanted to say that his car travels three times as fast as the Toyota, the correct way of saying it from the Toyota driver's perspective would be that the Toyota travels at one third of the velocity of the Porsche, not three times slower as our clueless Porsche braggart is ignorantly claiming.

The same applies everywhere; there are loads of TV commercials stating something is two times cheaper than before, or that some miraculous hair balsam will give you ten times less dandruff. Exactly what the hell is that supposed to mean - minus nine times the amount of dandruff? You not only get rid of all the dandruff you had but your scalp also sucks the dandruff out of the heads of the nine people nearest to you and makes it disappear? Why do I find that extremely hard to believe? Of course, the correct way of putting it would be that the balsam will make the amount of dandruff drop to one tenth - or by 90% - compared to the situation before using it.

Summary

  What's being said | What it actually means
--------------------+-----------------------
     two times more | X + 2 * X = 3X
      twice as much | 2 * X = 2X
  double the amount | 2 * X = 2X
                    | 
     two times less | X - 2 * X = -X
       half as much | (1/2) * X = (1/2)X
one half the amount | (1/2) * X = (1/2)X

where X is the original amount we have.

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