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Every now and then a sports coach will say, "You all have to give 110% this game!" And when they do, you can always count on someone replying that this is absurd, that it's meaningless to talk about a percentage of more than 100. I recall my elementary school teachers saying this many times.

This is simply wrong. Percentages more than 100 are perfectly meaningful in many contexts. Yet the idea that this is absurd is, apparently, quite popular. So let's think about this for a moment.

Sure, there are times when 110% is a meaningless number. If, for example, someone said that of the people who voted in the last election, 110% of them voted for candidate Smith, this makes no sense. It sounds like the beginning of a charge of vote fraud.

But it would be quite reasonable to say that voter turnout this year was 110% of what it was last year. If last year 100,000 people voted, and this year 110,000 people voted, than the number of voters this year is 110% of the number of voters last year. There's nothing impossible about that.

If by "give 110%" the coach means that each player should do 110% of the maximum amount that he is capable of, sure, that would be mathematically impossible. But if he means, say, that each player should do 110% of what he did last game, that's quite plausible. If each player put in 100 hours of practice before the last game and he puts in 110 hours of practice before this game, then he's practicing 110% as much. Or if last game he scored 10 times and this game he scores 11 times, then he's scoring 110% as often.

Of course the average coach probably isn't thinking it out this far. He's not doing a statistical analysis, he's just trying to encourage the team. I suspect that what he really means is something like, "Give 110% of the maximum that you think you are capable of." That is, you think the best you can do is X. But I think that if you work hard and practice and struggle, you can do better than that. There's nothing impossible about such a claim. Maybe the player really is already doing the maximum that he could possibly do. But probably not.

A percentage is just a ratio between two numbers. It's how many hundredths X is of Y. If X is 50 and y is 100, then X is 50% of Y. If X is 110 and Y is 100, then X is 110% of Y. A percentage of more than 100 just means that one number under discussion is bigger than the other. That's nothing startling or absurd about the idea that one number could be bigger than another. Or to put it another way, a percentage is a fraction: so many over 100. 50% means 50/100, or 1/2. 100% means 100/100, or 1. 150% means 150/100, or 3/2. There's nothing inherently absurd about a fraction bigger than 1. Yes, in some contexts this makes no sense, like if we are asking what fraction of the total meets some condition. But if we are comparing this year's total to last year's, it's not at all impossible that this year could be bigger than last year. Or if we are comparing how much Al did to how much Bob did, it's not abusrd to suppose that Al could have done more than Bob.

Whether or not a percentage of more than 100 makes sense depends on what we are counting.