14 ms·

This problem was enough hard for me that I want to post my solution. (a+1)/(b+1) = (a * (1 + 1/a))/(b * (1 + 1/a)) Which in turn, is the same as (a/b) * ((1+1

This problem was enough hard for me that I want to post my solution. (a+1)/(b+1) = (a * (1 + 1/a))/(b * (1 + 1/a)) Which in turn, is the same as (a/b) * ((1+1/a)/(1+1/b)) This can only be greater than a/b if (1+1/a) / (1+1/b) > 1. A fraction is greater than 1 if the numerator is greater than the denominator, which means 1/a > 1/b. Dividing by a greater number leads to a smaller result, so this happens if a < b. 12345 < 54321, so 12346/54322 is greater. Did I overlook a much easier way to solve this, or is third grade much better at math than I am now?

groaner 15y ago
I suck at pure math, so I tend to think of this in some sort of analogy. Suppose the fraction a/b is some statistic you are trying to measure, say, a batting average or percentage of correct notes played in Guitar Hero. (a+1)/(b+1) would be the new fraction after you got the next one right. By getting the next one right, did you improve your score? Of course, if you have a perfect record already, getting an additional 1-for-1 won't change anything. And if a>b, then you'd have to somehow score more than 1 point per attempt in order to maintain the same ratio, so (a+1)/(b+1) would be lower.
- cousin_it 15y ago
  I used to be a mathematician, and I think your comment is everything that math education in school should aspire to be :-) Another example in the same vein is Terry Tao's airport puzzle (http://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/ http://terrytao.wordpress.com/2008/12/09/an-airport-inspired...), scroll down to Harald Hanche-Olsen's comment for the best explanation.
- yequalsx 15y ago
  That is a very nice explanation. It shows why intuitively we expect (a+1)/(b+1) to be greater. But note that there are some unstated assumptions. Namely that both a, b > 0. And so we see that intuition at times carries unstated assumptions that can be a trap in a generalized situation. Tokenadult and codehotter didn't mention the case when a or b < 0.
Someone 15y ago
You got the gist, but get a point deducted for that typo :-) You also forgot to realize that 10/-5 is less than one, even though its numerator is larger than its denominator. Finally, it can be done somewhat easier, assuming a, b > 0: (a+1)/(b+1) > a/b <=> b (a+1)/(b+1) > a <=> b (a+1) > a (b+1) <=> ba + b > ab + a <=> b > a
ShardPhoenix 15y ago
It's not a proof, but you can figure it out by considering some simple cases such as 2/3 vs. 3/4.
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groaner 15y ago

I suck at pure math, so I tend to think of this in some sort of analogy. Suppose the fraction a/b is some statistic you are trying to measure, say, a batting average or percentage of correct notes played in Guitar Hero. (a+1)/(b+1) would be the new fraction after you got the next one right. By getting the next one right, did you improve your score? Of course, if you have a perfect record already, getting an additional 1-for-1 won't change anything. And if a>b, then you'd have to somehow score more than 1 point per attempt in order to maintain the same ratio, so (a+1)/(b+1) would be lower.

cousin_it 15y ago
I used to be a mathematician, and I think your comment is everything that math education in school should aspire to be :-) Another example in the same vein is Terry Tao's airport puzzle (http://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/ http://terrytao.wordpress.com/2008/12/09/an-airport-inspired...), scroll down to Harald Hanche-Olsen's comment for the best explanation.
yequalsx 15y ago
That is a very nice explanation. It shows why intuitively we expect (a+1)/(b+1) to be greater. But note that there are some unstated assumptions. Namely that both a, b > 0. And so we see that intuition at times carries unstated assumptions that can be a trap in a generalized situation. Tokenadult and codehotter didn't mention the case when a or b < 0.

Someone 15y ago

You got the gist, but get a point deducted for that typo :-) You also forgot to realize that 10/-5 is less than one, even though its numerator is larger than its denominator. Finally, it can be done somewhat easier, assuming a, b > 0: (a+1)/(b+1) > a/b <=> b (a+1)/(b+1) > a <=> b (a+1) > a (b+1) <=> ba + b > ab + a <=> b > a

ShardPhoenix 15y ago

It's not a proof, but you can figure it out by considering some simple cases such as 2/3 vs. 3/4.

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