[groaner]

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]

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...), scroll down to Harald Hanche-Olsen's comment for the best explanation.

[yequalsx]

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.