Hacker News Reader

Comment by codehotter | Hacker News Reader

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?

View on HN

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.

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.

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.

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

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

Reply on news.ycombinator.com