integer optimization

[Juha Lappi]

Given three real numbers a and b, a<b, and c, what integer values of n and m maximize n*a+m*b subject to n*a+m*b <=c?

 

[Cubist]

What methods have you been shown for tackling problems like this (I assume that you're on a course)?
Please show what you've tried and where you're stuck.

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One way of approaching this might be to rearrange the inequality so that the subject becomes m/n and then see what the other side looks like. The other side should have a term with something/n which would become small with a large value of n.

(But if you've been shown another method then please use that and show us your work)

 

[Cubist]

One way of approaching this might be to rearrange the inequality so that the subject becomes m/n and then see what the other side looks like. The other side should have a term with something/n which would become small with a large value of n.

...after a bit more thought I think my suggested method isn't valid, although it seemed very compelling. This is because it eliminates c for large values of n. And c is a very fundamental part of the original inequality. Without taking c into account how could the answer ever be close!

I'll give this a bit more thought, especially if OP can contribute to the thread.

Perhaps worth mentioning in your answer that there's a trivial, exact, solution for one value of c. And also there could be a different method (possibly with an exact solution) if a,b and c happen to be rational.