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.

Posting Guidelines (Summary)

Welcome to our tutoring boards! :) This page summarizes main points from our posting guidelines. As our name implies, we provide math help (primarily to students with homework). We do not generally post immediate answers or step-by-step solutions. We don't do your homework. We prefer to help...

www.freemathhelp.com

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.