I haven't taken calculus yet, but this sounds like a calculus problem (maybe precalculus or algebra 2)

 

I want to know any techniques to solving this problem.

 

Suppose there are variables A, B, C, D, E, F, G, and H, where A, B, and C are less than 25.

 

.3A + .8B = D

.2A + .1 B + .2C +.4F= E

(A+B+C)/3=F

G=D/F

H=E/F

 

Find values A, B, and C that maximize G + H.

any A,B,C that give A+B+C=0 will result in G+H going to infinity and I guess that is not what you are looking for, so please give us some more information. are there no other restrictions on A,B,C? for example, if they have to be positive numbers, there is an easy solution without any calculus involved.

Yeah, positive numbers for A, B, C

as I said, you can get away without calculus, it is just playing with fractions and some logic. start with the obvious - express the desired sum in terms of A,B,C only. it is a ratio of two linear combinations of A,B,C (expressions like xA+yB+zC, where x,y,z are some numbers). get rid of one of the variables in the numerator. it remains in the denominator, however. now you can realize what it takes to get a maximum value. i won't get further, try it out. just don't worry about calculus, this problem is simply advanced 5th grade stuff. (or whenever you study fractions)

It simplifies down to 1+ (.9A+2.1B)/(A+B+C). C goes away because increasing C would give a lesser value. You want to maximize the numerator and minimize the denominator, so in the end, all you need to do is to have B=25. Is that right?

exactly