This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with simultaneous diophantine equations, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes:
A fruit vendor sells fruit in three combinations:
Small bag: 1 apple and 2 bananas.
Medium bag: 4 apples and 3 bananas.
Large bag: 8 apples and 7 bananas.
Noticing that the large bags are selling better than the others, he decides to scrap the smaller sizes and just sell the big ones.
In what proportions can be combine small and medium bags to make large ones, without any fruit left over ?
If he combines $s$ small bags with $m$ medium bags to get $k$ large ones, we have $$s+4m=8k\\2s+3m=7k$$ From the first, $s$ must be a multiple of $4$, so call it $4t$ Then we have $$t+m=2k\\8t+3m=7k$$ Then if we subtract $3$ times the first fom the second we get $$5t=k$$ We can do it in units of $5$ large bags, which we make from $4$ small bags and $9$ medium bags.