How to adjust a ratio to create desired proportion Let's say I have a bunch of apples and oranges. Specifically 500 apples and 760 oranges. 
I need to create a big basket of fruit. For every 7 apples I include, I need to include 3 oranges. 
I want to make the biggest basket possible using as much of my fruit as I can while honoring the desired apple to orange ratio.
I'm not mathematically inclined and I've been just taking bunches of 7 and 3 at a time until I run out. There has to be some kind of formula I can use to tell me upfront the max number of each that I should pull.
Thanks in advance.

We assume that fractional fruits are not sensible things to include in our basket. We need $7$ apples per $3$ oranges, but we can see that we have more oranges than apples. So, we know that apples are the limiting reagent. Because we only want natural numbers of fruits, and $7,3$ are relatively prime, we know that the largest multiple of $7$ which fits in $500$ is the number of apples we can have. From there we can determine the number of oranges.
As it turns out, we can have $71\times 7=497$ apples in the basket, and so $71\times3=213$ oranges. That is, we can have at most $710$ pieces of fruit: $497$ apples and $213$ oranges. This maximizes the ratio-honoring basket size.
