Linear Programming: algorithm to check if ratios can be combined with n bottles to equal a given ratio
I don't know what class this is for, and how much linear programming you are learning in it, so I don't really know what you need. 

One way to do it: assuming $a_i + b_i + c_i = 1, \ \forall i$, use the linear program:

$$ \max x$$

such that

$$\eqalign{
\sum_i  a_i \lambda_i &amp;= a,\\
\sum_i  b_i \lambda_i &amp;= b,\\
\sum_i  c_i \lambda_i &amp;= c, \\
\lambda _i -x &amp;\geq 0 \quad \forall i.
}$$

If there is a solution with $x \geq 0$, then you're done. Why? If $x \geq 0$, then there is a set of positive $\lambda_i$ satisfying the first three equations above: $\sum_i a_i \lambda_i = a$. The $\lambda_i$ give you the proportions with which to mix the bottles.  If one of the $\lambda_i$ were less than $0$, then this would tell you to add a negative amount from that bottle, yielding an unphysical "solution". If there is no solution of the linear program with $x \geq 0$, then no combination will give the proportions $a:b:c$, as that would give a solution to the linear program with $x \geq 0$.