If $64$ is divided into three parts proportional to $2$, $4$ and $6$, what is the smallest part?
The problem I have is with understanding what the question means.
What does it mean for a number $N$ to be divided into $n$-parts proportional to $x_1,...,x_n$? Is it like this means that there exist constants $\xi_1,...,\xi_2$ such that $$N=x_1\xi_1+x_2\xi_2+...+x_{n-1}\xi_{n-1}+x_n\xi_n$$ or that for a single constant $\xi$ we have $$N=x_1\xi+...+x_n\xi=\xi(x_1+...+x_n)?.$$
If the number $N$ is divided into $n$ parts proportional to $x_1, \dots, x_n$, I would imagine that means
$$N= a_1+\cdots + a_n$$ such that $a_1:a_2:\dots:a_n = x_1:x_2:\dots:x_n$ or in other words, there exists such a positive constant $\lambda$ that $a_1=\lambda x_1$, $a_2=\lambda x_2,\dots, a_n = \lambda x_n$.