How to define "closer to proportion" $\def\prop#1#2#3{#1:#2:#3}$Let's say I have a proportion, $\prop 135$
And then a set of other ones,
$$ \begin{array}{l} \prop 235\\ \prop 145 \\ \prop 136 \end{array} $$
Which one is closer to $\prop 135$?
How to define it closer with mathematics?
(If you think it needs editing, edit away or comment)
 A: This depends on which metric you define. Possible metrics would, for example, result in converting the $a:b:c$ to a vector $$\left( \begin{matrix}a\\b\\c\end{matrix} \right)$$
And using some standard $\mathbb R^3$ metrics, like
$$d(x,y) := \Vert x-y \Vert_p$$
For $1\leq p\leq \infty$.
The results vary with the choice of $p$. For $p=\infty$, for example we have with enumeration $y_1, y_2, y_3$ of your alternatives and $x$ the first:
$$\begin{align*}
d(x,y_1) & = 1 \\
d(x,y_2) & = 1 \\
d(x,y_3) & = 1
\end{align*}$$
so they are all "equally close".

However, any of these $p$ will yield a distance of $1$, since the changes are always to add one to one "component".

Another choice for converting $a:b:c$ to a vector containing "proportions", would be
$$\vec{x} = \left(\begin{matrix}a/b\\b/c\\c/a\end{matrix}\right)$$

With the second option and $p=2$ we get, according to MATLAB with the code
x = [1 3 5];
Y = [x;x;x] + eye(3);
xx = x./x([2 3 1]);
YY = Y./Y(:,[2 3 1]);
Z = [norm(xx - YY(1,:)); norm(xx - YY(2,:)); norm(xx - YY(3,:))]
we have
$$Z \approx \left(\begin{matrix}2.5221\\0.2167\\1.0050\end{matrix}\right)$$
So $y_2$ is "closest" in this interpretation.
