Ratios and the Constant of Proportionality

Suppose two quantities

\[x, \: y\]

related by a ratio, say

\[A:B\]

.
An quantites related in this way are proportional to each other. Where does the constant of proportionality appear?
The meaning of the ratio is this: If

\[x\]

is a multiple of

\[A\]

then

\[y\]

is that multiple of

\[B\]

so that

\[\frac{x}{A}= \frac{y}{B}\]

then

\[y=\frac{B}{A} x\]

and the constant of proportionality relating

\[y\]

\[x\]

\[\frac{B}{A}\]