The motivation is an object which generalizes the notion of percentages.
Consider the the set $\mathbb{R}$ along with the usual binary addition operation $+$ and infinitely-many binary multiplication operations $\boldsymbol{\cdot}_\alpha$ where $\alpha \in \mathbb{R}$ and $a\boldsymbol{\cdot}_\alpha b = (\alpha a) \cdot b$.
For instance, $50\boldsymbol{\cdot}_{0.01} 6 = (0.01\cdot50) \cdot b$
You can easily prove that the set $\mathbb{R}$ with $+$ and any fixed $\boldsymbol{\cdot}_\alpha$ is a field. The $\alpha = 1$ case corresponds to the usual definition of field $\mathbb{R}$ and $\alpha=0.01$ corresponds to a field with a "percentage of" as it's product operation. If, for some twisted reason, one wanted to work with "perpentages" one would consider the $\alpha = 0.2$ case.
This structure represents arbitrarily scaled multiplication and satisfies the field axioms for any fixed scalar. Working with this structure was motivated by the following question: "How can I most effectively procrastinate studying for the GRE and punish myself for making stupid math mistakes on easy percentage problems?"
Are there references to comparable objects anywhere/what would you call this type of thing?