Given an equation, you may multiply (or divide) both sides by the same non-zero number without changing the solution set of the equation.
If you have $ pounds=float\cdot{1\over ratio}; $ and wish to solve for $float$; first multiply both sides by $ratio$: $ ratio\cdot pounds = float\cdot{1\over ratio} \cdot ratio. $ But ${1\over ratio} \cdot ratio =1$, so you have $ ratio\cdot pounds = float. $
The "flip" business, I presume, means to take the reciprocal of both sides of the equation. You can do this when it doesn't lead to division by zero, because it amounts to dividing both sides of the equation by each side. Given $\tag{1} {pounds\over float}={1\over ratio} $ "Flipping" yields: $\tag{2} {float\over pounds}={ ratio\over 1} $
You can see that you can get equation $(2)$ from equation $(1)$ by multiplying both sides of equation $(1)$ by $ratio$ first, then multiplying both sides of the resulting equation by $float/pounds$.
Generally, when solving equations of this type for a specific quantity, you first make sure that quantity is "upstairs" ("flip" if it isn't). Then divide both sides of the equation by what is necessary to get the quantity of interest on one side of the equation by itself.
For example, suppose you have equation $(1)$ and that you want to solve for $float"$.
First "flip" to get equation $(2)$.
To get $float$ by itself, multiply both sides of equation $(2)$ by $pounds$.
You obtain at the end $float=ratio\cdot pounds$.