Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A/A|$?

In the line above, $AA+A:=\{ab+c:a,b,c \in A \}$, while $A/A:=\{a/b:a,b \in A, b\neq 0 \}$ is the ratio set.

This is closely related to a previous question about the relative sizes of the sets $AA+A$ and $A+A$. See Is the set $ AA+A $ always at least as large as $ A+A $?.