  Given a finite set $A\subset \mathbb{R}\backslash \{0\}$, define
\begin{align*}&A\cdot A \ =\ \{a_i\cdot a_j\,|\, a_i,a_j\in A\},\\ &A/A \ =\
\{a_i/a_j\,|\,a_i,a_j\in A\},\\ &A + A \ =\ \{a_i + a_j\,|\, a_i,a_j\in A\},\\
&A - A \ =\ \{a_i - a_j\,|\,a_i,a_j\in A\}.\end{align*} The set $A$ is said to
be MPTQ (more product than quotient) if $|A\cdot A|>|A/A|$ and MSTD (more sum
than difference) if $|A + A|>|A - A|$. Since multiplication and addition are
commutative while division and subtraction are not, it is natural to think that
MPTQ and MSTD sets are very rare. However, they do exist. This paper first
shows an efficient search for MPTQ subsets of $\{1,2,\ldots,n\}$ and proves
that as $n\rightarrow \infty$, the proportion of MPTQ subsets approaches $0$.
Next, we prove that MPTQ sets of positive numbers must have at least $8$
elements, while MPTQ sets of both negative and positive numbers must have at
least $5$ elements. Finally, we investigate several sequences that do not have
MPTQ subsets.
