  We present a detailed Hausdorff dimension analysis of the set of real numbers
where the product of consecutive partial quotients in their continued fraction
expansion grow at a certain rate but the growth of the single partial quotient
is at a different rate. We consider the set \begin{equation*}
\FF(\Phi_1,\Phi_2) \defeq \EE(\Phi_1) \backslash \EE(\Phi_2)=\left\{x\in[0,1):
\begin{split}
  a_n(x)a_{n+1}(x) & \geq\Phi_1(n) \text{\,\, for infinitely many } n\in\N
  a_{n+1}(x) & <\Phi_2(n) \text{\,\, for all sufficiently large } n\in\N
\end{split} \right\}, \end{equation*} where $\Phi_i:\N\to(0,\infty)$ are any
functions such that $\lim\limits_{n\to\infty} \Phi_i(n)=\infty$. We obtain some
surprising results including the situations when $\FF(\Phi_1,\Phi_2)$ is empty
for various non-trivial choices of $\Phi_i$'s. Our results contribute to the
metrical theory of continued fractions by generalising several known results
including the main result of [Nonlinearity, 33(6):2615--2639, 2020]. To obtain
some of the results, we consider an alternate generalised set, which may be of
independent interest, and calculate its Hausdorff dimension. One of the main
ingredients is in the usage of the classical mass distribution principle;
specifically a careful distribution of the mass on the Cantor subset by
introducing a new idea of two different types of probability measures.
