We investigate two types of ratios of ergodic sums in (regular) continued fractions. Let $[a_1(x),a_2(x),a_3(x),\ldots]$ denote the (regular) continued fraction expansion of an irrational number $x\in [0,1)$. For any $n\in \mathbb N$, define

\[R_n(x):=\frac{\sum^n_{k=1}a_{k+1}(x)}{\sum^n_{k=1}a_{k}(x)}\quad\text{and}\quad\widehat{R}_n(x):= \frac{\sum^n_{k=1}a_{2k}(x)}{\sum^n_{k=1}a_{2k-1}(x)}.\]

We establish the limit inferior and limit superior of the sequences $\{R_n(x)\}_{n \geq 1}$ and $\{\widehat{R}_n(x)\}_{n \geq 1}$ for Lebesgue almost every $x\in [0,1)$. Furthermore, we conduct a multifractal analysis of the exceptional sets associated with the behavior of these sequences. As an application, we improve the multifractal analysis of the arithmetic means of backward continued fractions developed by Jaerisch and Takahasi (2021).