In the present paper, we prove lower and upper bounds for each of the ratios $GA/\delta$, as well as a lower bound on $GA/\sqrt{\delta}$, in terms of the order $n$, over the class of connected graphs on $n$ vertices, where $GA$ and $\delta$ denote the geometric-arithmetic index and the minimum degree, respectively. We also characterize the extremal graphs corresponding to each of those bounds. In order to prove our results, we provide a modified statement of a well-known lower bound on the geometric-arithmetic index in terms of minimum degree.