Spoiler:
$\text{dollars}:\quad 3.16,\quad 1.25,\quad 1.50,\quad 1.20$ $\text{cents}:\quad 2^2\cdot 79,\quad 5^3,\quad 2\cdot3\cdot5^2,\quad 2^3\cdot 3 \cdot5 $
I cheated; I just googled the problem and found that someone brute forced a solution. It's possible that one could do some form of case-by-case modular analysis, but even if that's feasible it sounds exhausting. (One pattern that jumps out to me, though, is how the factors of $2$ and $5$ are spread; perhaps one could prove the distribution of them among the solutions totally a priori?) You can also get pretty damn close by making $a=0.79, b=2.00$, $c=1.75$-$1.76$ and $d=2.57$-$2.56$.