4 items add up to and multiply to 7.11 what are the value of the items?

Question

A man walks into a 7-Eleven and buys four items, the items add up to $7.11$ and multiply to $7.11$. What are the prices for the $4$ items?

During a talk about truth in mathematics, the presenter asked this question to the audience. I'm just curious as to what the answer is.

Answer 1

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 $d=2.57$-$2.56$.