30% pizza to 10% people, 70% pizza to 90% people, what is the difference? Yesterday I was reading an article, which had a statistic exposing a certain disproportion in different groups of people. I will now reformulate this problem in a more "neutral" way.
If 10% of people get 30% of pizza, then how many times more pizza does an average person in the 10% group get than the average person in the 90% group?
I know we can solve this problem the following way:

EDIT: $$x/(x+1) = .3$$
  so $x=0.428571$, which means that the 10% group gets $x=0.428571$ pizzas every time the other group gets one pizza. 

But how can we solve this in a different way? I seem to have always had some issue with percentages, despite doing well in much more complicated math or physics problems. Here's what I've tried:
$30/10 = 70x/90$, so $x\approx 3.8571$, which is obviously false, and we can easily verify this.
Can someone please clarify why exactly my second equation is incorrect, and what a correct equation should look like? I mean an equation with proportions, but not like the first equation, which is easy to come up with, but I want a more "direct" approach.
 A: The proportion you need is $(30/10) = (1+x) (70/90)$ where $x$ is the additional "slice" of pizza-for-the-rich on top of the portion of pizza-for-the-poor.
A: What you get seems correct to me. I don't know where did $x/(x+1)=.7$ come from? Anyway, you get think in two (seemingly) different ways.
The first one is a straightforward calculation: let $P$ be total number of pizzas and $n$ total number of people. $0.3P$ goes to $0.1n$, while $0.7P$ goes to $0.9n$. A person from the first group gets $\frac{0.3P}{0.1n} = 3P/n$ pizzas, while from the second group a person gets $\frac{0.7P}{0.9n} = 7/9\,P/n$ pizzas, so 10-percenters get $3/(7/9) = 27/7$ times more.
The second way is to denote with $x$ and $y$ the number of pizzas a person from the first or second group gets. Obviously, number of pizzas any person gets is directly proportional to what the total number of pizzas is, while it is inversely proportional to the total number of people, so we can write:
$$\left.\begin{align}
x:y &= 0.3:0.7\\
&= \frac 1{0.1}:\frac 1{0.9}
\end{align}\right\}\implies x:y = \frac{0.3}{0.1}:\frac{0.7}{0.9} = 3:\frac 79 = 27:7$$
