How to solve for the parameters of the Gamma distribution given x for the 50th and 90th percentiles? How can I solve for the parameters alpha and beta given that x = 20 is the 50th percentile, and x = 300 is the 90th percentile? 
 A: I'll use the notation of the wikipedia article on the gamma distribution throughout.
Given that there's no simple closed form for the median of a gamma distribution, you'll want to do this numerically.
You can be smart about this, though.  First, note that the ratio of the 90th to 50th percentiles of a gamma distribution depends only on the shape parameter $k$, not the scale parameter $\theta$.  So we can ask: what is the shape parameter $k$ for which the 90th percentile divided by the 50th percentile is $300/20 = 15$?  This is a couple lines in R:
f = function(k){qgamma(.9, k, 1)/qgamma(.5, k, 1)}
k0 = uniroot(function(k){f(k)-15}, c(0.1, 1))$root

The first line defines a function f which returns the 90th percentile of the $gamma(x,1)$ distribution divided by its 50th percentile.    The second line finds a root of $f(x) - 15$ in the interval $[0.1, 1]$ and assigns it to k0.
This returns the root $\theta \approx  0.2672395$.
Now the median of a $Gamma(k_0, 1)$ random variable is given by qgamma(.5, k0, 1) in R and is equal to $m_0 = 0.05319006$.  So to get the median equal to 20, you need a scale parameter $\theta_0 = 20/m_0 \approx 376.0101$.
