I was hoping that I could find help solving this problem. I've been working on it for a while and haven't been able to solve it correctly.

So far I have:
[tex]
\frac{dQ}{dt} = r - \frac{rQ}{100}
[/tex]

Where Q is the amount of salt, r is the rate in and out.
I think that this may work for the first 10 minutes while the salt is part of what is getting mixed in, but after that I'm lost. Any help will be appriciated.
Thank you.

No, not quite. There are 2 gallons of water coming in every minute (I don't know why you call that "r") and each gallon carries 1/2 pound of salt so 2*(1/2) = 1 pound of salt comes in every minute ((1/2)r, not r). Each gallon of water going out contains [itex]\frac{Q}{100}[/itex] pounds of salt so the amount going out is [itex]\frac{rQ}{100}[/itex] as you have (with, of course, r= 2). The differential equation is:
[tex]\frac{dQ}{dt}= 1- \frac{Q}{50}[/tex]
Also, of course, Q(0)= 0.

Yes, so solve that differential equation for Q(t) and use it to find Q(10).
Now, start all over again. Now, there is NO salt coming in so the differential equation is
[tex]\frac{dQ}{dt}= -\frac{Q}{50}[/tex].
Q(10)= whatever you found before and then find Q(20) (ten more minutes).

(Since t does not appear explicitely in that equation, you could just "restart" the clock: Let Q(0)= whatever you found before and then find Q(10) from this new equation.)