Two people, A and B, with respective utility functions of:
$$U_a(X_a,Y_a) = X_a^2 Y_a\\ U_b(X_b,Y_b) = X_b Y_b^2$$
Total $X$ (that is, $X_a+X_b$) is fixed at $36$.
Total $Y$ ($Y_a+Y_b$) is fixed at $48$.
Income of A is $M_a = P_x X_a + P_y Y_a$ (also known as budget constraint).
Income of B is $M_b = P_x X_b + P_y Y_b$.
Graphically this is an "Edgeworth Box", example seen here: http://upload.wikimedia.org/wikipedia/en/thumb/b/b9/Contract-curve-on-edgeworth-box.svg/400px-Contract-curve-on-edgeworth-box.svg.png
Points that are "pareto optimal" have $\frac{dU/dx}{dU/dy}$ for A equal to for B.
Original endowment is $X_a=14, Y_a=32, X_b=22, Y_b=16$.
Find the line $-P_x/P_y$ that leads from the original endowment to the "competitive equilibrium" where $\frac{dU/dx}{dU/dy}$ for A equals $\frac{dU/dx}{dU/dy}$ for B.
It seems I had the right answer all along in what I did, so I am going to provide the answer here to help anyone else trying to solve similar problems.
All utility functions in the form of U(x,y)= x^c * y^d will have demands for X and Y of:
X = c/(c+d) * m/Px
Y = d/(c+d) * m/Py
(alternatively you can set up a lagrangrian and maximize each utility function for A and B subject to the budget constraint of Ma or Mb, which will derive the above equations for demands of X and Y)
For the m's of A and B we use the original endowments, so:
Ma = 14 * Px + 32 * Py
Mb = 22 * Px + 16 * Py
so just calculate Xa, Xb, Ya, Yb using the above equations. You can also normalize Py = 1 to make the calculations more simple.
Once you have each value, since X is fixed at 36 and Y fixed at 48, just set Xa+Xb = 36 and Ya+Yb = 48 and you can find Px/Py from these equations.
Use the MRS equation (dU/dx/dU/dy) for A and B, set equal to confirm the Px/Py is in fact pareto optimal.
The answer is Px/Py = 40/29 so you can verify.
Sorry I didnt put this answer into nice math type but I am unsure of how to do it. Hope this description helps other people who were stuck :)