Find Pareto-optima two good economy There's an exercise I cannot wrap my head on. I actually have the solution to the problem, but still I don't quite understand it.

Let an economy be composed of two consumers $A$ and $B$ and two goods $X$ and $Y$, such that $Y$ is produced using $X$ with production function $Y = \sqrt{5X}$. We suppose $A$ and $B$ have identical utility function $U(X,Y) = XY$. Initial endowment of the economy is $X = 1$ and $Y=0$ Find the Pareto-optima of this economy, and express them as a function of $X_A$, the quantity of good $X$ allocated to consumer $A$.

So what I did to find those optima is to put $MRS_{XY}^A = MRS_{XY}^B$ subject to constrains $X_A + X_B \leq 1$ and $Y_A + Y_B = \sqrt{5(1 - X_A - X_B)}$. But it seems to me that I got to few equations to actually solve the problem. What I find is, putting $X = 1- X_A - X_B$:
\begin{align*}
Y_A = \frac{\sqrt{5X}}{1 - X} \cdot X_A 
\end{align*}
but I can't seem to be able to go any further... In the solution, it reads:
\begin{align*}
\frac{Y_A}{X_A} = \frac{Y_B}{X_B} (=\frac{Y_A + Y_B}{X_A + X_B}) = \frac{\sqrt{5}}{2\sqrt{X}}
\end{align*}
without explaining how to derive the last equality.
The first equality, as you've already found, equates $A$'s and $B$'s MRSs. The last equality comes from setting the marginal rate of transformation $MRT_{XY}$ between $X$ and $Y$ to the common MRS. For a production function with a single input and a single output, $MRT_{XY}$ is equal to $MP_X$, the marginal product of the input good $X$. That is:

$$ MRT_{XY} = \frac{1}{2}\sqrt\frac{5}{X}. $$

For details, see Mas-Colell, Whinston and Green (1995), section 16.F.