# Determining water of crystallisation in a hydrate from the mass percentages

How would I find the amount of water in a hydrate if all I was given was the mass percentage of one element in the compound? For example, a question asks:

$\ce{Zn(ClO3)2.$x$H2O}$ contains $21.5\%$ zinc. Find the value of $x$.

After thinking for some time I now know that I need to find the mass of $\ce{Zn(ClO3)2}$, and then find the mass and amount of water, and then divide by the amount of $\ce{Zn(ClO3)2}$. But I can't seem to find the mass of $\ce{Zn(ClO3)2}$.

In this case you only have one variable to solve for ($x$) and you have one piece of information (Zn being 21.5% by mass) so you have enough information to solve the question.
The molecular mass of the hydrate can be expressed in terms of $x$:
\begin{align} M(\ce{Zn(ClO3)2.xH2O}) &= M(\ce{Zn}) + 2\cdot M(\ce{Cl}) + (6+x)\cdot M(\ce{O}) + 2x\cdot M(\ce{H}) \\ &= \pu{65.38 g mol-1} + 2(\pu{35.45 g mol-1}) + (6+x)(\pu{16.00 g mol-1}) + \,\\ &\qquad 2x(\pu{1.01 g mol-1}) \\ &= (232.28 + 18.02x)~\pu{g mol-1} \end{align}
\begin{align} \frac{M(\ce{Zn})}{M(\ce{Zn(ClO3)2.xH2O})} = \frac{\pu{65.38 g mol-1}}{(232.28 + 18.02x)~\pu{g mol-1}} &= 0.215 \\ 65.38 &= 0.215(232.28 + 18.02x) \\ x &= 3.99 \approx 4 \end{align}