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I found this problem:

The oxide of a metal contains 60.0% metal and the chloride of the same metal contains 25.26% metal (in mass percents). What are the possible formulas for the oxide and the chloride?

I looked up the solution and it states that the results are $\ce{MgO}$, $\ce{MgCl2}$, $\ce{TiO2}$ and $\ce{TiCl4}$, but I don't understand how they got it. How should I approach this kind of problem?

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Because an oxygen ion is $\ce{O^2-}$ and a chlorine ion $\ce{Cl^-}$, you know that the salts have to be of the form $\ce{M_xO_y}$ and $\ce{M_xCl_{2y}}$.

Let $o$ be the atomic mass of oxygen, $c$ of chlorine and $m$ of the unknown metal M. The problem states that

$$0.6(xm+yo)=xm$$

which implies

$$0.6yo = 0.4xm \implies m = \frac{3y}{2x}o$$

If you recall some formulas for salts, you know that $x$ can only be 1 or 2 (e.g. $\ce{M_5O_3}$ makes no sense). This means that $m$ is a multiple of $\frac34o$ which is (approximately) 12. So we only need to check the elements with an atomic weight which is a multiple of 12:

  • 12: Carbon (C) isn't a metal
  • 24: Magnesium (Mg) is, and you can check the oxide/chloride satisfy the equations
  • 36: n/a
  • 48: Titanium (Ti) checks out
  • 60: n/a
  • 72: n/a
  • 84: Krypton (Kr) isn't a metal

and so on. Even if the atomic weight corresponds to a metal, you can verify the value of $y$ doesn't match with the oxide that's formed.

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