Oxide and chloride of a metal
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.