Let the atomic weight of element $X$ be $x$ and that of element $Y$ be $y$. The molar mass of a substance is the mass of one mole of that substance. For $XY_2$: Molar mass $= x + 2y$ We are given that $0.1 \text{ mole}$ of $XY_2$ weighs $10 \text{ g}$. Therefore, the mass of $1 \text{ mole}$ of $XY_2 = \frac{10}{0.1} = 100 \text{ g}$. So, $x + 2y = 100$ --- (Equation 1)

For $X_3Y_2$: Molar mass $= 3x + 2y$ We are given that $0.05 \text{ mole}$ of $X_3Y_2$ weighs $9 \text{ g}$. Therefore, the mass of $1 \text{ mole}$ of $X_3Y_2 = \frac{9}{0.05} = 180 \text{ g}$. So, $3x + 2y = 180$ --- (Equation 2)

Subtracting Equation 1 from Equation 2: $(3x + 2y) - (x + 2y) = 180 - 100$ $2x = 80 \implies x = 40$

Substituting the value of $x$ in Equation 1: $40 + 2y = 100 \implies 2y = 60 \implies y = 30$

Therefore, the atomic weights of $X$ and $Y$ are $40$ and $30$ respectively.