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**Hint:**The atomic weights of $X$ and $Y$ can be calculated by determining the molecular weights of their compounds $XY_2$ and $X_2Y_3$. The molecular weight of a compound can be determined from its given number of moles and weight. The information provided in the question consists of the number of moles and weight of the compounds $XY_2$ and $X_3Y_2$ formed in a combination reaction. So, this question can be solved by applying the formula which establishes a relationship between the number of moles and the given weight of each compound.

**Complete step by step answer:**

Since the atomic weights of $X$ and $Y$ are to be calculated, let us consider the atomic weight of $X$ to be $x$, and that of $Y$ to be $y$.

According to the question,

$4X + 4Y\rightarrow XY_2 + X_3Y_2$

Given that,

Number of moles of $XY_2 = 0.1$

Weight of $XY_2$ formed in the reaction $= 10\ g$

Number of moles of $XY_2 = 0.05$

Weight of $XY_2$ formed in the reaction $= 9\ g$

The number of moles and weight of a compound are related to each other by the following formula:

$Number\ of\ moles = \dfrac{Given\ Weight}{Molecular\ Weight}$

Now, applying this formula, we can determine the molecular weight of the compounds $XY_2$ and $X_3Y_2$.

The molecular weight of $XY_2 = \dfrac{Given\ weight\ of\ XY_2}{Number\ of\ moles}$

\[ \Rightarrow Molecular\ weight\ of\ XY_2 = \dfrac{10\ g}{0.01} = 100\ g\]

The molecular weight of $X_2Y_3 = \dfrac{Given\ weight\ of\ X_2Y_3}{Number\ of\ moles}$

\[ \Rightarrow Molecular\ weight\ of\ X_2Y_3 = \dfrac{9\ g}{0.05} = 180\ g\]

Now, the molecular weight of a compound is the sum of the atomic weights of the elements forming the compound. In the case of both $XY_2$ and $X_2Y_3$, the elements forming the compounds are $X$ and $Y$.

Therefore, the molecular weight of $XY_2 = Atomic\ weight\ of\ X + 2\times Atomic\ weight\ of\ Y$

\[ \Rightarrow 100\ g = x + 2y\] …$(i)$

Molecular weight of $X_2Y_3 = 2\times Atomic\ weight\ of\ X + 3\times Atomic\ weight\ of\ Y$

\[ \Rightarrow 180\ g = 2x + 3y\] …$(ii)$

The equations $(i)$ and $(ii)$ can be solved simultaneously by subtracting equation

$(i)$ from $(ii)$ as follows:

$3x + 2y = 180$ …$(ii)$

$x + 2y = 100$ ….$(i)$

$3x – x = 180 – 100$ [$+2y$ and $-2y$ get cancelled]

Putting the value of $x$ in equation in $(i)$, we get;

$40 + 2y = 100$

\[ \Rightarrow 2y = 100-40 = 60\]

\[ \Rightarrow y = \dfrac{60}{2} = 30\]

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

**So, the correct answer is Option A.**

**Note:**Please take note that since two unknown values are to be found, two linear equations must be formed.

Do not forget to multiply the atomic weight of each element by the number of atoms of that particular element present in the compound, while calculating the molecular weight of the compound. For example- In $XY_2$, the atomic weight of $Y$ must be multiplied by its respective number of atoms, which is $2$. In $X_2Y_3$, the atomic weight of $X$ must be multiplied by its number of atoms, $2$, and the atomic weight of $Y$ must be multiplied by its number of atoms, $3$.

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