If $X_i \backsim \operatorname{Gamma}(\alpha,\beta)$ where $\alpha$ is the shape and $\beta$ is the scale parameter then
$$ \mathbb{E}\left[ X_i \right] = \alpha \beta \quad \quad \mbox{and} \quad \quad \mathbb{V}\mbox{ar}\left[ X_i \right] = \alpha \beta^2 $$
From the properties of the gamma distribution
$$ \overline{X} \backsim \operatorname{Gamma}\left(n \alpha, \beta/n \right) $$
which means
$$ \mathbb{E}\left[ \bar{X}\right] = \alpha\beta \quad \quad \mbox{and} \quad \quad \mathbb{V}\mbox{ar}\left[\bar{X}\right] = \alpha \beta^2/n $$
Then for
$$ Y_i | X_i \backsim \operatorname{Gamma}\left(\alpha, \beta X_i \right) $$
$$ \mathbb{E}\left[ Y_i | X_i \right] =\alpha \beta X_i \quad \quad \mbox{and} \quad \quad \mathbb{V}\mbox{ar}\left[Y_i | X_i \right] = \alpha (\beta X_i )^2 $$
From the law of total expectation we have
\begin{equation}
\begin{split}
\mathbb{E}\left[\frac{\bar{Y}}{\bar{X}}\right]&= \left.
\mathbb{E}\left[ \mathbb{E}\left[ \frac{\bar{Y}}{\bar{X}} \right| X_1, \ldots, X_n \right] \right] \\
&=
\mathbb{E}\left[ \frac{1}{\bar{X}} \frac{1}{n} \sum_{i=1}^n\mathbb{E} [ Y_i \big| X_1, \ldots, X_n ] \right] \\
&=
\mathbb{E}\left[ \frac{1}{\bar{X}} \frac{1}{n} \sum_{i=1}^n\mathbb{E}[ Y_i \big| X_i ] \right] \\
& = \mathbb{E}\left[ \frac{1}{\bar{X}} \frac{1}{n} \sum_{i=1}^n \alpha \beta X_i \right] \\
& = \alpha \beta \mathbb{E}\left[ \frac{1}{\bar{X}} \frac{1}{n} \sum_{i=1}^n X_i \right] \\
& = \alpha \beta \mathbb{E}\left[ \frac{1}{\bar{X}} \bar{X} \right] \\
& = \alpha \beta \mathbb{E}\left[ 1 \right] \\
& = \alpha \beta \\
\end{split}
\end{equation}
From the law of total variance we have
\begin{equation*}
\begin{split}
\mathbb{V}\mbox{ar}\left[\frac{\bar{Y}}{\bar{X}}\right] &= \left.
\mathbb{V}\mbox{ar}\left[ \mathbb{E}\left[ \frac{\bar{Y}}{\bar{X}} \right| X_1, \ldots, X_n \right] \right] + \left. \mathbb{E}\left[ \mathbb{V}\mbox{ar} \left[ \frac{\bar{Y}}{\bar{X}} \right| X_1, \ldots, X_n \right] \right] \\
&= \left.
\mathbb{V}\mbox{ar}\left[ \frac{1}{\bar{X} } \frac{1}{n} \mathbb{E}\left[ \sum_{i=1}^nY_i\right| X_1, \ldots, X_n \right] \right] + \left. \mathbb{E}\left[ \frac{1}{\bar{X}^2 } \frac{1}{n^2} \mathbb{V}\mbox{ar} \left[ \sum_{i=1}^nY_i \right| X_1, \ldots, X_n \right] \right] \\
&=
\mathbb{V}\mbox{ar}\left[ \frac{1}{\bar{X} } \frac{1}{n} \sum_{i=1}^n \mathbb{E}\left[ Y_i\big| X_i\right] \right] + \mathbb{E}\left[ \frac{1}{\bar{X}^2 } \frac{1}{n^2} \sum_{i=1}^n \mathbb{V}\mbox{ar} [ Y_i \big| X_i] \right] \\
&= \mathbb{V}\mbox{ar}\left[ \frac{1}{\bar{X} } \frac{1}{n} \sum_{i=1}^n \alpha \beta X_i \right] +\mathbb{E}\left[ \frac{1}{\bar{X}^2 } \frac{1}{n^2} \sum_{i=1}^n \alpha (\beta X_i )^2 \right] \\
&= \alpha^2 \beta^2 \mathbb{V}\mbox{ar}\left[ \frac{1}{\bar{X} } \bar{X} \right] +\mathbb{E}\left[ \frac{n^2}{ (\sum_{i=1}^n X_i)^2 } \frac{\alpha \beta^2}{n^2} \sum_{i=1}^n X_i^2 \right] \\
&= \alpha^2 \beta^2 \mathbb{V}\mbox{ar}\left[ 1 \right] + \alpha \beta^2 \mathbb{E}\left[ \frac{1}{ (\sum_{i=1}^n X_i)^2 } \sum_{i=1}^n X_i^2 \right] \\
&= \alpha \beta^2 \mathbb{E}\left[ \frac{ \sum_{i=1}^n X_i^2 }{ (\sum_{i=1}^n X_i)^2 } \right] \\
\end{split}
\end{equation*}
Best Answer
$\text{Var}\left(cX\right)=\mathbb{E}\left[\left(cX\right)^{2}\right]-\left(\mathbb{E}\left[cX\right]\right)^{2}=\mathbb{E}\left[c^{2}X^{2}\right]-\left(c\mathbb{E}\left[X\right]\right)^{2}=c^{2}\mathbb{E}\left[X^{2}\right]-c^{2}\left(\mathbb{E}\left[X\right]\right)^{2}=c^{2}\left[\mathbb{E}\left[X^{2}\right]-\left(\mathbb{E}\left[X\right]\right)^{2}\right]=c^{2}\text{Var}\left(X\right)$