Correct Answer: 4/5

Solution : The correct answer is (a) $4 / 5$

To find the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$, we need to calculate the ratio of the marginal utility of $\mathrm{X}$ to the marginal utility of $\mathrm{Y}$.

The utility function is $\mathrm{U}=\mathrm{XY}$.
To find the marginal utility of $\mathrm{X}$, we differentiate the utility function with respect to $\mathrm{X}$ :
$
\partial \mathrm{U} / \partial \mathrm{X}=\mathrm{Y}
$

To find the marginal utility of $\mathrm{Y}$, we differentiate the utility function with respect to $\mathrm{Y}$ :
$
\partial \mathrm{U} / \partial \mathrm{Y}=\mathrm{X}
$

Now we can calculate the MRS:
$
\begin{gathered}
\mathrm{MRS}=(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\
\quad=\mathrm{Y} / \mathrm{X}
\end{gathered}
$

Substituting $\mathrm{X}=5$ and $\mathrm{Y}=4$ :
$
\operatorname{MRS}=4 / 5
$

Therefore, the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ is $4 / 5$.