Answer
$2.5$
Work Step by Step
Recall:
An inverse variation is represented by the equation $xy=k$, where $k$ is the constant of variation.
As the value of $x$ increases, the value of $y$ decreases.
The equation that models the inverse variation can be determined by finding the value of $k$:
This can be done by taking any ordered pair $(x, y)$ then substituting the values of $x$ and $y$ into the inverse variation formula $xy=k$.
Since the variation contains the point $\left(2, 5\right)$, substitute these into the formula $xy=k$ to obtain:
\begin{align*}
xy&=k\\\\
2(5)&=k\\\\
10&=k\\\\
\end{align*}
Thus, the equation of the inverse variation is $xy=10$.
To find the value of $y$ when $x=4$, substitute $4$ to $x$ in the equation above to obtain:
\begin{align*}
xy&=10\\\\
4(y)&=10\\\\
\frac{4(y)}{4}&=\frac{10}{4}\\\\
y&=\frac{5}{2}=2.5
\end{align*}
