Let $a$ be a positive number and $b=\frac{4}{a}.$ What is the mean proportional of $a$ and $b$?

The correct answer is 2.

To find the mean proportional of two numbers, we use the definition of mean proportional. For two positive numbers, $a$ and $b$, their mean proportional (say, $x$) is given by the geometric mean:
$x=\sqrt{a·b}$

We are given that:
$b=\frac{4}{a}$

Substitute the expression for $b$ into the formula for the mean proportional:
$x=\sqrt{a·\left(\frac{4}{a}\right)}$

Since $a$ is a positive number, we can cancel $a$ in the numerator and the denominator:
$x=\sqrt{4}$

Simplifying the square root gives:
$x=2$

Thus, the mean proportional of $a$ and $b$ is 2.