If $x$ is subtracted from each of $17,14,22$ and $18$, then these numbers are in proportion, in this order. What is the mean proportion between $9x$ and $5x$?

According to the problem, after subtracting $x$ from each number, they form a proportion:
$$\frac{17-x}{14-x}=\frac{22-x}{18-x}$$
Cross-multiplying to solve for $x$:
$$(17-x)(18-x)=(22-x)(14-x)$$
$$306-17x-18x+x^2=308-22x-14x+x^2$$
$$306-35x=308-36x$$
$$-35x+36x=308-306$$
$$x=2$$
Now we need to find the mean proportion between $9x$ and $5x$:
$$\text{First value}=9x=9\times 2=18$$
$$\text{Second value}=5x=5\times 2=10$$
The mean proportion between two numbers $a$ and $b$ is given by $\sqrt{ab}$:
$$\text{Mean Proportion}=\sqrt{18\times 10}=\sqrt{180}=\sqrt{36\times 5}=6\sqrt{5}$$