y is inversely proportional to the square root of x. It is given that $y=5$ for a certain value of $x$. Find the value of $y$ when $x$ is increased by 300%. Answer $y$ = ...

1. Write the inverse proportional relationship.

Since $y$ is inversely proportional to the square root of $x$,
$y\propto \frac{1}{\sqrt{x}}$
so
$y=\frac{k}{\sqrt{x}}$
for some constant $k$.

2. Find how the square root changes when $x$ is increased by 300%.

Increasing $x$ by $300\%$ gives
$x_{\text{new}}=x+300\%\text{ of }x=4x.$
Hence
$\sqrt{x_{\text{new}}}=\sqrt{4x}=2\sqrt{x}.$

3. Use the inverse relationship to find the new value of $y$.

Originally,
$y=\frac{k}{\sqrt{x}}=5.$
When $x$ becomes $4x$,
$y_{\text{new}}=\frac{k}{\sqrt{4x}}=\frac{k}{2\sqrt{x}}=\frac{1}{2}\left(\frac{k}{\sqrt{x}}\right)=\frac{1}{2}(5)=2.5.$