Solving inverse proportion with a square root factor

Key concept: This is an inverse variation problem with a square-root factor in the denominator. The constant of proportionality can be found from one known pair, then reused for the new value of x.

Step by step

Identify the proportional relationship

We are given

$y \propto \frac{1}{\sqrt{x+1}}.$

That means we can write

$y = \frac{k}{\sqrt{x+1}}$

for some constant $k$.

The first data point is $x=8$ and $y=5$. Substitute these values:

$5 = \frac{k}{\sqrt{8+1}} = \frac{k}{3}$

So

$k=15.$

Find the new value of y

Now substitute $x=24$:

$y = \frac{15}{\sqrt{24+1}} = \frac{15}{\sqrt{25}} = \frac{15}{5}=3.$

Final answer

$\boxed{3}$

Key idea to remember

When a variable is proportional to the reciprocal of a square root, the first step is always to convert the proportion into an equation with a constant $k$. Once $k$ is found from one known pair, any other value can be computed by direct substitution. This is much safer than trying to compare the two x-values mentally, because the square root changes the scaling in a nonlinear way.

Pitfall alert

A frequent error is to forget the $+1$ inside the square root and write $\sqrt{x}$ instead of $\sqrt{x+1}$. Another common mistake is to treat the relationship as direct proportion and multiply by the ratio of x-values, which does not work here because the square root changes the pattern. Some students also compute $\sqrt{24+1}$ incorrectly or simplify before substituting the given point. The best habit is to find the constant first, then substitute the new x-value exactly as written.

Try different conditions

If the question changed to $y \propto \frac{1}{\sqrt{x-3}}$ with $x=12$ giving $y=4$, the method would be the same. You would write $y=\frac{k}{\sqrt{x-3}}$, use the known point to find $k$, and then evaluate at the new x-value. For example, if the new x were 28, then you would calculate $y=\frac{k}{\sqrt{25}}=\frac{k}{5}$. Any version with a shifted square root, such as $x+5$ or $x-3$, still follows the same constant-of-proportionality approach.

Further reading

constant of proportionality, inverse variation, square root function