Find the third proportional to the numbers 16 and 24.

The correct option is 36.

Understanding Proportionality:

When three numbers $a$, $b$, and $c$ are in continued proportion, the ratio of the first number to the second is equal to the ratio of the second number to the third.

This can be written as:

$a:b=b:c$

Or in fractional form:

$\frac{a}{b}=\frac{b}{c}$

Here, $c$ is called the third proportional to $a$ and $b$.

Step-by-Step Calculation:

We are given:

First number ($a$) = 16

Second number ($b$) = 24

Let the third proportional ($c$) be $x$.

Setting up the proportion:

$\frac{16}{24}=\frac{24}{x}$

Cross-multiplying to solve for $x$:

$16\times x=24\times 24$

$16x=576$

Now, divide both sides by 16:

$x=\frac{576}{16}$

$x=36$

Thus, the third proportional to 16 and 24 is 36.