Two numbers are in the ratio $5:7$. If the larger number is $21$, what is the smaller number?

Let the two numbers be $5x$ and $7x$, where $x$ is a common constant multiplier.
Given that the larger number corresponds to the higher ratio component ($7x$):

$$7x=21$$

$$x=\frac{21}{7}=3$$

Now, calculate the smaller number:

$$\text{Smaller number}=5x=5\times 3=15$$