Two numbers are in the ratio of $3:7$. If $11$ is subtracted from each, the ratio between the numbers becomes $7:20$. Find the smaller number.

Let the two numbers be $3x$ and $7x$, where $x$ is a common constant multiplier.
According to the question, when $11$ is subtracted from each number, the new ratio becomes $7:20$. This can be expressed as:

$$\frac{3x-11}{7x-11}=\frac{7}{20}$$

By cross-multiplying, we get:

$$20(3x-11)=7(7x-11)$$

$$60x-220=49x-77$$

Rearranging the terms to solve for $x$:

$$60x-49x=220-77$$

$$11x=143$$

$$x=\frac{143}{11}$$

$$x=13$$

The smaller number is represented by $3x$:

$$\text{Smaller number}=3\times 13=39$$

Therefore, the smaller number is $39$.