Two numbers are in the ratio $5:8$. If $4$ is added to each, the ratio becomes $2:3$. Find the smaller number.

Let the two numbers be $5x$ and $8x$.
According to the question:

$$\frac{5x+4}{8x+4}=\frac{2}{3}$$

Cross-multiplying:

$$3(5x+4)=2(8x+4)$$

$$15x+12=16x+8$$

$$16x-15x=12-8$$

$$x=4$$

The numbers are:
Smaller number $=5x=5(4)=20$
Larger number $=8x=8(4)=32$

The smaller number is $20$.