The ratio of two numbers is $5:8$ and their difference is $69$. Find the numbers.

Let the two numbers be $5x$ and $8x$.
The difference between the two numbers is:

$$8x-5x=3x$$

Given that the difference is $69$:

$$3x=69⟹x=23$$

Now, calculate the two numbers:

$$\text{First number}=5\times 23=115$$

$$\text{Second number}=8\times 23=184$$

Therefore, the numbers are $115$ and $184$.