The sum of two natural numbers is $8$. Determine the numbers if the sum of their reciprocals is $\frac{8}{15}$.

Let the two numbers be $a$ and $b$.Given:
$$a+b=8$$
$$\frac{1}{a}+\frac{1}{b}=\frac{8}{15}⟹\frac{a+b}{ab}=\frac{8}{15}$$
Substitute $a+b=8$ into the equation:
$$\frac{8}{ab}=\frac{8}{15}⟹ab=15$$
We need two numbers whose sum is $8$ and product is $15$. By factoring or inspection, the numbers are clearly $3$ and $5$.