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

Let the two natural numbers be $x$ and $y$.From the given conditions:$x+y=8$$\frac{1}{x}+\frac{1}{y}=\frac{8}{15}$ Simplifying the reciprocal equation:
$$\frac{x+y}{xy}=\frac{8}{15}$$
Substitute $x+y=8$ into the equation:
$$\frac{8}{xy}=\frac{8}{15}⟹xy=15$$
We need two numbers whose sum is $8$ and product is $15$. We can form a quadratic equation or solve this by inspection:The factors of $15$ that add up to $8$ are $3$ and $5$ ($3\times 5=15$ and $3+5=8$).Thus, the numbers are $3$ and $5$.