The sum of two numbers is $15$. If the sum of their reciprocals is $\frac{3}{10}$, then find the two numbers?

Let the two numbers be $x$ and $y$.Given:
$$x+y=15\text{— (Eq. 1)}$$
The sum of their reciprocals is given by:
$$\frac{1}{x}+\frac{1}{y}=\frac{3}{10}$$
$$\frac{x+y}{xy}=\frac{3}{10}$$
Substitute $x+y=15$ into the numerator:
$$\frac{15}{xy}=\frac{3}{10}$$
$$3xy=150⟹xy=50\text{— (Eq. 2)}$$
We need two numbers whose sum is $15$ and product is $50$.Solving the quadratic equation or checking factors reveals the numbers are $5$ and $10$ ($5+10=15$, $5\times 10=50$).