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

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Step 1: Let one number be $x$, then the other number $= 15 - x$. Given: $\frac{1}{x} + \frac{1}{15 - x} = \frac{3}{10}$.
Step 2: $\frac{15 - x + x}{x(15 - x)} = \frac{3}{10} \Rightarrow \frac{15}{x(15 - x)} = \frac{3}{10} \Rightarrow 150 = 3x(15 - x)$.
Step 3: $3x^2 - 45x + 150 = 0 \Rightarrow x^2 - 15x + 50 = 0 \Rightarrow (x - 10)(x - 5) = 0$.
Step 4: $x = 10$ or $x = 5$. The two numbers are $10$ and $5$.

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