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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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Let one number be $x \Rightarrow \text{another number} = 15 - x$
Therefore, $\frac{1}{x} + \frac{1}{15-x} = \frac{3}{10}$
$\frac{15-x + x}{x(15 - x)} = \frac{3}{10} \Rightarrow 150 = 3x(15 - x)$
$3x^2 - 45x + 150 = 0$
$x^2 - 15x + 50 = 0 \Rightarrow (x - 10)(x - 5) = 0$
$\Rightarrow x = 10, 5$
Numbers are $10, 5$ or $5, 10$
Therefore, $\frac{1}{x} + \frac{1}{15-x} = \frac{3}{10}$
$\frac{15-x + x}{x(15 - x)} = \frac{3}{10} \Rightarrow 150 = 3x(15 - x)$
$3x^2 - 45x + 150 = 0$
$x^2 - 15x + 50 = 0 \Rightarrow (x - 10)(x - 5) = 0$
$\Rightarrow x = 10, 5$
Numbers are $10, 5$ or $5, 10$