Question

A positive integer is twice another. the sum of the reciprocals of the two positive integers is 3/10. find the two integers.

Answer 1

Let's call the integers $x$ and $y$. The first information (one is twice the other) translates as $x = 2y$.

The second information about the sum of reciprocals can be written as

$\dfrac{1}{x}+\dfrac{1}{y} = \dfrac{3}{10}$

But since we know that $x = 2y$, we can rewrite this expression as

$\dfrac{1}{2y}+\dfrac{1}{y} = \dfrac{3}{2y} = \dfrac{3}{10}$

Since the numerators of both fractions equal 3, then the denominators must be equal as well. So, we have

$2y = 10 \implies y = 5$

And since we know that $x = 2y = 2\cdot 5 = 10$.