Question

The sum of two numbers is $8$ and the sum of their reciprocal is $\frac{8}{15}$. Find the numbers.

Answer 1

Let the numbers be x and y

$x+y=8\to 1$

$\frac{1}{x}+\frac{1}{y}=\frac{8}{15}$

$\therefore \frac{x+y}{xy}=\frac{8}{15}$

$\therefore xy=15$

$y=\frac{15}{x}$

Substituting in $1$

$x+\frac{15}{x}=8$

$x^2+15=8x$

$x^2-8x+15=0$

$(x-3)(x-5)=0$

$x=3,5$

$\therefore y=5,3$

The numbers are $5$ and $3$