Question

The sum of two numbers is 8. Determine the numbers, if the sum of their reciprocals is $\frac{8}{15}$.

Answer 1

Let the two numbers be x and y. Then,

$x+y=8$

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

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

$xy=15$

Now, $(x-y)=\sqrt{(x+y{)}^2-4xy}=\sqrt{64-60}=\pm 2$

Therefore,

$x+y=8andx-y=2$ or $x+y=8andx-y=-2$

Solving these, we get,

$x=5andy=3$ or $x=3andy=5$

Therefore, the required numbers are 3 and 5.