Question

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

Answer 1

Let one number be x. The other number be y.

Given

x+y=8

x=8-y ---------------(1)

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

By simplyfying this we get

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

From (1) we get.

$\frac{8}{(8-y)y}=\frac{8}{15}$

$\frac{8}{8y-y^2}=\frac{8}{15}$

$\frac{1}{8y-y^2}=\frac{1}{15}$​​​​​​​

$8y-y^2=15$​​​​​​​

$y^2-8y+15=0$​​​​​​​

$(y-3)(y-5)=0$​​​​​​​

$y=3or5$​​​​​​​

We know that x+y=8

When y=5, x=3

When y = 3, x = 5

Thus, the two numbers are 3 and 5