Question

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

Answer 1

Given: The sum of two natural numbers is 8, if the sum of their reciprocals is $\frac{8}{15}$

To find the numbers

Sol: let the numbers be $x$ and $y$

$x+y=8.......\left(1\right)$

$\frac{1}{x}+\frac{1}{y}=\frac{8}{15}................\left(2\right)$

i.e. $\frac{(x+y)}{\left(xy\right)}=\frac{8}{15}$

i.e $xy=15$ .......(put $x+y=8$ from 1)

i.e $y=\frac{15}{x}........\left(3\right)$

Put $\left(3\right)$ in $\left(1\right)$ to get

$x^2-8x+15=0⟹x^2-5x-3x+15=0⟹x(x-5)-3(x-5)=0⟹(x-3)(x-5)=0⟹x=3,x=5$

Hence the numbers are $3$ and $5$