Alg3.2.21

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

Let large number be $x\,$

Then,the small number is $8-x\,$

Given $\frac{1}{x}+\frac{1}{8-x}=\frac{8}{15}\,$

$\frac{8-x+x}{x(8-x)}=\frac{8}{15}\,$

Cross multiplying,we get

$15=x(8-x)\,$

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

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

$x=5,3\,$

The numbers are $5,3\,$ and $3,5\,$

Therefore,the large number is $5\,$ and the small number is $3\,$