The sum of two numbers is 9. The sum of their reciprocals is . Find the numbers.

Explanation

Let the required natural numbers be x and `(9-x)`. Then, `(1)/(x)+(1)/((9-x))=(1)/(2)`
`implies" "((9-x)+x)/(x(9-x))=(1)/(2)`
`implies" "x(9-x)=18" "["by cross multiplication"]`
`implies" "x^(2)-9x+18=0impliesx^(2)-6x-3x+18=0`
`implies" "x(x-6)-3(x-6)=0implies(x-6)(x-3)=0`
`implies" "x-6=0" or "x-3=0impliesx=6" or "x=3.`
Hence, the required natural numbers are 6 and 3.