The difference of two numbers is 5 and the difference of their reciprocals is `1/10` Find the numbers.

Explanation

Let the required natural numbers be x and `(x-5)`. Then,
`xgtx-5implies(1)/(x)lt(1)/(x-5)implies(1)/(x-5)gt(1)/(x).`
`:." "(1)/(x-5)-(1)/(x)=(1)/(10)`
`implies" "(x-(x-5))/((x-5)x)=(1)/(10)implies(5)/((x-5)x)=(1)/(10)" "["by cross multiplication"]`
`implies" "(x-5)x=50impliesx^(2)-5x-50=0`
`implies" "x^(2)-10x+5x-50=0impliesx(x-10)+5(x-10)=0`
`implies" "(x-10)(x+5)=0`
`implies" "x-10=0" or "x+5=0`
`implies" "x=10" or "x=-5`
`implies" "x=10" "[:." "-5" is not a natural number"]`
Hence, the required natural numbers are 10 and 5.