Sum of two natural numbers is 8 and the difference of their reciprocal is `2/15`. Find the numbers.

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#### Solution

Let the two natural numbers be x and (8 – x). Then, we have

`1/x - 1/(8 -x) = 2/15`

`=> (8-x-x)/(x(8 - x)) = 2/15`

`=> (8-2x)/(x(8 -x)) = 2/15`

`=> (4-x)/(x(8-4)) = 1/15`

`=> 15(4-x) = x(8 - x)`

`=> 60 - 15x - 8x - x^2`

`=> x^2 - 15x - 8x + 60 = 0`

`=> x^2 - 23x + 60 = 0`

`=> x^2 - 20x - 3x + 60 = 0`

`=> (x - 3)(x - 20) = 0`

`=> (x - 3) = 0 or (x - 20) = 0`

`=> x = 3 or x = 20`

Since sum of two natural numbers is 8, x cannot be equal to 20

`=> x = 3 and 8 - x = 8 - 3 = 5`

Hence, required natural numbers are 3 and 5.

Concept: Quadratic Equations

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