Sum of two natural numbers is 8 and the difference of their reciprocal is find the numbers. 2 15

Given Data:

Sum of two natural number is 8.

The sum of their reciprocals is 8/15.

Concept Used:

(a + b)2 = (a – b)2 + 4ab

Calculation:

Let the number be a and b.

a + b = 8  …(i)

1/a + 1/b = 8/15

⇒ (b + a)/ab = 8/15

⇒ 8/ab = 8/15

⇒ ab = 15

(a + b)2 = (a – b)2 + 4ab

⇒ 82 = (a – b)2 + 4 × 15

⇒ 64 = (a – b)2 + 60

⇒ (a – b)2 = 4

⇒ a – b = 2    ….(ii)

Adding (i) and (ii), 2a = 10

⇒ a = 5 and b = 3

∴ The two natural numbers are 5 and 3.

Shortcut:

Directly we can get the value from option 4, as a + b = 8 and ab = 15.

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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.