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.