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. India’s #1 Learning Platform Start Complete Exam Preparation
Video Lessons & PDF Notes Trusted by 2,78,24,673+ Students 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. |