Question

the sum of reciprocals of two consecutive odd natural numbers is 12/35. Find those numbers

Answer 1

$\textbf{Answer:}\\\\ 5\:\: and\:\: 7\\\text{}\\\textbf{Step by Step Explanation:}\\\\\text{Let the two consecutive odd natural numbers be x and x + 2}\\\\\implies \text{The reciprocals are:} \:\:\dfrac{1}{x} \:\: and \:\: \dfrac{1}{x+2}\\\\\\\star\: \text{Given that, Sum of reciprocals is} \: \dfrac{12}{35}\\\\\\\implies \dfrac{1}{x} + \dfrac{1}{x+2} = \dfrac{12}{35}$

$\text{Taking LCM we get,}\\\\\implies \dfrac{ x + 2 + x }{ x ( x + 2 )} = \dfrac{12}{35}\\\\\\\implies \dfrac{ 2x + 2 }{ x^2 + 2x } = \dfrac{12}{35}\\\\\text{Cross multiplying we get,}\\\\\implies 35 ( 2x + 2 ) = 12 ( x^2 + 2x )\\\\\implies 70x + 70 = 12x^2 + 24x\\\\\implies 12x^2 + 24x - 70x - 70 = 0\\\\\implies 12x^2 -46x - 70 = 0\\\\\text{Dividing by 2 we get,}\\\\\implies 6x^2 - 23x - 35\\\\\implies ( 6x + 7 )( x - 5 )\\\\\implies x = \dfrac{-7}{6} \:\:and\:\:5$

$\text{Since x is an odd number, we neglect -7/6 and consider only 5 as the value}\\\\\underline{\textbf{Hence value of x and x+2 are 5 and 7.}}$