Question

he difference of two natural numbers is 3 and the difference of their reciprocals is 3 /28 . Find the numbers.

Answer 1

Hey there !

Solution:

Let the two numbers be x and y where x > y.

According to the question,

x - y = 3 => Equation 1

$\dfrac{1}{y} - \dfrac{1}{x} = \dfrac{3}{28}$  => Equation 2

Solving Equation 2 we get,

$Taking \: LCM \: we \: get, \\ \\ \\ \implies \dfrac{ x - y }{xy} = \dfrac{3}{28} \\ \\ \\ We \: \: know \:\: that \: \: x - y = 3 \: \: from \: \: Equation \: 1. \\ Substituting \: \: in\:\: the\:\: above \: \: equation \: \: get, \\ \\ \dfrac{3}{xy} = \dfrac{3}{28} \\ \\ Cross \:\: Multiplying \:\: we \:\: get, \\ \\ 3 \times 28 = 3 \times xy \\ \\ \implies 28 = xy \\ \\ \implies \dfrac{28}{x} = y  => Equation \: 3$

$\text{ Substituting Equation 3 in Equation 1 we get,} \\ \\ \implies x - \dfrac{28}{x} = 3 \\ \\ \text{ Taking LCM we get, } \\ \\ \implies \dfrac{ x^2 - 28}{x} = 3 \\ \\ \implies x^2 - 28 = 3x \\ \\ \implies x^2 - 3x - 28= 0 \\ \\ \implies x^2 - 7x + 4x - 28 = 0 \\ \\ \implies x ( x - 7 ) + 4 ( x - 7 ) = 0 \\ \\ \implies x = -4, 7 \\ \\ \text{ But x is a natural number. Hence x is 7}$

$\implies x - y = 3 \\ \\ \implies 7 - y = 3 \\ \\ \implies 7 - 3 = y \\ \\ \implies y = 4 \\ \\ \boxed{ Hence \: the \: two \: numbers \: are \: 7 \: and \: 4}$

Hope my answer helped !