Respuesta :

Let the smaller number is x.

So the bigger number is x+4.

Reciprocal of smaller number is [tex] \frac{1}{x} [/tex] and reciprocal of larger number is [tex] \frac{1}{x+4} [/tex]

SO our equation is
[tex] \frac{1}{x} + 2* \frac{1}{x+4} = \frac{5}{6} [/tex]
[tex] \frac{1}{x} + \frac{2}{x+4} = \frac{5}{6} [/tex]
[tex] \frac{x+4 + 2x}{x(x+4)} = \frac{5}{6} [/tex]
[tex] \frac{3x+4}{x(x+4)} = \frac{5}{6} [/tex]

by cross multiplication rule
[tex]6(3x+4) = 5(x(x+4))[/tex]
[tex]18x+ 24= 5x^2 + 20x[/tex]
[tex]5x^2 + 20x - 18x - 24 = 0[/tex]
[tex]5x^2 +2x - 24 = 0[/tex]

Now we can solve it by factorization.

As we can see -24*5 = -120.

So we can write -120 as 12 * (-10) = -120 because 12 - 10 = 2

SO we can write expression as
[tex]5x^2 + 12x - 10x - 24 = 0[/tex]
[tex]x(5x+12) - 2(5x+12) = 0[/tex]
[tex](5x+12)(x-2) = 0[/tex]
So x = 2 or x = [tex] \frac{-12}{5} [/tex]

But we have to choose positive integer. SO we will choose x = 2.

SO the smaller number is 2 and bigger number is 6.

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