Answer:
12 and 6
Step-by-step explanation:
Let the numbers be x and y
Sum of two numbers is 18.
⇒ x + y = 18 ----------------(II)
[tex]\sf \text{Reciprocal of x = $\dfrac{1}{x}$}\\\\ \text{Reciprocal of y =$\dfrac{1}{y}$}[/tex]
Sum of reciprocals is 1/4
[tex]\sf \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{4}\\\\ \dfrac{1*y}{x*y}+\dfrac{1*x}{y*x}=\dfrac{1}{4}\\\\\dfrac{y}{xy}+\dfrac{x}{xy}=\dfrac{1}{4}\\[/tex]
[tex]\sf \dfrac{x +y}{xy}=\dfrac{1}{4}[/tex]
Plugin (x +y = 18)
[tex]\dfrac{18}{xy}=\dfrac{1}{4}\\\\[/tex]
Cross multiply,
18*4 = 1*(xy)
xy = 72
[tex]x = \dfrac{72}{y}[/tex]
Substitute x value in equation (I)
[tex]\dfrac{72}{y}+y=18\\\\\text{Multiply the entire equation by y}\\\\\\ 72 + y^2=18y\\[/tex]
y² - 18y + 72 = 0
y² - 12y - 6y + 72 = 0
y(y -12) - 6(y - 12) = 0
(y - 12)(y -6 ) = 0
y - 12 = 0 ; y - 6 = 0
y = 12 ; y = 6
