Question

The sum of two numbers is 18 and the sum of their reciprocals is Find the numbers. 1/4
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Answer 1

12 and 6

Explanation:

Let the numbers be x and y

Sum of two numbers is 18.

⇒ x + y = 18 ----------------(II)

`\sf \text{Reciprocal of x = $(1)/(x)$}\\\\ \text{Reciprocal of y =$(1)/(y)$}`

Sum of reciprocals is 1/4

`\sf (1)/(x)+(1)/(y)=(1)/(4)\\\\ (1*y)/(x*y)+(1*x)/(y*x)=(1)/(4)\\\\(y)/(xy)+(x)/(xy)=(1)/(4)\\`

`\sf (x +y)/(xy)=(1)/(4)`

Plugin (x +y = 18)

`(18)/(xy)=(1)/(4)\\\\`

Cross multiply,

18*4 = 1*(xy)

xy = 72

`x = (72)/(y)`

Substitute x value in equation (I)

`(72)/(y)+y=18\\\\\text{Multiply the entire equation by y}\\\\\\ 72 + y^2=18y\\`

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

The numbers are 12 , 6