Question

Two numbers have a sum of 4 and the sum of their reciprocal is 8. Find the number.

Answer 1

Answer :

2 + √7⁄2 and 2 - √7⁄2

Solution :

Let the two numbers be x and y .

Now ,

According to the question , the two numbers have a sum of 4 .

Thus ,

x + y = 4 --------(1)

Also ,

It is given that , sum of the reciprocals of the two numbers is 8 .

Thus ,

1/x + 1/y = 8 --------(2)

Considering eq-(2) , we have ;

=> 1/x + 1/y = 8

=> (y + x)/xy = 8

=> x + y = 8xy

=> 4 = 8xy

=> 1 = 2xy

=> 1 = 2x(4 - x) { using eq-(1) , y = 4 - x }

=> 1 = 8x - 2x²

=> 2x² - 8x + 1 = 0

Now ,

Let's solve this equation using completing square method .

Thus ,

=> 2x² - 8x + 1 = 0

=> 2(x² - 4x + ½) = 0

=> x² - 4x + ½ = 0

=> x² - 4x + 2² - 2² + ½ = 0

=> x² - 2•x•2 + 2² = 2² - ½

=> (x - 2)² = 4 - ½

=> (x - 2)² = (8 - 1)/2

=> (x - 2)² = 7/2

=> x - 2 = ± √7⁄2

=> x = 2 ± √7⁄2

Now ,

Using eq-(1) , we have ;

=> x + y = 4

=> y = 4 - x

• If x = 2 + √7⁄2 , then ;

=> y = 4 - (2 + √7⁄2)

=> y = 4 - 2 - √7⁄2

=> y = 2 - √7⁄2

• If x = 2 - √7⁄2 , then ;

=> y = 4 - (2 - √7⁄2)

=> y = 4 - 2 + √7⁄2

=> y = 2 + √7⁄2

In both the cases , we got the same pair of numbers . ie ; 2 + √7⁄2 and 2 - √7⁄2 .

Hence ,

Required numbers are :

2 + √7⁄2 and 2 - √7⁄2 .