Question

Divide two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.

Answer 1

Let the two natural numbers be x and y. According to the question:  

`x^2+y^2=25(x+y)`                              ..................(1) 

x^2+y^2=50(x-y)                                 ...................(2) 

From (1) and (2), we get: 

`25(x+y)=50(x-y)` 

⇒`x+y=2(x-y)` 

⇒`x+y=2x-2y` 

⇒`y+2y=2x-x` 

⇒`3y=x` 

From (2) and (3), we get: 

`(3y)^2+y^2=50(3y-y)` 

⇒`9y^2+y^2=100y` 

⇒`10y^2=100y` 

⇒`y=10`  

From (3), we have: 

` 3xx10=x` 

⇒`30=x`  

Hence, the two natural numbers are 30 and 10