Question

The sum of two numbers is 1000 and the difference between their squares is 256,000. Find the numbers.

Answer 1

Let the numbers are x and y. One of them must be greater than or equal to the other. Let us assume that x is greater than or equal to y.

The sum of the two numbers is 1000. Thus, we have `x + y = 1000`

The difference between the squares of the two numbers is 256000. Thus, we have

`x^2 - y^2 = 256000`

`⇒ (x + y)(x - y) = 256000`

`⇒ 1000(x - y) = 256000/1000`

`⇒ x - y = 256`

So, we have two equations 

` x + y = 1000`

` x - y = 256`

Here x and y are unknowns. We have to solve the above equations for x and y.

Adding the two equations, we have

`(x + y) + (x - y) = 1000 + 256`

`⇒ x + y + x - y = 1256`

`⇒ 2x = 1256/2`

`⇒ x = 628`

Substituting the value of x in the first equation, we have

`628 + y = 1000`

`⇒ y = 1000 - 628`

`⇒ y = 372`

Hence, the numbers are 628 and 372.