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Question
The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.
Solution
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.
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