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

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

$x^{2}-y^{2}=256000$

$\Rightarrow(x+y)(x-y)=256000$

$\Rightarrow 1000(x-y)=256000$

$\Rightarrow x-y=\frac{256000}{1000}$

$\Rightarrow x-y=256$

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

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

$\Rightarrow x+y+x-y=1256$

$\Rightarrow 2 x=1256$

$\Rightarrow x=\frac{1256}{2}$

$\Rightarrow x=628$

Substituting the value of in the first equation, we have

$628+y=1000$

$\Rightarrow y=1000-628$

$\Rightarrow y=372$

Hence, the numbers are 628 and 372.