Let x be the larger and y be the smaller.

x² + y² = 25(x + y) = 50(x - y)

25(x + y) = 50(x - y) 

Divide both sides by 25

x + y = 2(x - y)

x + y = 2x - 2y

3y = x

Substitute 3y for x in:

x² + y² = 25(x + y)

(3y)² + y² = 25(3y + y)

9y² + y² = 75y + 25y

10y² = 100y

 y² = 10y

y² - 10y = 0

y(y - 10) = 0

y = 0,  y - 10 = 0
             y = 10

0 is not a natural number, so y = 10 

and since

3y = x
 
3(10) = x

30 = x

So the two natural numbers are 30 and 10.

Checking:

The sum of their squares is 30²+10² = 900+100 = 1000

Their sum is 30+10 = 40

25 times 40 = 1000.  That checks.

Their difference is 30-10 = 20

50 times 20 = 1000.  That checks and 
so the answers are indeed 30 and 10.

Edwin