To not bury the lead, the original check was $31.63.



Typing while I work this out..

Original check is for x dollars and y cents, total intended value = 100*x + y cents
Check is cashed for y dollars and x cents, cash in pocket = 100*y + x cents
Newspaper costs 5 cents, cash now in pocket = 100*y + x - 5 cents

Cash in pocket is now double the original check value.

100y + x - 5 = 2 * (100x + y)
100y + x - 5 = 200x + 2y
 98y             =  199x + 5

At this point, I'll pause, and consider real world restrictions. 

1) First, x and y must be integers.  No fraction of a cent, no fraction of a dollar.
2) Second, x and y probably should be restricted to the range of 0 to 99, inclusive.

Anyone who's been following me with previous puzzlers, I'll probably revisit those, folding, twisting and mutilating for fun and profit.   

At this point, I could try brute force, through the 10,000 combinations.  And, I don't remember any special tricks for integer-restricted math or the like, but I'm going to take a stab at being cleverer at this.

First, y is greater than x.  That eliminates 5,050 combinations, leaving 4,950.

I'm going to try a substitution,  y = x + z and see what that gets me...

 98y             =   199x + 5
 98(x + z)     =  199x + 5
 98x + 98z    =  199x + 5
           98z    =   101x + 5
           
                           101x + 5
               z    =   -------------
                                98

Working in integers, (101x + 5) must be a multiple of 98.  With x constrained to 98 or less (y has to be greater then x, x can't be 99), we're down to 99 possible values for x, and this still looks complex.  With integer math, however, the 101 can be simplified (modulo) to 3, so restated:

                             3x + 5
              z'    =   -------------
                                98

This looks more managable.  Looking for values of x where (3x + 5) = multiples of 98.  The max allowed value of x, 98, gives a max (3x + 5) of 299, so looking for integer solutions of:

       (3x + 5) =   98
       (3x + 5) = 196
       (3x + 5) = 294

The solutions are x=31, 63.666... and 96.333..., the only integer solution is x=31.  Which makes z = 32 (when you plug back to the original formula, not the simplified one.  I edited back and added the z' to keep it straighter), and y = 63.  So, the original check was $31.63.

And, as a test, the amount cashed would have been $63.31, the amount after buying a $0.05 newspaper would be $63.26, which is double the original check value, as 2*$31.63= $63.26.

(Spent 3 minutes rechecking everything, as I mentally forgot to carry the 1 from dimes to dollars when doing my double check   )