Workings

Hint
The answer is 4 digits long, so what must G equal?

Answer
423 + 675 = 1098.

Reasoning
Remembering that:
   even + even = even
   odd  + odd  = even
   even + odd  = odd

To discuss individual letters, it's easiest to represent the sum as:
   A B C
   D E F +
————————
 G H I J

A + D has to be over 9, which means that G = 1.

B + E = I, is even + odd = odd, which means that we can't have a carry from C + F (otherwise it would have been even + odd + 1, which is even).

The 1 has already gone, so the smallest possible value for either C or F is 3, which means that the other can't be 7 or 9 (otherwise we'd have a carry).

Therefore, C and F are 3 and 5, but we don't know which is which. But we do now know that J = 8.

A + D = H, is even + even = even, which means that we can't have a carry from B + E. Therefore, E can't be 9, as this would force a carry. So E = 7.

I is the only remaining odd number, so I = 9.

Which means that B = 2.

Neither A nor D can be 0 (otherwise we would have two of the same digit). So, H = 0.

Therefore, A and D are 4 and 6 (but we don't yet know which is which).

Since the top row's digits have to add to 9, A can't be 6, so A = 4, making C = 3.

This makes the sum 423 + 675 = 1098.