** asd+fgh=jkl**
Believe me - there are numerous solutions of the above alphametic.

I. What is the minimal possible sum of participating digits?

ii. Same question for **as+df=gh**

(In reply to

Both at once? by Brian Smith)

I found 10 solutions for solving both alphametics simultaneously (x is the unused digit):

asdfghjklx

1627439052

1926458370

2415397806

3145768902

3942816750

4152937086

4613598207

5142938076

5413679082

5712698403

To minimize the sum of the used digits, the unused digit must be as large as possible. This leaves the 7th solution as the solution with the smallest sum of digits used. **461+359=820 46+13=59**