1. **State the problem:** Find the sum of the largest and smallest 5-digit numbers formed using the digits 3, 0, 4, 1, 9. 2. **Important rule:** A 5-digit number cannot start with 0 because then it would be a 4-digit number. 3. **Find the largest number:** Arrange digits in descending order, but the first digit cannot be 0. Digits: 9, 4, 3, 1, 0 Largest number = 94310 4. **Find the smallest number:** Arrange digits in ascending order, but the first digit cannot be 0. Digits: 0, 1, 3, 4, 9 Smallest number must start with 1 (smallest non-zero digit), then 0, 3, 4, 9 Smallest number = 10349 5. **Sum the two numbers:** $$94310 + 10349 = 104659$$ **Final answer:** The sum of the largest and smallest 5-digit numbers formed is $104659$.