Final answer:
To solve the equation ABC + ACB = BCA, we apply the commutative property of addition, and through trial and error, we determine that ABC equals 589 by analyzing the effects of addition without carrying over.
Step-by-step explanation:
In the equation ABC + ACB = BCA, where the same letters correspond to the same numbers, and different letters correspond to different numbers, we must determine the value of ABC. The clue provided with the question is A+B=B+ A, which is an example of the commutative property of addition, stating that numbers can be added in any order, and the sum will remain the same.
Let's analyze the equation step by step. Since A, B, and C are different digits and they combine to form a three-digit number in different orders, our first observation should be that the sum of two three-digit numbers cannot exceed 999. This eliminates the possibility of carrying over, which would change the values of A, B, or C. Next, we notice that in the sum BCA, the hundreds place is B, which can only occur if A + A, or 2A, equals 10 or more (since there is no carry over from the previous sum). This would mean A is at least 5.
Similarly, considering the ones place in BCA, it indicates that C + B must be less than 10 (as there's no carrying over) and must end with a digit A. Now we iterate through the possible values for A, and we quickly find that A cannot be more than 5 as it would contradict the condition for B and C. By trial and error, we find that A is indeed 5, which makes 2A equal to 10, allowing B to take the hundreds place in the answer.
Going further with substitution and similar logic, we can eventually deduce that B must be 9, and C must be 8, solving the puzzle and finding that ABC equals 589.