Final answer:
To solve the problem, set up equations based on the given conditions (sum of digits is 12, and interchanging increases number by 18), solve the equations simultaneously to find the digits of the number, which leads to the answer being 57 (option c).
Step-by-step explanation:
The sum of a two-digit number is 12, and the number obtained by interchanging the digits exceeds the original number by 18. Let the tens digit be x and the units digit be y. The original number can be expressed as 10x + y, and the number after interchanging the digits would be 10y + x. From the problem, we know:
- x + y = 12 (because the digits sum to 12)
- 10y + x = 10x + y + 18 (interchanging digits gives a number 18 more than the original)
Solving the system of equations:
- Subtract the first equation from the second one to get 9y - 9x = 18
- Simplify to get y - x = 2
- Use the first equation (x + y = 12) to find x = 5 and y = 7
So, the original number is 10x + y = 10(5) + 7 = 57, which is option (c).