1. **State the problem:** We need to find a two-digit number such that when we add 18 to the number formed by interchanging its digits, the sum of the digits of the original number is 12. 2. **Define variables:** Let the two-digit number be $10x + y$, where $x$ is the tens digit and $y$ is the units digit. 3. **Sum of digits condition:** Given that the sum of the digits is 12, we have: $$x + y = 12$$ 4. **Interchanged number:** The number formed by interchanging the digits is $10y + x$. 5. **Addition condition:** Adding 18 to the interchanged number equals the original number: $$18 + (10y + x) = 10x + y$$ 6. **Simplify the equation:** $$18 + 10y + x = 10x + y$$ $$18 + 10y + x - 10x - y = 0$$ $$18 + 9y - 9x = 0$$ $$9y - 9x = -18$$ $$y - x = -2$$ 7. **Solve the system:** From step 3: $x + y = 12$ From step 6: $y - x = -2$ Add the two equations: $$(x + y) + (y - x) = 12 + (-2)$$ $$2y = 10$$ $$y = 5$$ Substitute $y=5$ into $x + y = 12$: $$x + 5 = 12$$ $$x = 7$$ 8. **Find the number:** The original number is: $$10x + y = 10 \times 7 + 5 = 75$$ **Final answer:** The number is 75.