1. **Problem statement:** We have two numbers that add up to 415. One number is a two-digit number (between 10 and 99), and the other is a three-digit number (between 100 and 999). We want to find the smallest possible difference between these two numbers. 2. **Set variables:** Let the two-digit number be $x$ and the three-digit number be $y$. 3. **Given equation:** $$x + y = 415$$ 4. **Express $y$ in terms of $x$:** $$y = 415 - x$$ 5. **Difference between the numbers:** $$|y - x| = |(415 - x) - x| = |415 - 2x|$$ 6. **Goal:** Minimize $|415 - 2x|$ with $x$ in $[10, 99]$ and $y = 415 - x$ in $[100, 999]$. 7. **Check constraints for $y$:** Since $y = 415 - x$, for $y$ to be three-digit: $$100 \leq 415 - x \leq 999$$ From the left inequality: $$415 - x \geq 100 \Rightarrow x \leq 315$$ From the right inequality: $$415 - x \leq 999 \Rightarrow x \geq -584$$ Since $x$ is two-digit, $10 \leq x \leq 99$, which fits the constraints. 8. **Minimize $|415 - 2x|$:** The expression is minimized when $415 - 2x = 0$ or as close as possible. Solve for $x$: $$2x = 415 \Rightarrow x = 207.5$$ But $x$ must be two-digit, so closest two-digit integers to 207.5 are 99 and 10. 9. **Calculate differences for $x=99$ and $x=10$:** - For $x=99$: $$|415 - 2(99)| = |415 - 198| = 217$$ - For $x=10$: $$|415 - 2(10)| = |415 - 20| = 395$$ 10. **Try values near the boundary of two-digit numbers:** Try $x=99$ (largest two-digit number) gives difference 217. Try $x=100$ (not two-digit) is invalid. Try $x=99$ is best so far. 11. **Try $x= 99$:** Then $y = 415 - 99 = 316$ (three-digit number). Difference: $$|316 - 99| = 217$$ 12. **Try $x= 99$ is the smallest difference possible under constraints.** **Final answer:** The smallest possible difference is $217$.