Finding the 3-Digit Number

Let the 3-digit number be represented as $100h + 10t + u$, where $h$, $t$, and $u$ are the hundreds, tens, and units digits, respectively.

Setting Up the Equations

Based on the problem statement, we can establish the following equations:

• Sum of digits is 18: $h + t + u = 18$
• Middle digit is the sum of the other two: $t = h + u$
• Reversed number increases by 297: $100u + 10t + h = (100h + 10t + u) + 297$

Solving the Equations Step-by-Step

1. Substitute the second equation into the first: Replace $t$ in the first equation: $h + (h + u) + u = 18$ Simplify: $2h + 2u = 18$ Divide by 2: $h + u = 9$
2. Determine the middle digit ($t$): From the second equation, $t = h + u$. Since we found $h + u = 9$, the middle digit $t = 9$.
3. Simplify the third equation (reversal condition): $100u + 10t + h = 100h + 10t + u + 297$ Subtract $10t$ from both sides: $100u + h = 100h + u + 297$ Rearrange terms: $100u - u - 100h + h = 297$ $99u - 99h = 297$ Divide by 99: $u - h = 3$
4. Solve for $h$ and $u$: We now have a system of two linear equations:
   • $h + u = 9$
   • $u - h = 3$
   Add the two equations: $(h + u) + (u - h) = 9 + 3$ $2u = 12 \implies u = 6$ Substitute $u=6$ into $h + u = 9$: $h + 6 = 9 \implies h = 3$
5. Determine the number: The digits are $h=3$, $t=9$, and $u=6$. The number is $100(3) + 10(9) + 6 = 300 + 90 + 6 = 396$.

Verification

Let's check if the number 396 satisfies all conditions:

• Sum of digits: $3 + 9 + 6 = 18$ (Condition 1 met)
• Middle digit sum: $9 = 3 + 6$ (Condition 2 met)
• Reversed number: 693. Difference: $693 - 396 = 297$ (Condition 3 met)

The number 396 meets all the given criteria.