The sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is 27 less than the original. What is the original number?
["### Finding the Two-Digit Number Where the Sum of Digits is 9 and Reversed Number is 27 Less", "Have you ever come across a number puzzle that combines simple arithmetic with logic? One classic problem asks: What is the two-digit number such that the sum of its digits is 9, and reversing the digits results in a number 27 less than the original? This article explores how to solve this problem step by step and reveals the original number.", "---", "#### Step 1: Represent the Two-Digit Number Algebraically", "Let the two-digit number be represented as ( 10a + b ), where:\n- ( a ) is the tens digit (1 ≤ ( a ) ≤ 9),\n- ( b ) is the units digit (0 ≤ ( b ) ≤ 9).", "From the problem:\n1. Sum of digits is 9:\n [\n a + b = 9\n ]", "2. Reversed number is 27 less:\n Reversing the digits gives ( 10b + a ), so:\n [\n 10a + b = 10b + a - 27\n ]", "---", "#### Step 2: Simplify the Equation", "Start with the second equation:\n[\n10a + b = 10b + a - 27\n]", "Bring all terms to one side:\n[\n10a - a + b - 10b + 27 = 0 \implies 9a - 9b + 27 = 0\n]", "Factor out 9:\n[\n9(a - b + 3) = 0 \implies a - b + 3 = 0 \implies a - b = -3\n]", "---", "#### Step 3: Solve the System of Equations", "Now you have two simplified equations:\n1. ( a + b = 9 )\n2. ( a - b = -3 )", "Add both equations:\n[\n(a + b) + (a - b) = 9 + (-3) \implies 2a = 6 \implies a = 3\n]", "Substitute ( a = 3 ) into ( a + b = 9 ):\n[\n3 + b = 9 \implies b = 6\n]", "---", "#### Step 4: Verify the Solution", "Original number: ( 10a + b = 10(3) + 6 = 36 )\nReversed number: ( 10b + a = 10(6) + 3 = 63 )", "Check the difference:\n[\n63 - 36 = 27 \quad \ ext{(correct)}\n]", "Sum of digits: ( 3 + 6 = 9 ) (also correct)", "---", "#### Conclusion", "The two-digit number that satisfies both conditions is 36. This problem beautifully combines basic algebra with logical reasoning — a great exercise for budding math enthusiasts and puzzle solvers alike.", "---", "#### Key Takeaways:\n- Represent digits algebraically using place values.\n- Set up equations based on given constraints.\n- Solve the system step by step.\n- Always verify your solution with the original conditions.", "If you love this type of number puzzle, try similar ones involving other digit sums or digit-swap differences — the world of two-digit numbers holds many more intriguing challenges!"]
