Two-digit number from digit conditions: The number obtained by interchanging the digits of a two-digit number is 18 more than the original number, and the sum of the digits is 8. Find the original number.
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A50
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B51
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C35
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D53
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E44
Answer
Correct Answer: 35
Explanation
Introduction / Context:Digit puzzles translate directly into small linear equations. Represent the two-digit number as 10a + b with a as the tens digit and b as the units digit. Use the reverse condition and the digit sum to form two equations and solve for a and b.
Given Data / Assumptions:
- Reverse number = original number + 18.
- Sum of digits = 8.
- Digits a, b are integers from 0–9 with a ≥ 1 for a two-digit number.
Concept / Approach:Let original = 10a + b; reverse = 10b + a. Then 10b + a = 10a + b + 18 and a + b = 8. Solve simultaneously for a and b, ensuring valid digit constraints.
Step-by-Step Solution:
10b + a = 10a + b + 18 ⇒ 9b − 9a = 18 ⇒ b − a = 2 a + b = 8 Add equations: (b − a) + (a + b) = 2 + 8 ⇒ 2b = 10 ⇒ b = 5 Then a = 8 − b = 3 Original number = 10a + b = 30 + 5 = 35Verification / Alternative check:Reverse is 53; 53 − 35 = 18 (fits). Digit sum 3 + 5 = 8 (fits).
Why Other Options Are Wrong:50, 51, 53, 44 do not satisfy both the reverse-difference and digit-sum conditions together.
Common Pitfalls:Mixing up which number is larger (original vs reverse) and sign errors when rearranging 10b + a = 10a + b + 18 are common mistakes.
Final Answer:35