Reversing digits increases value by 45: When the digits of a two-digit number are interchanged, the resulting number is greater than the original by 45. If the difference between the digits is 5, what is the original number?
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A16
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B27
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C38
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DCannot be determined
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E49
Answer
Correct Answer: Cannot be determined
Explanation
Introduction / Context:This question tests modeling two-digit numbers and understanding what information is sufficient to determine a unique answer. We will translate the statements into equations and examine whether a single solution or multiple solutions exist.
Given Data / Assumptions:
- Original number has tens digit a and units digit b.
- Reversed number is 10b + a and is larger than original by 45.
- The difference between the digits is 5.
Concept / Approach:Form two relations: 10b + a = (10a + b) + 45 and |a − b| = 5. Because the reversed number is larger, b > a, hence b − a = 5. We will see that several valid (a, b) pairs satisfy both relations, leading to multiple original numbers.
Step-by-Step Solution:
10b + a = 10a + b + 45 ⇒ 9(b − a) = 45 ⇒ b − a = 5This is identical to the stated digit difference, so both conditions reduce to b − a = 5.Valid pairs with a ≥ 1 (tens digit cannot be 0): (1,6), (2,7), (3,8), (4,9)Corresponding originals: 16, 27, 38, 49Verification / Alternative check:For each, reversing increases value by 45 (e.g., 61 − 16 = 45; 72 − 27 = 45, etc.). Thus multiple answers exist.
Why Other Options Are Wrong:
- 16, 27, 38, 49: Each is possible, but the question asks for a unique original number.
Common Pitfalls:Assuming uniqueness without checking the digit constraints. Always ensure the system of equations yields a single solution before selecting a numeric option.
Final Answer:Cannot be determined