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?

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:

Verification / 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