Two-digit number with digit swap: When the digits of a two-digit number are interchanged, the new number is smaller by 45 than the original. Also, the ratio (new : original) is 3 : 8. Find the original number.

Introduction / Context: Let the original number be 10a + b and the swapped number be 10b + a. The problem gives both a difference and a ratio condition, which combine to determine the original number uniquely using algebraic substitution.

Given Data / Assumptions:

• New = Original − 45.
• New : Original = 3 : 8.

Concept / Approach: Translate the ratio to New = (3/8)*Original and combine with the difference condition. Solve for Original. Then verify that the digits indeed swap correctly to satisfy both conditions.

Step-by-Step Solution:

Verification / Alternative check: New = 27. Difference: 72 − 27 = 45. Ratio: 27 : 72 = 3 : 8. Both conditions hold, digits 7 and 2 swap as expected.

Why Other Options Are Wrong: 61, 94, and 54 do not satisfy the combined difference and ratio conditions upon swapping digits; only 72 meets both.

Common Pitfalls: Using 8 : 3 instead of 3 : 8, or forgetting that the swapped number must be smaller according to the problem statement.

Final Answer: 72