Two-digit number with digit relation and swap effect In a two-digit number, the unit's digit equals (2 × ten's digit) − 1. After interchanging the digits, the difference between the new and original number is less than the original number by 20. What is the original number?

Introduction / Context:
This is a classic digits problem involving a two-digit number N with a link between the digits and a condition after swapping them. We translate words into equations in terms of the ten's digit (t) and unit's digit (u) and solve logically.

Given Data / Assumptions:

• N = 10t + u, with t from 1 to 9 and u from 0 to 9.
• u = 2t − 1.
• New number after swap, N' = 10u + t.
• “Difference is less than the original by 20” ⇒ N' − N = N − 20.

Concept / Approach:
Express the difference using digits, then equate to N − 20 and solve for t. Finally compute u from the digit relation and form the number.

Step-by-Step Solution:
N = 10t + u; N' = 10u + t.Given u = 2t − 1 ⇒ N' − N = (10u + t) − (10t + u) = 9(u − t).Condition: 9(u − t) = (10t + u) − 20.Substitute u = 2t − 1: 9((2t − 1) − t) = 10t + (2t − 1) − 20 ⇒ 9(t − 1) = 12t − 21.9t − 9 = 12t − 21 ⇒ 12 = 3t ⇒ t = 4; hence u = 2(4) − 1 = 7.Original number = 10t + u = 47.

Verification / Alternative check:
N = 47, N' = 74. Difference N' − N = 27. N − 20 = 27. Condition holds exactly.

Why Other Options Are Wrong:
59, 23, 35, 41 do not satisfy u = 2t − 1 and the difference requirement simultaneously.

Common Pitfalls:
Interpreting the sentence incorrectly; using N − N' instead of N' − N; forgetting to restrict t to 1–9.

Final Answer:
47