The numbers 585, 546, 514, 404, 206 and 369 are each transformed by interchanging their first and third digits. After this transformation, the new numbers are arranged in ascending order. Which original number will be second from the end in this order?

Introduction / Context:
This question tests place value understanding and careful manipulation of digits. You need to perform a digit swap on several three digit numbers, then order the resulting numbers and relate the sorted list back to the original numbers.

Given Data / Assumptions:

• Original numbers: 585, 546, 514, 404, 206 and 369.
• For each number, the first and third digits are interchanged.
• The transformed numbers are then arranged in ascending order.
• We must identify which original number is second from the end (second largest) in this sorted list of transformed numbers.

Concept / Approach:
We systematically perform the digit interchange for each number and record the new value. After that, we sort the transformed values in ascending order, identify the second largest transformed number, and finally map that transformed number back to its original form.

Step-by-Step Solution:
Step 1: For 585, swapping the first and third digits gives 585 (it remains the same).Step 2: For 546, swapping first and third digits (5 and 6) gives 645.Step 3: For 514, swapping first and third digits (5 and 4) gives 415.Step 4: For 404, swapping the first and third digits (both 4) leaves 404 unchanged.Step 5: For 206, swapping first and third digits (2 and 6) gives 602.Step 6: For 369, swapping first and third digits (3 and 9) gives 963.Step 7: Now list transformed numbers with their originals: 585 (from 585), 645 (from 546), 415 (from 514), 404 (from 404), 602 (from 206) and 963 (from 369).Step 8: Arrange these transformed numbers in ascending order: 404, 415, 585, 602, 645, 963.Step 9: The largest transformed number is 963, the second largest is 645. The transformed number 645 came from the original number 546.

Verification / Alternative check:
Double check each swap to ensure no digit was misread: note that only numbers with different first and third digits actually change.Confirm that the sorted order is correct, paying attention that 602 is less than 645 and both are below 963.

Why Other Options Are Wrong:
404 remains 404 and is the smallest transformed value, not second from the end.585 remains 585 and lies in the middle of the sorted order.369 becomes 963, which is the largest transformed number, not the second largest.

Common Pitfalls:
Some students may confuse the position after sorting with the position in the original list.Others may incorrectly swap digits or forget that repeated digits such as 4 and 4 in 404 produce no change.Careful bookkeeping of each transformation and a clear sorted list avoids these errors.

Final Answer:
The original number that is second from the end is 546.