For each of the numbers 517, 325, 639, 841 and 792, only the second and third digits within the number are interchanged. What is the first digit of the second highest number obtained after this change?

Introduction / Context:
Digit interchange questions check your comfort with place value and simple manipulation of three digit numbers. Here you are asked to swap only the second and third digits of each number, then compare the resulting numbers. The final objective is to identify the first digit of the second highest transformed number, not the number itself.

Given Data / Assumptions:
- Original numbers: 517, 325, 639, 841, 792. - Operation: interchange only the second digit (tens place) and third digit (units place) of each number. - After transformation, we must find the second highest number and then identify its first digit.

Concept / Approach:
A three digit number has the form 100 * a + 10 * b + c, where a is the hundreds digit, b the tens digit, and c the units digit. Swapping b and c changes the number to 100 * a + 10 * c + b. We apply this to each given number, list all the new values, sort them in descending order, and then inspect the first digit of the second largest value. This systematic approach minimizes mistakes.

Step-by-Step Solution:
Step 1: Transform 517. Digits are a = 5, b = 1, c = 7. Swapping b and c gives 571. Step 2: Transform 325. Digits 3, 2, and 5 become 3, 5, and 2 after swapping, giving 352. Step 3: Transform 639. Digits 6, 3, and 9 become 6, 9, and 3, giving 693. Step 4: Transform 841. Digits 8, 4, and 1 become 8, 1, and 4, giving 814. Step 5: Transform 792. Digits 7, 9, and 2 become 7, 2, and 9, giving 729. Step 6: List all transformed numbers: 571, 352, 693, 814, 729. Step 7: Arrange these in descending order: 814, 729, 693, 571, 352. Step 8: The second highest transformed number is 729. Its first digit is 7.

Verification / Alternative check:
To verify, you can check that no transformed value exceeds 814 or lies between 814 and 729. Since 814 clearly has the greatest hundreds digit 8, it is the highest. Among the remaining numbers, 729 has hundreds digit 7, which is greater than 6 in 693 and 5 in 571, and greater than 3 in 352. Therefore 729 is indisputably the second highest. The first digit read from 729 is 7, which confirms the result.

Why Other Options Are Wrong:
8: This is the first digit of the highest number 814, not the second highest. 2: No transformed number that could be second highest begins with 2, so this is not possible. 9: No transformed number begins with 9, so this option cannot be correct.

Common Pitfalls:
One error is to swap the first and second digits or to reverse the entire number instead of swapping only the last two digits. Another frequent mistake is to misorder the transformed numbers, especially if one relies only on quick visual comparison. Writing all transformed numbers clearly and comparing first digits, then second digits if required, ensures correct ranking and a reliable answer.

Final Answer:
The first digit of the second highest number after interchanging the second and third digits is 7.