In a row of students, John is 16th from the left and Johnson is 8th from the right. If they interchange their positions and John then becomes 33rd from the left, how many students are there in the row?

Introduction / Context:
This is a slightly more involved rank and position problem. You must use the information about changes in positions when two students interchange their seats in a row. The question tests your understanding of how left and right ranks relate to the total number of students in the row.

Given Data / Assumptions:

• Initially, John is 16th from the left.
• Initially, Johnson is 8th from the right.
• They interchange positions.
• After interchange, John becomes 33rd from the left.
• We assume a single straight row with no empty positions and no other movements.

Concept / Approach:
The key idea is that when John and Johnson swap positions, John takes the exact position previously occupied by Johnson. Therefore, after the swap, John's new rank from the left is the same as Johnson's old rank from the left. Knowing the rank of a student from both sides of the row allows us to use the standard formula: total students = position from left + position from right - 1. We apply this to Johnson using his left and right positions.

Step-by-Step Solution:
Step 1: Before interchange, John is 16th from the left.Step 2: Before interchange, Johnson is 8th from the right.Step 3: After interchange, John takes the position that Johnson previously had.Step 4: The question says that after interchange John becomes 33rd from the left, which means Johnson's original position from the left is 33rd.Step 5: Now, for Johnson before interchange, position from left is 33 and from right is 8.Step 6: Use the formula total students = left position + right position - 1.Step 7: Total students = 33 + 8 - 1 = 40.

Verification / Alternative check:
To verify, imagine a row of 40 students. A student who is 33rd from the left is at position 40 - 33 + 1 = 8th from the right, which matches Johnson's given initial right rank. Therefore, the number 40 satisfies all position data. If the total were 39, then the 33rd from the left would be 7th from the right, not 8th. For 41, 33rd from the left would be 9th from the right. Thus, the total must be exactly 40 for the given ranks to be consistent.

Why Other Options Are Wrong:
38, 39, 41 and 42 all lead to an inconsistency between the left rank 33 and the right rank 8 for the same student.For example, with 38 students, a student who is 33rd from the left would be 6th from the right, not 8th.With 39 students, 33rd from the left gives 7th from the right.With 41 or 42 students, the right ranks would be 9th or 10th, again contradicting the problem statement.

Common Pitfalls:
One common mistake is to forget that after the interchange John takes the exact original position of Johnson, and thus John's new left rank equals Johnson's old left rank. Another error is to add the two positions from the left directly or to misapply the total formula. Carefully linking the statements and explicitly using total = left rank + right rank - 1 is the correct and reliable method.

Final Answer:
The total number of students in the row is 40.