In a row of boys, A is 10th from the left and B is 9th from the right. After they interchange their positions, A becomes 15th from the left. How many boys are there in the row in total?

Introduction / Context:
This time-sequence and ranking question deals with positions in a row from both ends. The twist is that two boys interchange their positions, which changes their ranks. You must relate positions from the left and right to deduce the total number of boys. Such questions are standard in reasoning sections and test comfort with positional arithmetic.

Given Data / Assumptions:

- In the original arrangement, A is 10th from the left.- In the original arrangement, B is 9th from the right.- After they interchange positions, A becomes 15th from the left.- We must find the total number of boys in the row.

Concept / Approach:
For any person in a row, if we know their position from one side and the total number of persons, we can find their position from the other side using: position from right = total - position from left + 1. Here, we use that relationship in reverse. After interchanging, A goes to B's original place. That information allows us to equate A's new left position with B's original left position and then solve for the total number of boys.

Step-by-Step Solution:
Step 1: Let the total number of boys be N.Step 2: A is originally 10th from the left, so his position from the right is N - 10 + 1 = N - 9.Step 3: B is originally 9th from the right, so his position from the left is N - 9 + 1 = N - 8.Step 4: After interchange, A moves to B's original position, which is N - 8 from the left.Step 5: According to the question, after the interchange A becomes 15th from the left. Hence N - 8 = 15.Step 6: Solve for N: N = 15 + 8 = 23.Step 7: Therefore, there are 23 boys in the row.

Verification / Alternative check:
With N = 23, B's original left position is N - 8 = 15. So after the interchange, A is indeed 15th from the left, matching the given information. Similarly, B moves to the position that was 10th from the left, which will become 14th from the right (23 - 10 + 1), consistent with the swap. This confirms that N = 23 is correct.

Why Other Options Are Wrong:
If N were 27, 28 or 31, the calculated left positions after interchange would not match 15th from the left for A. Each of these values produces a contradiction when we apply the left and right rank formulas. Hence, they cannot be the correct total number of boys.

Common Pitfalls:
Students sometimes mix up the formulas or forget to add 1 when converting between left and right positions. Another mistake is assuming that the sum of the ranks from left and right remains constant for different students, which is not generally true unless you know the total. Always set up the equations carefully and solve for the total explicitly.

Final Answer:
The total number of boys in the row is 23.