In a row of boys, Rahul is 19th from the top and Ajay is 21st from the bottom. They interchange their positions and after swapping Ajay becomes 28th from the bottom. How many boys are there in the row in total?

Introduction / Context:
This is a classic ranking and ordering puzzle involving two students who exchange their positions in a single row. You are given their positions from different ends before and after the interchange. The aim is to use these positions to determine the total number of boys in the row. Such problems test your ability to convert between ranks from the top and from the bottom and to set up simple equations.

Given Data / Assumptions:
- Rahul is 19th from the top initially. - Ajay is 21st from the bottom initially. - After they interchange positions, Ajay becomes 28th from the bottom. - There is a fixed total number of boys N in the row, with no vacant places.

Concept / Approach:
If the total number of boys is N, then a boy who is k-th from the bottom is at position N - k + 1 from the top. When two boys swap positions, each takes the other's original place. Here, after the swap, Ajay sits at Rahul's original position from the top. The information about Ajay's new position from the bottom gives a direct equation involving N that we can solve. This relation is the core of the method.

Step-by-Step Solution:
Step 1: Let N be the total number of boys in the row. Step 2: Initially, Rahul is 19th from the top; his top rank is 19. Step 3: Initially, Ajay is 21st from the bottom. His top rank is N - 21 + 1 = N - 20. Step 4: After they interchange positions, each boy occupies the other's original place. Step 5: So, after the swap, Ajay is now at Rahul's original position from the top, which is 19th from the top. Step 6: We are told that after the swap Ajay is 28th from the bottom. Step 7: The relation between Ajay's new bottom rank and the total N is: bottom rank = N - top rank + 1. Step 8: Substitute: 28 = N - 19 + 1, so 28 = N - 18. Step 9: Solve for N: N = 28 + 18 = 46. Step 10: Therefore, there are 46 boys in the row.

Verification / Alternative check:
We can confirm consistency by computing Ajay's original top position using N = 46 and his original bottom rank of 21. Top rank = 46 - 21 + 1 = 26. Thus initially Rahul is at position 19 and Ajay at position 26. After swapping, Ajay becomes 19th from the top and, with N = 46, his bottom rank is 46 - 19 + 1 = 28, which matches the given information. This cross-check supports the correctness of N = 46.

Why Other Options Are Wrong:
47: For N = 47, Ajay's new bottom rank would be 47 - 19 + 1 = 29, not 28. 45: For N = 45, Ajay's new bottom rank would be 27, which again does not match the question. 48: For N = 48, Ajay's new bottom rank becomes 30, which contradicts the given data.

Common Pitfalls:
A frequent error is to forget the +1 in the conversion formula, writing N - k instead of N - k + 1. Another mistake is misidentifying which boy's position changes are being described after the swap, leading to the wrong equation. Clearly labelling positions “before” and “after” for each person prevents confusion and helps systematically reach the correct total.

Final Answer:
The total number of boys in the row is 46.