In a class of 31 students, Manoj and Satya are ranked seventh and eleventh from the top. What are their respective ranks from the bottom of the class?

Introduction / Context:
Ranking and ordering problems are very common in reasoning and aptitude examinations. They test your ability to relate positions from the top and from the bottom within a fixed total. In this question, you are given two ranks from the top in a class of 31 students and you must find the corresponding ranks from the bottom for both students.

Given Data / Assumptions:
- Total number of students in the class = 31. - Manoj is ranked seventh from the top. - Satya is ranked eleventh from the top. - There is exactly one student at each rank, with no shared positions.

Concept / Approach:
The core relationship between a rank from the top and a rank from the bottom in a group of N students is bottom rank = N - top rank + 1. This works because you count how many students are after a given student plus the student himself. Applying this formula separately to Manoj and Satya gives their positions from the opposite end. Once this direct formula is clear, such questions become straightforward calculations.

Step-by-Step Solution:
Step 1: Note that N, the total number of students, is 31. Step 2: Manoj is seventh from the top, so his top rank T1 = 7. Step 3: Use the relation for Manoj: bottom rank B1 = N - T1 + 1 = 31 - 7 + 1 = 25. Step 4: Therefore, Manoj is twenty fifth from the bottom. Step 5: Satya is eleventh from the top, so his top rank T2 = 11. Step 6: Use the same relation for Satya: bottom rank B2 = N - T2 + 1 = 31 - 11 + 1 = 21. Step 7: Therefore, Satya is twenty first from the bottom. Step 8: Combine the results: Manoj and Satya are respectively 25th and 21st from the bottom.

Verification / Alternative check:
A quick check is to imagine the positions in reverse. If Manoj is 25th from the bottom, then there are 24 students below him and 6 above him. That makes 6 + 1 + 24 = 31 students, and his top rank is 7, which matches the given data. Similarly, for Satya at 21st from the bottom, there are 20 students below and 10 above, so 10 + 1 + 20 = 31 and his top rank is 11. This confirms that our conversion between top and bottom ranks is consistent.

Why Other Options Are Wrong:
26th and 22nd: These values come from subtracting one incorrectly or misapplying the formula. 20th and 24th: This option reverses and miscalculates the positions, not matching either student correctly. 24th and 20th: Both ranks are off by one for each student and do not satisfy the conversion back to the top ranks.

Common Pitfalls:
Students often forget to add 1 in the relation bottom rank = N - top rank + 1 and instead use only N - top rank, which gives an incorrect result. Another error is to confuse whose rank is being converted and to accidentally swap the positions of Manoj and Satya. Writing the formula clearly each time and substituting step by step helps avoid these mistakes in exam conditions.

Final Answer:
The respective ranks of Manoj and Satya from the bottom of the class are 25th and 21st.