In a row of people, Manu is 7th from the bottom end of the row. Shrey is 10 ranks above Manu. If Shrey is 8th from the top end, then how many people are there altogether in this row?

Introduction / Context:
This question involves ranks from the top and bottom in a single row. You are given Manu’s position from the bottom, the vertical distance between Manu and Shrey, and Shrey’s position from the top. Using this information, you must determine the total number of people in the row. Such problems test your understanding of how positions from opposite ends relate to each other and to the total count.

Given Data / Assumptions:

• Manu is 7th from the bottom end of the row.
• Shrey is 10 ranks above Manu.
• Shrey is 8th from the top end of the row.
• We must find the total number of people in the row.

Concept / Approach:
The key idea is to express both Manu’s and Shrey’s positions from the top and from the bottom in terms of the total number of people. If the total number of people is N, then a person who is k-th from the bottom is at position N - k + 1 from the top. We can use this to find Manu’s position from the top, then apply the information that Shrey is 10 ranks above Manu, and finally equate this to Shrey’s given position from the top to solve for N.

Step-by-Step Solution:
Let the total number of people in the row be N. Manu is 7th from the bottom. So Manu’s position from the top is N - 7 + 1 = N - 6. Shrey is 10 ranks above Manu, meaning Shrey is 10 positions closer to the top than Manu. Therefore, Shrey’s position from the top is (N - 6) - 10 = N - 16. We are also told that Shrey is 8th from the top, so N - 16 = 8. Solve for N: N = 8 + 16 = 24. Thus, there are 24 people in the row.

Verification / Alternative check:
Check the consistency of positions with N = 24. Manu is 7th from the bottom, so his position from the top is 24 - 7 + 1 = 18. Shrey is 10 ranks above Manu, so his position from the top is 18 - 10 = 8, which matches the given information. This confirms that the total number of people must be 24 and that our interpretation of “10 ranks above” is correct.

Why Other Options Are Wrong:
If N were 23, Manu’s position from the top would be 23 - 7 + 1 = 17, and Shrey would then be 17 - 10 = 7 from the top, not 8. Similar contradictions occur for N = 25, 26 or 27. None of these values lead to Shrey being both 10 ranks above Manu and 8th from the top. Only N = 24 satisfies all the conditions simultaneously.

Common Pitfalls:
A common mistake is to misunderstand “10 ranks above” and incorrectly add instead of subtracting, or to miscompute the conversion from bottom to top. Another pitfall is forgetting the +1 term when converting positions between ends, leading to off by one errors. Writing all relations algebraically and solving systematically helps avoid these mistakes.

Final Answer:
The total number of people in the row is 24.