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Difficulty: HardOrder and Ranking

During a summit, a single straight row of delegates is seated facing South. Mr. Patel is seated 21st21^{\text{st}} from the right end of the row. Ms. Quinn is seated 28th28^{\text{th}} from the left end of the row. Dr. Ray is seated 7th7^{\text{th}} to the left of Mr. Patel and 5th5^{\text{th}} to the right of Ms. Quinn. What is the total number of delegates in the row?

  1. A
    60
  2. B
    50
  3. 36Answer
  4. D
    37

Answer

The total number of delegates in the row is 36.
Because the delegates are facing South, their personal left corresponds to the right end of the row, and their personal right corresponds to the left end of the row. Dr. Ray is 77 seats to the left of Mr. Patel (21st21^{\text{st}} from the right), making him 21−7=14th21 - 7 = 14^{\text{th}} from the right end. Dr. Ray is 55 seats to the right of Ms. Quinn (28th28^{\text{th}} from the left), making him 28−5=23rd28 - 5 = 23^{\text{rd}} from the left end. The total number of delegates is the sum of Dr. Ray's position from both ends minus one (to avoid double-counting him): 23+14−1=3623 + 14 - 1 = 36.

Step-by-Step Solution

1
Determine the relative left and right directions for delegates facing South.
For someone facing South, their 'Left' points toward the Right End of the row, and their 'Right' points toward the Left End of the row.
Correctly mapping personal left/right to the absolute row ends is required before applying positional offsets.
2
Calculate Dr. Ray's absolute position from the right end of the row.
Mr. Patel is 21st21^{\text{st}} from the right end. Dr. Ray is 7th7^{\text{th}} to his left (toward the right end). Dr. Ray is 21−7=14th21 - 7 = 14^{\text{th}} from the right end.
Because Dr. Ray is to Mr. Patel's left, and left points toward the right end, Dr. Ray is closer to the right end than Mr. Patel.
3
Calculate Dr. Ray's absolute position from the left end of the row.
Ms. Quinn is 28th28^{\text{th}} from the left end. Dr. Ray is 5th5^{\text{th}} to her right (toward the left end). Dr. Ray is 28−5=23rd28 - 5 = 23^{\text{rd}} from the left end.
Because Dr. Ray is to Ms. Quinn's right, and right points toward the left end, Dr. Ray is closer to the left end than Ms. Quinn.
4
Calculate the total number of delegates using Dr. Ray's known positions from both ends.
Total = (Position from Left) + (Position from Right) - 1 = 23+14−1=3623 + 14 - 1 = 36.
The sum of a person's rank from both ends counts that person twice, so 1 must be subtracted to find the accurate total.

Key Concept

Positional Ranking with Directional Inversion
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Topics

Order and Ranking

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