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Difficulty: MediumOrder and Ranking

During a graduation ceremony, graduates are standing in a single file line facing the stage. Liam is standing 18th18^{\text{th}} from the front, while Sophia is standing 31st31^{\text{st}} from the rear. After exchanging their places, Liam becomes 42nd42^{\text{nd}} from the front. What is Sophia's new position from the rear?

  1. A
    54th54^{\text{th}}
  2. B
    72nd72^{\text{nd}}
  3. 55th55^{\text{th}}Answer
  4. D
    7th7^{\text{th}}

Answer

Sophia's new position from the rear is 55th55^{\text{th}}.
When Liam and Sophia exchange places, Liam moves from the 18th18^{\text{th}} position to the 42nd42^{\text{nd}} position from the front, representing a shift of 2424 places backward (42−18=2442 - 18 = 24). Since Sophia takes Liam's old spot, she shifts 2424 places forward (towards the rear end of the line). Therefore, her new position from the rear is her old position plus the shift: 31+24=5531 + 24 = 55. Alternatively, finding the total number of people (42+31−1=7242 + 31 - 1 = 72) and applying the formula for rank from the opposite end (72−18+1=5572 - 18 + 1 = 55) yields the exact same correct result.

Step-by-Step Solution

1
Determine the total number of graduates in the line.
Total = 4242 (Liam's new position from front) + 3131 (Sophia's initial position from rear) - 11 = 7272.
After swapping, Liam occupies Sophia's original position. The sum of positions from both ends minus one gives the total count.
2
Identify Sophia's new position from the front.
Sophia is now at Liam's original position, which is 18th18^{\text{th}} from the front.
Because they exchanged places, Sophia takes over Liam's initial spot.
3
Calculate Sophia's new position from the rear.
New position = 7272 (Total) - 1818 (Position from front) + 11 = 55th55^{\text{th}}.
The position from the rear is calculated by taking the total number of people, subtracting the position from the front, and adding one.

Key Concept

Position Interchange and Total Queue Calculation
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Topics

Order and Ranking

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