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Difficulty: MediumOrder and Ranking

In a processing queue of a mainframe computer, Job Alpha is positioned 28th28^{\text{th}} from the front of the queue, and Job Beta is positioned 34th34^{\text{th}} from the back of the queue. If there are exactly 1212 jobs currently waiting between Job Alpha and Job Beta, what is the minimum possible total number of jobs in this queue?

Answer: 48

Answer

48
To find the minimum possible number of jobs, we assume an overlapping sequence. The formula for the minimum total in a ranking problem is: Total = (Rank from Front) + (Rank from Back) - (Number of items in between) - 2. Substituting the values gives: 28 + 34 - 12 - 2 = 62 - 14 = 48.

Step-by-Step Solution

1
Identify the queue configuration required for a minimum total.
The configuration must be overlapping, meaning Job Beta is ahead of Job Alpha in the queue.
An overlapping configuration results in the smallest possible total number of items because the counted items from both ends overlap.
2
Apply the minimum total formula for overlapping cases.
Total = Rank of Alpha from front + Rank of Beta from back - Jobs between them - 2
This formula deducts the overlapping region (the jobs between them and the two jobs themselves) which were counted twice.
3
Substitute the provided numerical values into the formula.
Total = 28 + 34 - 12 - 2
Job Alpha is 28th, Job Beta is 34th, and there are 12 jobs in between.
4
Calculate the final value.
Total = 62 - 14 = 48
Basic arithmetic yields the minimum possible number of jobs in the queue.

Key Concept

Order and Ranking (Overlapping/Minimum Capacity Queue)
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Topics

Order and Ranking

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