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Question

Difficulty: MediumFactors, Multiples, and Prime Factorization

A security guard has two patrol routes. Route A takes 18 minutes to complete, and Route B takes 24 minutes to complete. The guard starts both routes at the same time at 6:00 PM. Assuming the guard works continuously without any breaks, how many times will both routes be completed at the same time between 6:01 PM and 11:59 PM on the same evening?

  1. A
    3
  2. 4Answer
  3. C
    5
  4. D
    6
  5. E
    10

Answer

4
The correct answer is 4. The least common multiple (LCM) of 18 and 24 is 72, which means both routes are completed at the same time every 72 minutes (1 hour and 12 minutes). Starting from 6:00 PM, the simultaneous completions occur at 7:12 PM, 8:24 PM, 9:36 PM, 10:48 PM, and 12:00 AM. Only the four times between 7:12 PM and 10:48 PM fall strictly between 6:01 PM and 11:59 PM.

Step-by-Step Solution

1
Find the Least Common Multiple (LCM) of the two cycle times, 18 minutes and 24 minutes.
LCM(18, 24) = 72 minutes
Both routes are completed at the same time at intervals that are common multiples of their individual durations. The first time they align after the start is the LCM.
2
List the times when both routes are completed simultaneously, starting after 6:00 PM.
7:12 PM, 8:24 PM, 9:36 PM, 10:48 PM, and 12:00 AM
Add multiples of 72 minutes (1 hour 12 minutes) to the starting time of 6:00 PM.
3
Determine which of the calculated times fall strictly within the window from 6:01 PM to 11:59 PM.
4 times (7:12 PM, 8:24 PM, 9:36 PM, and 10:48 PM)
The times 7:12 PM, 8:24 PM, 9:36 PM, and 10:48 PM are within the specified window. The completion at 12:00 AM is outside the window.

Key Concept

Least Common Multiple (LCM) in scheduling word problems
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Topics

Factors, Multiples, and Prime Factorization

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