High School

8. Buses on three routes run on different schedules. The bus on route 1 runs every 25 minutes, the bus on route 2 runs every 30 minutes, and the bus on route 3 runs every 40 minutes. If all three buses depart at the same time now, after how long will they all depart together again?

9. Four athletes run on a circular track with different lap lengths: 120 m, 150 m, 180 m, and 240 m. If they all start from the same point, after how many meters along the track will they all be together again?

10. In a school, three different classes are scheduled to start every 28 days, 35 days, and 42 days. If all classes start today, after how many days will they all start on the same day again?

Answer :

To answer these questions, we need to calculate the Least Common Multiple (LCM) of the given numbers. The LCM of a set of numbers is the smallest number that is divisible by each of them.

  1. Bus Departure Schedule

    • The buses on routes 1, 2, and 3 run every 25 minutes, 30 minutes, and 40 minutes, respectively.
    • To find when they will all depart together again, we calculate the LCM of 25, 30, and 40.
    • First, find the prime factorization of each number:
      • 25 = [tex]5^2[/tex]
      • 30 = [tex]2 \times 3 \times 5[/tex]
      • 40 = [tex]2^3 \times 5[/tex]
    • The LCM is obtained by taking the highest power of each prime present in the factorizations:
      • LCM = [tex]2^3 \times 3^1 \times 5^2 = 600[/tex]
    • Therefore, all three buses will depart together every 600 minutes, or 10 hours.

  2. Athletes on a Circular Track

    • The lap lengths are 120 m, 150 m, 180 m, and 240 m.
    • Find the LCM of these lengths to determine when they will be together again.
    • Prime factorizations:
      • 120 = [tex]2^3 \times 3 \times 5[/tex]
      • 150 = [tex]2 \times 3 \times 5^2[/tex]
      • 180 = [tex]2^2 \times 3^2 \times 5[/tex]
      • 240 = [tex]2^4 \times 3 \times 5[/tex]
    • LCM = [tex]2^4 \times 3^2 \times 5^2 = 3600[/tex]
    • The athletes will all be together again after 3600 meters.

  3. School Classes Schedule

    • Classes start every 28 days, 35 days, and 42 days.
    • We need to find the LCM of 28, 35, and 42.
    • Prime factorizations:
      • 28 = [tex]2^2 \times 7[/tex]
      • 35 = [tex]5 \times 7[/tex]
      • 42 = [tex]2 \times 3 \times 7[/tex]
    • LCM = [tex]2^2 \times 3^1 \times 5^1 \times 7^1 = 420[/tex]
    • Therefore, all classes will start on the same day again after 420 days.

By finding the LCM, we can determine when events that occur at different intervals will coincide again.

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