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Three bells ring at intervals of 20, 30, and 40 seconds. If they all ring together at 10:00 am, when will they next ring together?

Mathematics

ScanSolve AI Answer

Answer

10:02 am\text{10:02 am}

Step-by-step solution

5 steps

Here's how to solve this problem:

  1. Identify the ringing intervals of the three bells.

    The bells ring at intervals of 20 seconds, 30 seconds, and 40 seconds. To find when they will ring together again, we need to find the Least Common Multiple (LCM) of these intervals.

  2. Find the prime factorization of each interval.

    20=2×2×5=22×520 = 2 \times 2 \times 5 = 2^2 \times 5 30=2×3×530 = 2 \times 3 \times 5 40=2×2×2×5=23×540 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5

  3. Calculate the LCM of 20, 30, and 40.

    To find the LCM, we take the highest power of each prime factor present in the numbers:

    LCM(20,30,40)=23×31×51\text{LCM}(20, 30, 40) = 2^3 \times 3^1 \times 5^1 LCM(20,30,40)=8×3×5\text{LCM}(20, 30, 40) = 8 \times 3 \times 5 LCM(20,30,40)=120 seconds\text{LCM}(20, 30, 40) = 120 \text{ seconds}

  4. Convert the LCM from seconds to minutes.

    Since there are 60 seconds in a minute:

    120 seconds=12060 minutes=2 minutes120 \text{ seconds} = \frac{120}{60} \text{ minutes} = 2 \text{ minutes}

  5. Determine the next time they will ring together.

    The bells rang simultaneously at 10:00 am10:00 \text{ am}. They will ring together again after 2 minutes.

    10:00 am+2 minutes=10:02 am10:00 \text{ am} + 2 \text{ minutes} = 10:02 \text{ am}

    The bells will next ring together at 10:02 am\text{10:02 am}.

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