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What is the least possible number of planks of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length?

Mathematics

ScanSolve AI Answer

Answer

5\text{5}

Step-by-step solution

3 steps
  1. Find the least common multiple (LCM) of the tolling intervals.

    The bells toll after 7, 8, 11, and 12 seconds.
    First, find the prime factorization of each number:
    7=717 = 7^1
    8=238 = 2^3
    11=11111 = 11^1
    12=22×3112 = 2^2 \times 3^1

    The LCM is the product of the highest powers of all prime factors present:

    LCM(7,8,11,12)=23×31×71×111LCM(7, 8, 11, 12) = 2^3 \times 3^1 \times 7^1 \times 11^1 LCM=8×3×7×11LCM = 8 \times 3 \times 7 \times 11 LCM=24×77LCM = 24 \times 77 LCM=1848 secondsLCM = 1848 \text{ seconds}

    This means the bells will toll together every 1848 seconds.

  2. Convert the given time duration into seconds.

    The question asks about the next 3 hours.
    1 hour = 60 minutes
    1 minute = 60 seconds
    1 hour = 60×60=360060 \times 60 = 3600 seconds
    3 hours = 3×3600=108003 \times 3600 = 10800 seconds

  3. Calculate how many times the bells will toll together again in the next 3 hours.

    The bells toll together at 9:00 a.m. We need to find how many times they toll together after this initial time within the 3-hour period.
    Divide the total time in seconds by the LCM:

    Number of times=Total timeLCM\text{Number of times} = \frac{\text{Total time}}{\text{LCM}} Number of times=108001848\text{Number of times} = \frac{10800}{1848} Number of times≈5.844\text{Number of times} \approx 5.844

    Since we are looking for the number of complete tolling cycles within the 3 hours, we take the floor of this value.
    The bells will toll together 5 more times within the next 3 hours.

    The final answer is 5\text{5}.

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