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Zorluk: OrtaHCF and LCM

A meteorological research station operates three automated data collection buoys in the ocean. The buoys transmit complete environmental telemetry packages to a satellite every 821\frac{8}{21} hours, 1135\frac{11}{35} hours, and 1549\frac{15}{49} hours, respectively. If all three buoys initiate a transmission simultaneously at a given moment, how many hours will pass before they all initiate a transmission together again?

  1. A
    1735\frac{1}{735}
  2. B
    8849\frac{88}{49}
  3. 13207\frac{1320}{7}Cevap
  4. D
    17\frac{1}{7}

Cevap

The buoys will transmit together again after 13207\frac{1320}{7} hours.
To find the time when all three buoys will transmit together again, we must calculate the Lowest Common Multiple (LCM) of the given time intervals. For fractions, the applicable formula is the LCM of the numerators divided by the highest common factor (HCF) of the denominators. The LCM of the numerators (8, 11, 15) is 1320. The HCF of the denominators (21, 35, 49) is 7. Therefore, the LCM of the fractions is 1320/7.

Adım Adım Çözüm

1
Identify that the problem requires finding the Lowest Common Multiple (LCM) of the given fractional time intervals.
The mathematical goal is to compute LCM(821,1135,1549)\text{LCM}(\frac{8}{21}, \frac{11}{35}, \frac{15}{49}).
Simultaneous periodic events synchronize again at the least common multiple of their individual periods.
2
Apply the standard formula for the LCM of fractions.
The formula requires calculating LCM of numeratorsHCF of denominators\frac{\text{LCM of numerators}}{\text{HCF of denominators}}.
This is the established algebraic rule to compute the LCM for rational numbers.
3
Calculate the LCM of the numerators (8, 11, 15).
Since 8, 11, and 15 share no common prime factors, their LCM is simply their product: 8×11×15=13208 \times 11 \times 15 = 1320.
To find the smallest whole number that is evenly divisible by all three numerators.
4
Calculate the HCF of the denominators (21, 35, 49).
Breaking them down into prime factors gives 21=3×721=3\times7, 35=5×735=5\times7, and 49=7×749=7\times7. The highest common factor is 7.
To find the largest whole number that divides evenly into all three denominators.
5
Combine the calculated numerator and denominator results.
13207\frac{1320}{7}
Substituting the calculated values back into the fraction LCM formula yields the final synchronized interval.

Anahtar Kavram

The lowest common multiple (LCM) of multiple fractions is calculated as the LCM of their numerators divided by the highest common factor (HCF) of their denominators.
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Konular

HCF and LCM

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