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Zorluk: OrtaHCF and LCM

An environmental agency uses three autonomous drones to patrol a protected reserve. The drones fly on continuous looping routes. Drone X completes one full route in 4215\frac{42}{15} hours, Drone Y completes a route in 3520\frac{35}{20} hours, and Drone Z completes a route in 6330\frac{63}{30} hours.

If all three drones depart simultaneously from the base station, after how many hours will they all meet at the base station again for the first time?

Cevap: 42 hours

Cevap

42
The drones will meet again at a time that is a common multiple of all their individual cycle times. The very first time this happens is represented by the Least Common Multiple (LCM). When calculating the LCM of fractions, it is mathematically required to reduce them to their simplest terms first: 145\frac{14}{5}, 74\frac{7}{4}, and 2110\frac{21}{10}. Using the correct formula, LCM(14,7,21)HCF(5,4,10)\frac{\text{LCM}(14, 7, 21)}{\text{HCF}(5, 4, 10)}, we get 421=42\frac{42}{1} = 42 hours.

Adım Adım Çözüm

1
Identify the mathematical operation required.
The convergence time is the Least Common Multiple (LCM) of the three cycle times.
The drones will meet again at a time that is a common multiple of all their individual route completion times.
2
Simplify the given fractions to their lowest terms.
4215→145\frac{42}{15} \rightarrow \frac{14}{5}; 3520→74\frac{35}{20} \rightarrow \frac{7}{4}; 6330→2110\frac{63}{30} \rightarrow \frac{21}{10}
The formula for the LCM of fractions requires all fractions to be in their simplest form to yield the correct result.
3
Calculate the LCM of the simplified numerators.
LCM(14,7,21)=42\text{LCM}(14, 7, 21) = 42
The numerator of the resulting fraction must be divisible by all original numerators.
4
Calculate the HCF of the simplified denominators.
HCF(5,4,10)=1\text{HCF}(5, 4, 10) = 1
The denominator of the resulting fraction must evenly divide all original denominators.
5
Compute the final fraction.
421=42\frac{42}{1} = 42 hours
Applying the formula LCM of NumeratorsHCF of Denominators\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}} gives the exact time of the next simultaneous meeting.

Anahtar Kavram

Calculating the Least Common Multiple (LCM) of fractions, emphasizing the critical prerequisite of simplifying the fractions first.
Tahmini Süre:2m 30s
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Konular

HCF and LCM

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