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Difficulty: Very hardHCF and LCM

Three automatic signaling beacons in a traffic control system flash at regular time intervals of 415\frac{4}{15} hours, 920\frac{9}{20} hours, and 1435\frac{14}{35} hours, respectively. If all three beacons flash simultaneously at 12:00 noon, after how many hours will they all flash together again for the first time?

Answer: 7.2 hours

Answer

7.2 hours
To find when events occurring at fractional time intervals coincide, compute the LCM of the fractions. Crucially, each fraction must first be simplified to its lowest terms. 1435\frac{14}{35} reduces to 25\frac{2}{5}. Taking LCM(4,9,2)HCF(15,20,5)\frac{\text{LCM}(4, 9, 2)}{\text{HCF}(15, 20, 5)} gives 365=7.2\frac{36}{5} = 7.2 hours.

Step-by-Step Solution

1
Simplify all given fractional time intervals to their lowest terms.
The reduced fractions are 415\frac{4}{15}, 920\frac{9}{20}, and 25\frac{2}{5}.
The standard LCM formula for fractions LCM of numeratorsHCF of denominators\frac{\text{LCM of numerators}}{\text{HCF of denominators}} is mathematically valid only when all fractions are reduced to co-prime numerator-denominator pairs.
2
Calculate the LCM of the numerators.
LCM(4,9,2)=36\text{LCM}(4, 9, 2) = 36.
The least common multiple of 4=224 = 2^2, 9=329 = 3^2, and 2=212 = 2^1 is 22×32=362^2 \times 3^2 = 36.
3
Calculate the HCF of the denominators.
HCF(15,20,5)=5\text{HCF}(15, 20, 5) = 5.
The highest common factor dividing 1515, 2020, and 55 is 55.
4
Divide the numerator LCM by the denominator HCF to find the simultaneous flashing interval.
365=7.2\frac{36}{5} = 7.2 hours.
The LCM of the fractional time intervals determines the minimum duration before all events synchronize.

Key Concept

LCM of Fractions with Mandatory Simplification
Estimated Time:2m 0s
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Topics

HCF and LCM

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