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Difficulty: MediumHCF and LCM

An urban traffic control system manages three independent electronic toll gates. Based on their internal sensor loops, Gate A completes its automated scanning cycle every 125\frac{12}{5} seconds, Gate B every 1825\frac{18}{25} seconds, and Gate C every 2735\frac{27}{35} seconds. If all three gates reset their cycles simultaneously, how many seconds will it take for all three gates to reset simultaneously again? Express your answer as an exact decimal.

Answer: 21.6 seconds

Answer

21.6
Simultaneous repeating events require finding the Lowest Common Multiple (LCM) of their individual periods. For fractional periods, the rule is LCM=LCM of numeratorsHCF of denominators\text{LCM} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}. The LCM of 1212, 1818, and 2727 is 108108, and the HCF of 55, 2525, and 3535 is 55. Therefore, the LCM of the fractions is 1085\frac{108}{5}, which evaluates precisely to 21.621.6.

Step-by-Step Solution

1
Identify the mathematical operation required.
Calculate the LCM of the fractions 125\frac{12}{5}, 1825\frac{18}{25}, and 2735\frac{27}{35}.
Simultaneous cyclic events coincide at the Lowest Common Multiple of their individual time intervals.
2
Determine the LCM of the numerators.
The LCM of 1212, 1818, and 2727 is 108108.
The formula for the LCM of fractions requires calculating the LCM of their respective numerators.
3
Determine the HCF of the denominators.
The HCF of 55, 2525, and 3535 is 55.
The formula for the LCM of fractions requires calculating the HCF of their respective denominators.
4
Apply the fraction LCM formula.
1085=21.6\frac{108}{5} = 21.6
Dividing the LCM of numerators by the HCF of denominators yields the final LCM of the given fractions.

Key Concept

Lowest Common Multiple (LCM) of Fractions
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Topics

HCF and LCM

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