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Difficulty: MediumFractions and Decimals

At a metropolitan central station, three express commuter trains depart simultaneously at 6:00 AM on Monday. The Green Line completes its round trip every 2142 \frac{1}{4} hours, the Blue Line every 3383 \frac{3}{8} hours, and the Red Line every 4124 \frac{1}{2} hours. Assuming continuous operation without delays, how many hours will it take for all three trains to depart simultaneously from the central station again?

  1. 131213 \frac{1}{2}Answer
  2. B
    1181 \frac{1}{8}
  3. C
    3383 \frac{3}{8}
  4. D
    1212

Answer

The correct time is 131213 \frac{1}{2} hours, found by taking the Least Common Multiple (LCM) of the three fractional time intervals.
To find the next simultaneous departure, the Least Common Multiple (LCM) of the round-trip times is needed. Converting the times to improper fractions gives 94\frac{9}{4}, 278\frac{27}{8}, and 92\frac{9}{2}. The LCM of fractions is found by dividing the LCM of the numerators (which is 27) by the Highest Common Factor (HCF) of the denominators (which is 2). This gives 272\frac{27}{2}, or 131213 \frac{1}{2} hours.

Step-by-Step Solution

1
Convert the mixed numbers to improper fractions.
Green Line: 94\frac{9}{4} hours, Blue Line: 278\frac{27}{8} hours, Red Line: 92\frac{9}{2} hours.
Improper fractions are required to accurately calculate the Least Common Multiple (LCM) of fractions.
2
Identify the formula for finding the LCM of fractions.
LCM of fractions=LCM of numeratorsHCF of denominators\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}
This formula allows us to find the lowest common multiple for non-integer values.
3
Calculate the LCM of the numerators.
The numerators are 9, 27, and 9. Their LCM is 27.
27 is the smallest number perfectly divisible by 9, 27, and 9.
4
Calculate the HCF of the denominators.
The denominators are 4, 8, and 2. Their HCF is 2.
2 is the largest number that perfectly divides 4, 8, and 2.
5
Apply the values to the fraction LCM formula.
272=1312\frac{27}{2} = 13 \frac{1}{2} hours.
Dividing the LCM of numerators by the HCF of denominators yields the final LCM of the fractions.

Key Concept

Calculating the Least Common Multiple (LCM) of fractions to solve periodic synchronization problems.
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Topics

Fractions and Decimals

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