Pregunta
Suppose a train to NYC leaves the station every 9 minutes and a train to Trenton leaves the station every 15 minutes. If both trains just left the station at the same time, in how many minutes will they leave at the same time again? (Hint: Find the LCM)
Ask by Carter Joseph. in the United States
Feb 28,2025
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Both trains will leave the station together again in 45 minutes.
Solución
To solve the problem, we first find the least common multiple (LCM) of the two time intervals.
The train to NYC leaves every 9 minutes and the train to Trenton every 15 minutes.
Step 1: Factor each number into primes.
• 9 = 3²
• 15 = 3 × 5
Step 2: The LCM is the product of the highest powers of all prime factors.
• For prime 3: the highest power is 3².
• For prime 5: the highest power is 5.
So, LCM = 3² × 5 = 9 × 5 = 45.
Thus, both trains will depart at the same time again in 45 minutes.
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Simplifique esta solución Extra Insights
To find out when the trains will leave at the same time again, we need to calculate the least common multiple (LCM) of the two intervals: 9 minutes for the NYC train and 15 minutes for the Trenton train. The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270... The multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270... The smallest common multiple is 45. Thus, both trains will leave the station at the same time again in 45 minutes! Happy train traveling!
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