Two cars on a circular track traveling at the rate of 10 and 15 laps per minute. If both the cars starts at a same time, after how long the cars cross at the strating point?
Question
Two cars on a circular track traveling at the rate of 10 and 15 laps per minute. If both the cars starts at a same time, after how long the cars cross at the strating point?
Solution
Para resolver este problema, necesitamos encontrar el tiempo en el que ambos coches se encontrarán nuevamente en el punto de partida. Esto se puede determinar encontrando el mínimo común múltiplo (MCM) de los tiempos que cada coche tarda en completar una vuelta.
- El primer coche viaja a una velocidad de 10 vueltas por minuto.
- El segundo coche viaja a una velocidad de 15 vueltas por minuto.
Primero, determinamos el tiempo que cada coche tarda en completar una vuelta:
- El primer coche tarda 1/10 de minuto por vuelta.
- El segundo coche tarda 1/15 de minuto por vuelta.
Para encontrar el MCM de 1/10 y 1/15, primero encontramos el MCM de los denominadores 10 y 15.
Descomponemos los números en factores primos:
- 10 = 2 * 5
- 15 = 3 * 5
El MCM de 10 y 15 es el producto de los factores primos elevados a la mayor potencia de cada uno:
- MCM(10, 15) = 2^1 * 3^1 * 5^1 = 30
Por lo tanto, el MCM de 1/10 y 1/15 es 1/30.
Esto significa que ambos coches se encontrarán nuevamente en el punto de partida después de 1/30 de minuto.
Sin embargo, para encontrar el tiempo en minutos, tomamos el recíproco de 1/30:
- Tiempo = 30 minutos
Por lo tanto, ambos coches cruzarán el punto de partida nuevamente después de 30 minutos.
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