Anna practices piano every 15 minutes and Leah practices every 20 minutes. If they both start practicing at the same time, how long will it take before they are practicing at the same time again?
Answer: B) 60 minutes
Solution: List the multiples of 15 and 20 to find the least common one:
Two traffic lights on a road change at intervals of 40 seconds and 75 seconds respectively. If they both change simultaneously at a certain point in time, how many seconds will pass before they change simultaneously again?
Answer: B) 300 seconds
Solution: To find the first time both lights will change simultaneously, look for common multiples:
Sarah goes to the gym every 9 days, and Mike goes every 12 days. If they both go to the gym on the same day, how many days will pass before they both go to the gym on the same day again?
Answer: C) 36 days
Solution: Find the least common multiple by listing the multiples of each number:
The Least Common Multiple (LCM) of two or more integers is the smallest integer that is divisible by all the numbers without any remainder. LCM problems typically involve situations where you need to find a common time or quantity that involves multiples of given numbers synchronizing or matching up. Here’s why the problems I provided are LCM problems:
Synchronization: When you are looking for a point in time or a measure where events align or repeat together, you are seeking the LCM. For instance, when two people starting exercises at different intervals want to find out when they will next exercise together, they are looking for the LCM of their exercise intervals. This is because the LCM represents the smallest interval at which both original intervals will align.
Repetition or Alignment: Similarly, in scenarios involving traffic lights changing at different intervals or episodes of TV shows of different lengths finishing at the same time, you seek the LCM to find the earliest time both events will coincide.
The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is the largest number that divides two or more integers without leaving a remainder. Here are scenarios typically associated with GCD:
Dividing into groups or parts: If you are dividing a set of items into smaller groups and need to know the largest group size that allows all items to be evenly distributed without leftovers, you use the GCD. For example, if you have a certain number of pencils and markers and want to package them in such a way that each package has the same number of pencils and markers without any left over, you look for the GCD of the number of pencils and markers.
Reduction or Simplification: When reducing fractions to their simplest form, you divide the numerator and the denominator by their GCD.
In summary, use LCM when you need to synchronize or align intervals or sequences, and use GCD when dividing things into uniform groups without leftovers or simplifying ratios. This distinction helps apply the right mathematical concept to practical problems effectively.
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