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April 27th, 2016, 09:26 AM   #1
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Unhappy A 'bus' combination/probability problem.

This is problematic. Considering i'm bad at math ( only thing i am good at is arithmetic ) i need help.

So i'm working through a textbook, and i run into a problem to do with buses leaving at different times. Basically i have to figure out what time do both buses leave.

I'll post the question:

"Red buses leave the bus garage every 12 minutes. Blue buses leave the bus garage every 20 minutes, a red bus and a blue bus both leave at 9am. At what time will a red bus and a blue bus next leave the garage together"

Please for the love of science enlighten me on how to figure this out and why it works that way? ( optional )

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April 27th, 2016, 11:30 AM   #2
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This problem may be solved without the use of combinations/probabilities. It involves the least
common multiple (lcm) of two numbers, 12 and 20, the lcm of which is 60. So the next time
that a red bus and a blue bus leave the station simultaneously is 10 am.

The leat common multiple of two numbers is the smallest number that has both numbers as a
factor. E.g. lcm(3, 7) = 21, lcm(4, 5) = 20, lcm(4, 8) = 8, lcm(6, 8) = 24.

Does that help?
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April 27th, 2016, 06:05 PM   #3
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It might help to visualize it if you write it out.

Red buses leave every 12 minutes, and one left at 9:00 So the next few Red buses will leave at:
9:12, 9:24, 9:36, 9:48, 10:00, 10:12, 10:24 etc

Similarly with the blue buses. One left a 9:00, so the next few will leave at:
9:20, 9:40, 10:00, 10:20, 10:40, etc.

You'll notice that two of those times match, and this is at 10:00

So at the most basic level, the question is asking if you add 12+12+12+12 etc what numbers will this list share with when you add 20+20+20+20 etc.

Another way to look at the sums of adding 12 to itself over and over is to look at 12x1, 12x2, 12x3...
Similarly with 20 you can instead look at 20x1, 20x2, 20x3...

And so this is where the concept of the least common multiple (LCM) comes in. In each list of multiples, there will be numbers that are shared by both (meaning numbers that appear in both lists). If you make a long list, you'll see the common multiples will be 60, 120, 180... and so on. You want the least of these common multiples (60) because it's asking when the next time two buses leave at the same time. And 60 minutes after 9:00am is 10:00am.

This all sounds very complicated until you realize it's an LCM problem and how to approach it. Until you get the hang of it, perhaps start by writing the times out as I did in the beginning and the whole LCM thing will start to make sense later.
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April 28th, 2016, 03:29 AM   #4
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I am so stupid

Oh, my god... Thank you guys so much for helping my stupid brain out. I cannot believe it was that simple. I think i thought the question was related to the piece of text above it haha.

No seriously though, thanks a lot. I am now quite embarrassed


It's funny as well as i had just skimmed over LCM and HCF ( already knowing those terms i didn't bother to think )

^ thanks.
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