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I want to make a quick note in this video because I realized
that some of these ratio and rate algebraic problems can
get a little bit confusing.
So I just want to make sure you get the
intuition behind it.
So if I were to tell you that Sal takes-- I don't know--
let's say he takes 1 hour to eat a pie.
Let me pick a number better than 1.
Let's say he takes 2 hours.
2 hours to eat a pie.
So he takes 2 hours per pie.
And let's say Dean takes 3 hours to eat a pie.
So he takes 3 hours per pie.
And if I were to ask you, how long does it take for both of
them, if they're both working together, how long will it
take them to eat a pie?
You might be wondering, well, why can't I just add the 2
hours per pie and the 3 hours per pie?
Why can't I just do 2 hours per pie plus 3 hours per pie?
Why would this be incorrect?
Why wouldn't this be their combined rate?
I guess the best way to think about it is think about what's
happening here.
You're saying that Sal takes 2 hours a pie, Dean takes 3
hours a pie.
If they're both working on the same pie, it's going to take
them less time.
They're going to do it faster.
But when you add the hours, what are you doing?
You're going to get 5 hours per pie.
This is not answering the question, how many hours will
it take for them to finish one pie?
This is answering the question, how much total time
would be spent if they each independently ate a pie?
Sal would spend 2 and then Dean would spend 3.
Their total time would be 5.
So we're just adding times here.
If you really want to combine their effort, you want to
combine their rates, you have to express both of these in
terms of rates.
And if you want to express 2 hours per pie in terms of
rate, you would say that this is the same thing as-- you
would just write the ratio the other way around.
Instead of 2 hours per pie you would say, 1 pie per 2 hours,
for 2 hours.
Or you could view it as 1/2 pie per hours.
Now it is expressed as a rate.
Do the same thing for Dean.
Instead of 3 hours per pie, you could say that he would do
1 pie every 3 hours.
Or another way to think about it is he does 1/3
of a pie per hour.
Now we're speaking in a language of rates.
How many pies per hour I can do, how many Dean can do.
If we combine it, then we have our combined rate.
So if we add 1/2 plus 1/3, that is our combined rate in
pies per hour.
And what is this equal to?
1/2 is 3/6.
Plus 1/3 is 2/6, which is equal to 5/6.
All the units here are in pies per hour.
And then if we wanted to say, how many hours?
So we would do literally 5/6 of a pie per hour when we're
eating together.
Now if you want to say how many hours per pie, you would
just write this ratio.
You would just flip it.
So this is the same thing as 6/5 hours per pie.
So hopefully that makes sense.
I don't want this to seem like some type of voodoo where I
just decide arbitrarily to flip both of these numbers.
I'm flipping them so I can express them as rates.
Which hopefully makes intuitive sense because you
can only add rates.
If you add the times, you're not saying that they're
working on the same pie.
You're saying how much total time is it taking them.
Only when you're flipping them and combining you're saying,
hey, their rates are being combined on the same effort.
I can combine them.
And that'll tell you their combined rate.
And then you can take the flip of that and say, on a combined
basis, how many hours?
And it should make intuitive sense.
If we're both eating the same pie, it's going to take fewer
hours than either of us.
And 6/5 is 1 and 1/5 hours or 1.2 hours.
Or 1 hour and 12 minutes, which makes sense.
It's a little bit more than if you had two Sals.
Two Sals would take 1 hour.
And two Deans would take an hour and a half.
So it's someplace in between those two numbers.
Which hopefully, make sense.