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Studying for a test? Prepare with these 9 lessons on Verhältnisse, Raten und Proportionen.

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# Einführung in Verhältnisse

Video-Transkript

- [Voiceover] What I want
to explore in this video is the notion of a "rate." So, let's look at some examples of rates that you've probably encountered
in your everyday life. So, if you're driving in
your car down the road, and you're looking at the speedometer, you might see that it
says that you are going 35 M-P-H, where the M-P-H stands for 35 miles per, per hour. Well, what's that saying? That's saying, well,
every hour, how many miles are you going if you were to
stay at that current rate. So, it could be a measure of speed. How much distance are you
covering per unit time? And, most typically, when
people talk about rates, that's what they're talking about. They're talking about
how much of something that is happening per unit time. And, it doesn't have to be
even distance per unit time, you might have a, you
might have your hourly rate for someone who is doing
some type of a job. They might say that they're making, they're making $10, so they're making $10. And, actually, let me
write the dollars out so the units become a
little bit more obvious, 10 dollars, dollars per hour, dollars per, dollars per hour. And so, once again,
this is how much money. It's not talking about distance anymore. How much money is being
earned per unit time? And, so, even though rates
are often associated with how much something is
happening per some unit time, and it could be miles per hour, or it could be meters per second, or, in this case, it could be a wage, it could be dollars per hour. Rates don't have to be
just in those terms. In fact, you might say, "All right, "I have a dessert that I really enjoy, "but I'm very conscientious "about, about the number of
calories that I consume." And, you might, you
might see something like, there are 200 calories, calories per serving, per serving. And, so, this is telling
us the number of calories per a serving. And they'll tell us what a serving is. A serving might be a cup or
eight ounces or whatever else. And, so, I could say, "Okay,
look, if I have two servings, "then I'm gonna have 400 calories. "Same way, if I work two hours,
I'm gonna have 20 dollars. "If I, or if I go two hours, "I'm gonna go 70 miles." So, rates give you a sense. It's like, how fast is
something happening? Or how much of one thing is happening for every time something else happens? Now, I can write rates so they look an awful lot like a ratio. And, these words are,
actually, very related, 'cause you see that even
how they're written. R-A-T, R-A-T. Their roots are coming
from the exact same idea. In fact, this rate over
here, 35 miles per hour, it could come from, "Hey, I just, I just went
35 miles in one hour, "what's the ratio?" So, the ratio of miles to hours. And, then, you could
say, "Well, I went 35, "the ratio miles to hours "was 35 to one." Or it could have been, maybe it was 70 to two or something like that. But, that could have been reduced to 35 to one. So, as a ratio, you would typically see
it written like this... Or maybe see it written like,
see it written like this... And, sometimes, you might
even see it written like this, 35 miles to one hour. But, now it's starting
to resemble more of the special case of a ratio,
which we call a "rate." Because, this is the same thing as 35. Instead of writing it
out "miles per hour," you'll often see it written like this, miles per, miles per hour. So, these are very, very related ideas. If you find the ratio between
calories and servings, well, then, you're going
to be able to write, you're going to be able
to express it as a rate and vice versa. Now, why do we care about rates? Well, especially if we're
thinking about things like speed, without rates, it would
be hard to quantify how fast things are happening. Otherwise, we'd be in a
world where we're saying, "Hey, I'm faster than you," or "She's faster than me." But we wouldn't be able to quantify exactly how fast they are. But with rates, we can
say, "Hey, that person ran "a hundred meters in 10 seconds, "they run 10 meters per second." We can quantify exactly how
fast that thing is happening, the rate at which it is happening. Here, instead of saying,
"Hey, a cup of that "is gonna give you, is
gonna give you more energy, or, maybe, contribute more to your weight than a cup of that, and making
these relative comparisons, here, you can actually, you
can actually quantify things. And when we study rate, we're gonna study rate
a lot in mathematics. It's gonna be essential in algebra when we look at the rate of change of a line, how far it moves
in the vertical direction relative to the horizontal direction. We're gonna call that "slope". And you can even imagine
the slope of a hill as how fast is it climbing for
as much as you move forward. But we're also gonna study rates in detail when we go to calculus. In fact, the whole basis
of differential calculus, that you might see later in
high school and early college, is all about measuring instantaneous rate. How fast is something going right now? So, rates are really, really interesting, really, really important. And, I would guess that, if you just look around your life, even over the next few hours, you're going to encounter many, many, many rates.