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Proportionality

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Proportionality

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Let's talk about proportionality.
All this means, if we say that two variables are
proportional, it just means that they are constant
multiples of each other.
So if we say that y is equal to 3 times x, we can say that
y is-- and we can be a little bit more specific-- we could
say y is directly proportional.
y is directly proportional to x.
And of course, we could divide both sides of this equation by
3, and we would get 1/3 y is equal to x.
So once again, x is a constant multiple of y.
We could write this as x is equal to 1/3 y.
So we could also write that x is directly proportional to y.
If y is directly proportional to x, then x is also directly
proportional to y, by a different
proportionality constant.
So this term right here, this is the constant of
proportionality between y and x, or between x and y.
This is the constant of proportionality.
So in general, if I say that a is directly proportional--
sometimes you'll just see it written as a is proportional--
a is directly proportional to b.
In general, from this statement, we could write that
a is equal to some constant times b, where this constant
is equivalent to that constant or that constant.
It's equal to some number.
And you could also say that b is directly proportional to a,
where if you divide both sides of this equation by k, you
would get b is equal to 1/k a.
This is direct proportionality.
Now another thing you might see-- and we'll see it later
in this video-- is being inversely proportional.
So let me draw a little line here.
So let me give you a few examples of inverse
proportionality.
Let's say that y is equal to 5 times 1/x.
Notice what's going on here.
In this situation, y is proportional to the
inverse of x, right?
Or you could say that y is inversely proportional-- I'll
abbreviate it-- proportional to x.
It is directly proportional to the inverse of x.
So here is our proportionality constant, but it is directly
proportional to the inverse of x.
Here's another example, and I'm just playing
with letters here.
I could write a is equal to 10/b.
Once again, a is inversely proportional to b.
You could rewrite this as a is equal to 10 times 1/b.
a is proportional to the inverse of b, or it's
inversely proportional to b.
So now that we have that out of the way, let's do some
examples using our newly-found knowledge of direct
proportionality and inverse proportionality.
So here I have a problem.
It says that x is proportional to y.
So that means that x is equal to some constant times y.
And just so you're familiar with other notation that you
might see when you're talking about proportionality.
Other ways to write this statement here. x is
proportional to y.
You could write it like this. x is equal to some
constant times y.
Or you could write it as x is proportional to y.
That little, I don't know, left facing fish looking thing
means proportional.
This and this mean the same thing.
This means that x is equal to some constant times y.
Or sometimes you'll even see it x squiggly line y. x is
equal to some constant times y.
This, this, and this, and that are all equivalent statements.
But this is the most useful because you say, oh, there's
some constant, maybe we can solve for that constant.
Then the next statement here says, x is proportional to y
and inversely proportional to z.
So that second statement says that x is equal to k.
It's going to be a different constant, not necessarily the
same constant.
So let me put a different-- well, I'll call it k2, I'll
call this k1-- is equal to some other constant-- not
necessarily some other, but I'm guessing it's going to be
some other constant-- times the inverse of z, times 1/z.
That's what inversely proportional to z means.
And other ways we could write this is, x is proportional to
the inverse of z.
x is inversely proportional to z, or x is inversely
proportional to z.
These things in orange are all equivalent.
I just wanted to make you familiar with it in case you
ever see it.
And then they tell us, OK, if x is proportional to y,
inversely proportional to z-- we wrote that already-- and x
is equal to 2 when y is equal to 10.
So let's do that.
x is equal to 2 when y is equal to 10.
So x is equal to 2 when k1 is multiplied by y.
That's what that statement tells us, and y is 10 when k1
is multiplied by 10.
x is 2 when y is 10.
I just put those numbers in there.
And then we can actually use this now to solve for k1.
But before that, let's see what this next
statement tells us.
It says x is 2 when y is 10 and z is 25.
So x is 2 when k2 is multiplied by 1/z or 1/25.
So that's what that second statement tells us.
So they say find x when y is equal to 8 and
z is equal to 35.
So essentially what they're asking us, hey, why don't you
use this information that we gave you in the green and the
red, solve for the different proportionality constants, and
then use that to write, I guess, the specific equations
for the proportionality or the inverse proportionality, and
then solve for y or z.
Let's just do it.
So here, up here in green, let me rewrite it over here.
We know that x is 2 when k1 is multiplied by y where y is 10.
Divide both sides of this equation by 10.
You get 2 over 10 is 1/5.
You get k1 is equal to 1/5.
So that tells us that this first equation right here--
I'll write it in that original color-- is x is
equal to 1/5 y.
Now, that second statement in red, we know that 2 is equal
to k2 times 1 over 25.
Let's multiply both sides of this equation by 25.
This cancels out.
And we're left with k2-- k2 is equal to 50.
k2 is equal to 50, so we can write this equation right here
that x is equal to 50 times 1/z or is equal to 50/z.
So we've now solved for the two constants of
proportionality for these two equations, so now we can
answer the second part of this question.
Find x when y is equal to 8 and z is equal to 35.
So the situation when y is equal to 8-- we just go right
here-- x is equal to 1/5 times 8, which is equal to 8/5.
So that's what x is equal to when y is equal to 8.
When z is equal to 35, once again, x is equal to 50/z, is
equal to 50/35.
Now let's see, we can divide the numerator and the
denominator here by 5, so we get 10/7.
So when z is equal to 35, we get 10/7.
When y is equal to 8, x is equal to 8/5.
So those are our two answers, right there.
Let's do another one.
Ohm's Law.
Ohm's Law states that current flowing in a wire is inversely
proportional to resistance of the wire.
So current flowing in a wire is inversely proportional to
resistance of a wire.
So let's use I for current.
I is equal to current.
And we'll use R for resistance.
R is equal to resistance.
And you might be wondering why I picked I, but later on when
you start studying electricity and maybe you become an
electrical engineer, you'll see that I is the conventional
letter used for current.
And I won't go into the details of why, just yet.
But it tells us that current, that I, is inversely
proportional to resistance.
So that first statement I underlined in yellow says that
current is inversely proportional.
It's equal to some constant times the inverse of
resistance.
It's inversely proportional to resistance of the wire.
If the current is 2.5 amperes when the resistance is 20
ohms. So the current is 2.5 when k is multiplied by 1 over
a resistance of 20 ohms, 1/20 ohms. And I'm assuming,
because they're going to keep the units in amperes and ohms
later on, so we could write the units here if we wanted
to, but I'll keep it simple and not write the units.
They ask, find the resistance when the current is 5 amperes.
So this first statement, right here.
The current is 2.5 amperes when the resistance is 20
ohms. That's this equation, right here.
So we can use this to solve for k.
You multiply both sides of this by 20.
These cancel out, and you're left with k is equal to--
what's 20 times 2.5?
20 times 2 is 40.
20 times 0.5 is 10.
So it's going to be 50, 40 plus 10.
So k is equal to 50.
So in this situation, this equation is I is equal to 50
times 1/R or 50/R.
That's our relationship, right there.
And then they ask, find the resistance when the
current is 5 amperes.
So when our current is 5 amperes, so 5 is equal to 50
over the resistance.
Let's multiply both sides of this equation by the
resistance, and you get 5R is equal to 50.
I just multiplied both sides of this equation by R.
These canceled out, so I got 5R is equal to 50.
Divide both sides of this equation by 5.
We get R is equal to 50 divided by 5 is 10.
So the answer is, when the current is 5 amperes, the
resistance will be equal to 10 ohms. And we could write it
like this, 10 ohms, just like that.
Let's do one more.
The intensity of light is inversely proportional to
the-- now let's be careful here-- to the square of the
distance between the light source and the object being
illuminated.
So let's just say, well, we could use I again.
Let's say I is equal to intensity of light.
And let's say that D-- I'm going to do it in a different
color-- D is equal to the distance between the light
source and object being illuminated.
So that's what?
D, the distance between the light source and the object
being illuminated.
Now what does this first sentence tell us?
The intensity of the light is inversely proportional.
So the intensity of the light, I, is inversely proportional.
So it's going to be some proportionality constant times
the inverse.
But notice, the intensity of light is inversely
proportional to the square of the distance.
Not just to the distance.
So to the square of the distance.
So to distance squared.
If it just said to the distance, we would
just have a D here.
But it says the intensity of light is inversely
proportional.
So 1 over the square of the distance between the light
source and the object being illuminated.
So that's what this first statement is going to give us,
this equation right here.
Now, they tell us, a light meter that is 10 meters-- so
they're saying, when distance is equal to 10 meters-- a
light meter that is 10 meters from a light source
registers 35 lux.
So when the distance is equal to 10 meters, they're telling
us that the intensity-- the lux is a measure of
intensity-- the intensity is equal to 35 lux.
So what intensity would register 25 meters from the
light source?
So this first statement they gave us, that I wrote down the
information in this purple color, we can use that to
solve for the proportionality constant.
And then once we have that constant, we can solve for I
or D, given one or the other.
So that statement told us, they told us, that when D is
equal to 10, I is equal to 35.
So I is equal to 35, so 35 is equal to K times 1 over D
squared, 1 over 10 squared.
Or we could say that 35 is equal to k times 1/100,
multiply-- 10 squared is just 100-- multiply both sides of
this equation by 100.
These cancel out.
And we're left with k is equal to-- what is this?
3,500.
So we've solved, we've used this statement here where they
gave us some values to solve for our
proportionality constant.
So now we know that the equation is-- the intensity is
equal to 3,500-- that's k times 1 over D squared.
Or you can just write over D squared.
Now, they say, what intensity would it register 25 meters
from the light source?
So they're saying D is 25.
So intensity is equal to 3,500 divided by 25 squared.
Our distance is 25 meters.
So the intensity is going to be equal to 3,500 divided by--
well, actually, let's just keep it simple.
Let me see, well, this is divided by 25 times 25.
It's 625.
Let's just get the calculator out.
So we get, it's going to be 3,500 divided by 25 times 25,
which is 625, which is equal to 5.6.
It's equal to 5.6 lux, which is the
unit for light intensity.
Hopefully you found that useful.

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Proportionality

Proportionality