找到比值上面的未知數 (英)

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找到比值上面的未知數 (英) : 找到比值上面的未知數

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We are asked to solve the proportion and we have 8/36 = 10/?
Now there is a bunch of different ways to solve this.
And I will explore all of them.
So one way to think about it is these two need
to be equivalent fractions. So whatever happened
to the numerator also has to happen to the denominator.
So what do you have to multiply 8 by to get 10?
You could multiply 8 * 10/8 will definitely give you 10.
So we're multiplying by 10/8 over here.
Or another way to write 10/8 is the same thing as 5/4.
So we're multiplying by 5/4 to get from 8 to 10.
If we did that to the numerator in order to have an equivalent fraction
you have to do the same thing to the denominator.
You have to multiply it times 5/4.
And so we can say that this n is going to be equal to 36 * 5/4.
Or you could say that this is going to be equal to 36*5 divided by 4.
And now 36 divided by 4, we know what that is.
We can divide both the numerator and denominator by 4.
You divide the numerator by 4, you get 9.
Divide the denominator by 4, you get 1. You get 45.
So that's one way to think about it.
8/36 = 10/45. Another way to think about it is,
what do we have to multiply 8 by to get its denominator?
How much larger is the denominator 36 than 8?
Let's just divide 36 over 8.
So 36/8 is the same thing as-- so we can simplify, dividing the numerator and denominator by 4.
That's their greatest common divisor. That's the same thing as 9/2.
If you multiply the numerator by 9/2, you get the denominator.
Then we'll have to do the same thing over here.
If 36 is 9/2 times 8, let me write this. 8 * 9/2 = 36.
That's how we go from the numerator to the denominator.
Then to figure out what the denominator here is,
if we want the same fraction, we'll have to multiply by 9/2 again.
So then we'll get 10 * 9/2 = n, is going to be equal to this denominator.
And so this is the same thing as saying 10*9 / 2,
divide the numerator and denominator by 2,
you get 5/1 which is 45. So 45 = n.
Once again we got the same way,
completely legitimate way to solve it.
Now sometimes when you see a proportion like this,
sometimes you'll say oh you could cross multiply.
And you can cross multiply and
I'll teach you how to do that.
And that's sometimes the quick way to do it.
But I don't like teaching it the first time
you look at proportions
because it's really just something mechanical.
You really don't understand what you're doing and
it really comes out of a little bit of algebra.
And I'll show you the algebra as well.
But if you don't understand it and
if it doesn't make as much sense to you
at this point, don't worry too much about it.
So we have 8/36 = 10/n. When you cross multiply
you're saying that the numerator here
times the denominator over here
is going to be equal to,
so 8*n is going to be equal to
the denominator over here--
let me do this in a different colour--
the denominator over here
times the numerator over here.
This is what it means to cross multiply.
So this is going to be equal to 36*10.
Or you could say, let me do this
in a neutral colour now,
you could say that 8n = 360.
And so you're saying 8 times what
is equal to 360. Or to figure out what that
times what is, you divide 360 divided by 8.
So we can divide, and this is a little bit of
algebra here, we're dividing both sides
of the equation by 8, and we're getting n= 360/8.
And you know, you can do that
without thinking in strict algebraic terms.
You say 8 times what is 360? Well, 8 times 360/8.
If I write 8 x ? = 360, well ? could definitely be 360/8.
If I multiply these out, this guy and that guy
cancel out and it's definitely 360.
And that's why it's 360/8.
But then we want to actually divide this
to actually get our right answer
or a simplified answer. 8 goes into 360.
8 goes into 36 4 times. 4 times 8 is 32.
You have a remainder of 4. Bring down the 0.
8 goes into 40 5 times. 5 times 8 is 40.
And then you have no remainder. And you're done.
Once again we got n=45. Now the last way I am going
to show you involves a little bit of algebra.
If any of the ways before this worked, that's fine.
And where this is sitting in the playlist you're
not expected to know the algebra.
But I want to show you the algebra just because
I want to show you that this
cross-multiplication isn't some magic.
That using algebra we will get this exact same thing.
But you could stop watching this, if you find this part
confusing. So let's rewrite our proportion.
8/36 = 10/n. And we want to solve for n.
The easiest way to solve for n is maybe multiply
both-- this thing on the left is equal to
this thing on the right. So we can multiply them both
by the same thing and the equality will still hold.
So we can multiply both of them by n.
On the right hand side, the n's cancel out.
On the left hand side we have 8/36 * n = 10.
Now if we want to solve for n, we can literally multiply
if we want just an n here, we would want to multiply
this side times 36- I'll do that in a different colour-
we would want to multiply this side times 36 times 8.
Because if you multiply these guys out,
you get 1 and you just have an n. But since
we're doing it to the left hand side, we also
have to do it to the right hand side. So times 36/8.
These guys cancel out and we're left with
n= 10 * 36 is 360/8. And notice we're getting the
exact same value that we got with cross multiplying.
And with cross multiplying, you're actually
doing 2 steps. You're actually doing an
extra step here. You're multiplying both
sides by n so that you get 8n.
And then you're multiplying both sides
by 36 so that you get your 36
on both sides and you get this value here.
But at the end when you simplify it, you'll get
the exact same answer. Those are
all different ways to solve this proportion.
Probably the most obvious way or the easiest
way to do it in your head was either
just looking at what you have to multiply
the numerator by and then doing the same thing
to the denominator or maybe by cross multiplication.

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