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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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