Ratios with algebra
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The Golden Ratio
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Ratio problem with basic algebra (new HD)
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Writing proportions
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More advanced ratio problem--with Algebra (HD version)
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Advanced ratio problems
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Alternate Solution to Ratio Problem (HD Version)
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Another Take on the Rate Problem
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Adding and Subtracting Rational Expressions
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Find an Unknown in a Proportion
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Find an Unknown in a Proportion 2
More advanced ratio problem--with Algebra (HD version) More advance ratio problem with algebra.
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- Let's do another ratio problem that's a bit of a classic.
- So, let's say that I start off with a certain number of fruit.
- So, I have some number of fruit.
- But I'll get back to that in a second.
- But we know that the ratio.
- We know that the ratio -- we know that the ratio of apples
- -- I'll do apples in this reddish color.
- The ratio of apples to oranges.
- To oranges, those colors are very similar.
- Let's say, it is equal to -- let's say it is equal to five:eight.
- So it's equal to five:eight.
- five:eight.
- Or I could also write it as five:eight, just like that.
- That's the ratio of apples to oranges.
- Now, the next thing that I know about this problem is that when
- I take away fifteen -- when I take away fifteen apples.
- So, take away fifteen apples, and the ratio becomes 1 to 4
- The ratio of apples to oranges becomes 1 to 4
- How many apples -- let me write it this way. how many total, how many total not how many apples
- how many, How much, let me say how much total fruit do I have, I have after taking away the fifteen apples? he fifteen apples
- How much total fruit do I have?
- So let's do a little bit of algebra.
- Let's say that x is the number, x is equal to the number of
- fruit that we start with.
- How much fruit are we going to end up with at the end?
- We're taking away fifteen apples. We're not doing anything to the oranges.
- So we know that x minus thirty, I'll do that in a different
- color, x minus oh x fifteen, we take away fifteen apples, xminus 15 is equal
- to the number of fruit we end up with.
- We know these two things are true.
- And now, we know that we start off with a ratio of five:eight.
- So we know that the ratio of apples to oranges is equal to
- five:eight that we start off with.
- Or, that's just another way of saying that for every group --
- assuming that we can divide our group of fruit into even -- or
- I can divide it into a set of groups, and every group
- has the same number of apples and oranges.
- That means that for every group of, what, thirteen fruits, thirteen fruits,
- there are five apples and eight oranges.
- That's what this tells me. For every group of thirteen fruits, there are five
- apples and eighteen oranges.
- And how many groups of thirteen fruit are there?
- Well, I have x total fruit.
- So if I divide x divided by thirteen, if I take x divided by thirteen,
- that is equal to -- this is thirteen thirteen right here, this is the thirteen
- at thirteen fruits per group, I have x total fruit.
- So this is the total number of groups, So this is equal to the number of groups.
- Now, for every group, I have five apples.
- So how many apples am I starting off with?
- There is a number of groups.
- For every group, I have five apples.
- So I'm starting off with five times that apples.
- fivex over thirteen apples.
- 5 x over thirteen apples
- Now, for every one of these groups, I also have eight oranges.
- So, how many oranges do I have?
- I have eight times that.
- Because for every one of these, I have eight oranges.
- So eight times that is eightx over thirteen oranges.
- Fair enough.
- That's what I start off with.
- Now, when I'm done we said that the ratio, the rato of apples to oranges
- the ratio of apples to oranges becomes 1 to 4.
- Becomes 1 to 4
- Or, another way of saying that is that, for every five -- right,
- if you add these two numbers together, for every five fruit
- there are one apple and four oranges.
- Fair enough.
- So, how many groups of five fruit do we have?
- Well, this is the total number of fruit we're left with
- after we took fifteen away.
- So how many groups of five do we have?
- We take the total number of fruit.
- x minus fifteen.
- Divide it by how many fruit we have per group.
- And this is how many groups of -- this is equal to
- the groups of five fruit.
- Of five fruit.
- Now, for each of these groups, I'm going to have
- one apple and four oranges.
- So how many apples and oranges do I now have after I took
- away fifteen of the apples?
- I took away fifteen fruit?
- Let's think about this a little bit.
- How many apples am I going to have?
- I have one apple for every one of these groups.
- So, that's how many -- so the number of groups is the
- number of apples I have.
- So, I'm going to have x minus fifteen over five apples.
- And then how many oranges am I going to have?
- For every one of these groups, I'm going to have four oranges.
- So, I'm going to have four times this many oranges.
- So we can write that as 4 over 5 times x minus fifteen oranges, right
- That's just five times the number of groups of five fruit.
- Now, what doesn't change?
- I'm messing with the number of apples, but I never changed
- the number of oranges.
- So, this number which is the number of oranges we
- started off with, has to equal this number.
- The number of oranges we ended up with.
- There's many ways to do this problem, but this is the way that's popping out of me that might be straightforward.
- So let's see if we can solve for x.
- This has to be equal to that.
- That never changed.
- So you could say 8 x over 13 is equal to 4 fifth times x minus fifteen.
- And, maybe, the best way to do this, let's multiply both sides
- of this equation times five / four.
- So, times five / four times five / four.
- These cancel out. That just becomes one.
- What does this become?
- This becomes, you see you get a four, divide it by four you get a two there.
- This becomes tenx over thirteen.
- tenx over thirteen, that's the left-thand side, is
- equal to x minus fifteen.
- Now, let's subtract x from both sides.
- And we get tenx over thirteen minus x is equal to minus fifteen.
- Let me scroll down a little bit.
- And so, what is this?
- This is the same thing as tenx over thirteen minus thirteenx over thirteen, all right
- That's just x.
- Is equal to minus fourteen.
- And then, ten minus thirteen, this is equal to -- so this is minus threex
- over thirteen, is equal to minus fifteen.
- Now, we can just multiply both sides of this equation
- by minus thirteen over three.
- Multiply this side by minus thirteen over three.
- I picked that number so that this and that cancel out.
- And that and that cancel out, and the negatives cancel out.
- So I'm just left with x on that side.
- And then, the right-hand side becomes what?
- The negatives cancel out.
- I can divide the fifteen and the three by three, so that becomes a five.
- And then this becomes a one.
- And I'm just left with five times thirteen.
- This was just, times plus thirteen, I wrote that just to make sure
- that you understood that the negatives canceled out.
- So, five times thirteen is what?
- five times twelve is sixty, so x is equal to sixty-five.
- But that is not our answer.
- Remember what x was.
- x was the number of fruit we started with.
- We just figured out that this is equal to sixty-five.
- But what the question asked, the question wanted to know is,
- how much total fruit do I have after taking away the fifteen apples.
- So the question wants to know, how many after taking the fifteen apples.
- If I started with sixty-five, I give away fifteen of the fruit, all of
- them are apples, then I'm left with sixty-five minus fifteen,
- which is equal to?
- fifty apples, oh not, fifty pieces of fruit.
- fifty pieces of fruit.
- That's what they're asking for.
- fifty pieces of fruit.
- If you want, you can go on and figure out how many apples you
- started with, you finished with, the number of oranges, or
- whatever else, but this is what they're asking.
- How much total fruit, not the number of apples, total fruit,
- do I have after taking away the fifteen apples?
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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