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- It's always good to see the same problem done more than one
- way, and it's especially true with these ratio problems
- because they, in general, can be done in many, many different
- ways, and different types of ways gel with different people.
- So I just did a video on this where I said that the
- ratio of apples to oranges starts off at five to eight.
- And we take away fifteen.
- We take away fifteen apples, and the ratio becomes one to four.
- And then the question that we have to answer is how much
- total fruit do we have after taking away the fifteen apples?
- And we figured it out.
- We figured out that it's fifty pieces of fruit.
- But let's see if we can do it a different way.
- The last time I kind of talked through it and
- used a lot of words.
- Let see if we can do it a little bit more algebraically.
- So let a is equal to the number of apples we are starting with,
- and let o equal the number of oranges.
- That doesn't change throughout the whole problem.
- So we're starting off with a apples and o oranges.
- So the ratio of a to o is equal to what?
- The ratio of a-- let me actually-- well, instead of
- trying to fit this whole problem into this part of the
- screen, let me just copy and paste this part, and do it at
- the bottom part of the screen.
- Let me copy it, and then let me scroll down here where I have
- some extra real estate because I think I'm going to need more
- than that little pocket of area right there.
- So I copied and pasted just like that.
- OK, so there we have it.
- a is the number of apples to start with, and o is
- the number of oranges.
- So we could write-- let me do it in a color that
- I haven't used before.
- I actually haven't used white.
- So we can say that the ratio of apples to oranges
- is equal to five to eight.
- That's what they tell us, that the number of apples we start
- with divided by the number of oranges, or the ratio of them--
- i have to be a little bit more particular-- is five to eight.
- Then we removed fifteen apples.
- So if we remove fifteen apples, how many apples do we have?
- We have a minus fifteen apples.
- And they tell us that the ratio-- then the ratio of
- apples to oranges becomes one to four.
- So a minus fifteen, which is the number of apples we start with
- minus fifteen, the ratio of that to oranges is equal to one over four.
- And let's see if we can solve these two equations.
- I have two equations here, and I have two unknowns.
- Let's see what we can do here.
- And I should probably shouldn't have labeled-- there's a reason
- why o isn't used in algebra a lot, because it looks just
- like a zero, but I'll try to be careful.
- So this first-- let me write down my two
- equations over here.
- It never hurts to rewrite it.
- The original apples to oranges ratio is five to eight.
- Then when we remove fifteen apples, the ratio of our new number of
- apples to oranges is equal to one over four.
- This is just an algebraic way of stating our problem.
- Let's see if we can solve this.
- If we cross-multiply both sides of this equation,
- we get eighta is equal to fiveo.
- That's not a zero.
- It's equal to fiveo.
- Or if we subtract fiveo from both sides, get eighta
- minus fiveo is equal to zero.
- So this is just a linear equation with two unknowns.
- Now, what do we get on this right-hand side?
- If we cross-multiply, we get foura minus sixty.
- I'm just multiplying four times a minus fifteen.
- That is equal to one times o, or we could just write--
- well, I'll write oneo.
- That's not a ten.
- That's oneo, or one orange.
- And if we rearrange this, if we subtract the oneo from both
- sides, we get foura minus oneo minus sixty is equal to zero.
- Or if we add sixty to both sides, we get foura minus
- oneo is equal to sixty.
- And here, you might recognize that these are just two
- linear equations right here.
- That's one of them, and that's the second of them.
- And I have two unknowns.
- I have a and o.
- I probably should have called them x and y to prevent this o
- from looking so confusing because that's a zero
- and that's an o.
- But I think you get what I'm saying.
- So how do we solve this?
- Well, what we can do is we can multiply one of these times a
- convenient factor, and then add them or subtract them to
- get rid of one of the two variables, so let's do that.
- Let's multiply this equation times minus two, so it becomes
- minus eighta plus twoo is equal to minus one hundred and twenty.
- I just multiplied this equation right here times minus two.
- If I do that and then if I add it to this equation right here,
- this equation is eighta minus fiveo is equal to zero, what do I get?
- Let's just add these two equations.
- This becomes zeroa.
- Minus five plus two is minus threeo is equal to minus one hundred and twenty, or o is
- equal to-- you divide both sides-- you can ignore this.
- You divide both sides by minus three, you get o is equal to forty.
- So let's see.
- So the oranges is equal to forty.
- And that's before and after.
- The o never changes.
- o is equal to forty.
- So what are our apples equal to?
- We could go to either of these equations.
- We could go to any of these that we have.
- Maybe we can go to this one.
- So we have eight times our number of apples is equal to five times
- the number of oranges, or five times forty.
- eight times the number of apples is equal to two hundred, or the apples is
- equal to two hundred over eight, which is equal to one hundred over four,
- which is equal to twenty-five.
- So remember what a and o are.
- O is the number of oranges we have throughout the problem.
- a is the number of apples we start with.
- That's what we start with.
- So you can say that we start with twenty-five apples, forty oranges,
- or we start with sixty-five pieces of fruit.
- Then if you remove fifteen fruit, you'll end up with fifty fruit,
- which is the exact same answer we got the other way of doing
- this problem, and it's nice to see it done two ways.
- If you want to know the number of apples we end up with,
- we started with twenty-five apples.
- If you were to take fifteen away from them, you're going
- to get ten apples.
- So you start with twenty-five apples, forty oranges, sixty-five fruit.
- You end with ten apples, forty oranges, or fifty fruit.
- Either way.
- Hopefully, it was reasonably useful to see this
- done another way.