Advanced ratio problems More advanced ratio problems
Advanced ratio problems
- Welcome to the presentation on more advanced ratio problems.
- Let's get started with some problems.
- So let's say that I have a class, and then-- oh the pen is
- messed up --OK, so in the class the total number of
- students is fifty-seven.
- And I would also tell you that the ratio of girls
- to boys is equal to four:fifteen.
- So now this the interesting part, so far it
- doesn't look to tough.
- My question is how many boys need to leave the room, so how
- many boys need to leave for the ratio of girls to
- boys to be four:eleven.
- This is fascinating.
- So, a good place to start is just to figure out how many
- girls and how many boys there are in this classroom.
- And we already learned how to do that in the introduction
- to ratio problems.
- We know that the girls plus the boys is equal to fifty-seven, right,
- because there are fifty-seven kids in the room.
- And we also know, just multiplying-- taking this
- equation --and multiplying both sides by b, we also know that
- the girls are equal to four / fifteen times the boys, right?
- And then we can just substitute that back into this equation,
- and then we get four / fifteenb plus b is equal to fifty-seven, is the same thing
- as nineteen / fifteenb is equal to fifty-seven.
- Let me clean this up a little bit.
- That's separate, and then let me go here.
- And we say b is equal to fifty-seven-- oh, woops --it's actually fifty-seven
- times fifteen all of that over nineteen, right?
- I just multiply both sides by fifteen / nineteen.
- So fifty-seven divided by nineteen is three.
- So b is equal to forty-five.
- And we know there're a total of fifty-seven kids in the class-- g plus
- b is fifty-seven --so we know that there are twelve girls, right?
- fifty-seven minus forty-five.
- So now we know that the current boys and girls
- are forty-five boys and twelve girls.
- So let's write that down.
- So there's twelve girls and forty-five boys.
- Now, the question says, how many boys need to leave for
- the ratio of girls to boys equals four four:eleven?
- So this is the number of girls right now, twelve,
- this is the number boys.
- Let's say x is a number of boys that need to leave the room.
- So if x boys leave the room the new ratio will be twelve girls to
- the forty-five boys minus the x boys that leave, right?
- If that confuses you, sit and look at that for a second.
- We start off with twelve girls and forty-five boys in the room.
- And we're saying x boys are going to leave, so the new
- ratio is going to be twelve:forty-five minus x.
- And we know from this part of the problem that that new
- ratio is going to equal four:eleven.
- There, we just set up a equation with one unknown
- and we can solve for x.
- I hope that doesn't confuse you much.
- All we did is we figured out how many boys, how many
- girls are in the room now.
- We said x is the number boys that need to leave.
- And we said the new ratio is going to be girls to the new
- number of boys, which is forty-five minus x, and that's going to
- be equal to the new ratio.
- So let's solve for x.
- Well, twelve times eleven is what that's one hundred and thirty-two.
- one hundred and thirty-two is equal to four times forty-five, one hundred and sixty, one hundred and eighty, minus fourx.
- And then if you solve for x, I think you know how to do this
- right now, and we can say minus fourx is equal to minus forty-eight.
- x is equal to twelve.
- There we solved it.
- So we say that if twelve boys left the room, the new ratio of
- girls to boys would be four:eleven.
- And does that make sense?
- Well if twelve boys left the room, then the new ratio of girls to
- boys would be twelve:thirty-three, right?
- Because forty-five minus twelve is thirty-three.
- And that's the same thing as if you divide the
- top and bottom by three.
- That's four:eleven.
- So there, we got it right.
- So what looked like a very hard problem actually wasn't so bad
- when you just sit down and work through the algebra.
- Let's do another problem.
- Let's say --this thing sometimes malfunctions --OK,
- let's say that the ratio of apples to bananas in a
- basket is equal to five:nineteen.
- And when we add twenty-three bananas the ratio of apples to bananas--
- and actually let's write it right now, we now have twenty-three
- bananas more --is equal to ten:sixty-one.
- So the question is, what is the total amount of
- fruit in the basket?
- Amount of fruit-- --ah, that's so messy --after
- adding the bananas.
- So I actually gave you a hint just when I wrote
- down initial problem.
- We're saying the ratio of a to b-- so let a equal the number
- of apples, and b equal the number bananas --so the ratio
- of apples to bananas equals five:nineteen.
- When I add twenty-three bananas, now the new ratio's going to be the
- number of apples to b plus twenty-three.
- The new ratio is ten:sixty-one.
- So how do we solve this?
- Well, once again we have two equations and two unknowns.
- We know that-- I guess let's take this equation first,
- because it's a little more complicated --we know if we
- cross multiply that sixty-onea is equal to tenb plus two hundred and thirty and if we
- divide both sides by sixty-one, we know that a is equal to
- ten / sixty-oneb plus two hundred and thirty / sixty-one.
- And we could take this equation and multiply both sides by b
- and we could say that a is equal to five / nineteenb.
- Well both of these are equal to a so we could set them
- equal to each other.
- And you get five / nineteenb is equal to ten / sixty-oneb plus two hundred and thirty times sixty-one.
- And we solve for b.
- While this might seem complicated to you at first,
- but it's just a basic linear equation.
- And for the sake of time, because I only have two minutes
- left in this youtube.
- I'm just going to solve for b, and you get b is equal to thirty-eight.
- if b is equal to thirty-eight, we know that the initial ratio is five:nineteen.
- So that's pretty easy.
- We just say a is equal to five / nineteen times thirty-eight is equal to ten.
- So the initial number of apples was ten, the initial number
- bananas is thirty-eight, right?
- So initially we started off with forty-eight pieces of fruit, and
- then we're going to add twenty-three more pieces of fruit, right?
- And forty-three plus twenty-eight, that's what, seventy-one pieces of fruit.
- So, let me review real quick what we said.
- We said the ratio of apples to bananas is five:nineteen.
- That's a is the number of apples, b is the
- number of bananas.
- When I add twenty-three bananas, I now have b plus twenty-three bananas, the
- new ratio of apples to total number bananas is ten:sixty-one.
- And I just used both of these equations.
- Two equations and two unknowns.
- Solved for a, and then substituted, and
- I solved for b.
- Nothing fancy here.
- I know there's a lot of fractions here, but if you
- just work through this, the fractions actually work out.
- And I was able to solve for a and b.
- Add the twenty-three pieces and I got seventy-one total pieces of fruit.
- I think you're now ready to try some of the more
- difficult ratio problems.
- Have fun!
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