Welcome to the presentation on ratios.
Now, I'm going to start just giving you a definition
of ratios, and this I got from Wikipedia.
"A ratio is a quantity that denotes the proportional amount
of magnitude of one quantity relative to another." So I'm
going to tell you from the beginning that I think a ratio
is something that's easier to understand than to give a
definition for because I don't think that Wikipedia
definition is that useful.
Let me give you some examples.
If there are-- let's say there are-- let me make
this pen size is right.
Let's say there 10 boys and 2 girls in a class.
So the ratio of boys to girls would be 10:2 or 10/2.
Those are two different ways of writing it.
And we know from fractions that that's also the same
thing as 5:1 or 5/1.
We want to keep the 1 there because we know that it's a
ratio of one thing to another thing.
So what does that mean?
All that means is that for every 5 boys there's 1 girl.
And so if we told you that the ratio of boys to girls in a
room is 5:1, and we told you that there are-- let's say that
we told you that there are 100 girls, then we'd know that,
well, for every 1 of those girls, there's 5 boys, so
that means that there'd be 500 boys, right?
Or you could also look at that as the ratio of boys to girls
is 500/100, which equals 5/1.
And this is the typical way that a ratio is written:
500:100 of boys girls.
Now, let me ask you a couple of questions based on that.
I think you get the general idea.
If I told you that the ratio of, let's say, red balls to
green balls in a bag is 2:3.
And then if I also told you that there are 40 red balls,
how many green balls are there?
Well, what we say is the ratio of red balls to green balls--
so we know that there are 40 red balls, and then we want to
solve for the number of green balls, that that
is equal to 2/3.
And then we could just solve this.
We just cross multiply.
40 times 3 is 120 is equal to 2g, and then we just solve.
We just say g equals 60.
And there's an easier way of doing this kind of in your
head, and this is the algebraic way that'll always work.
But you could also just say-- let me write this a little bit.
This is a 3 down here.
You can also say, well, to get from 2 to 40, you have to
multiply by 20, so to get from 3 to g, I'm also going
to multiply by 20.
And so 3 times 20 is 60.
That's another way to do it.
A lot of you might actually find it more intuitive
just to think about it.
Well, if for every 2 red balls, there are 3 green balls.
Then if there are 40 red balls, then it makes sense that there
would be 60 green balls because for every 20, there'd be 30,
for every 40, there'd be 60.
I hope I'm not completely confusing you.
Let me give you another example.
Let's say the ratio of boys to girls is equal to 2 to 7.
And if I were tell you that the total class has 180 kids in it,
can we figure out how many boys and girls there
are in the class?
Well, let's think about it.
Well, we know that the boys to girls is equal to 2/7, and we
also know that the boys plus girls is equal to 180.
So here, we have a system of two equations and two unknowns.
And you could actually, if you really think about it,
you could actually solve this without algebra.
But I'll show you the algebraic way, because when problems get
complicated, this'll always work.
So what we can do is we can do substitution.
We know that b is equal to 2/7g, right?
I just multiplied both sides of this equation by g.
It cancels out there, and then times g, and you get this.
And then we can just substitute that back in for b.
So then we have 2/7g plus g is equal to 180.
And what's 2/7g plus-- we could 1g or 7/7g?
Well, you could do the fraction, but it's 2/7 plus 1
is the same thing as-- that's equal to 2/7 plus 7/7, right,
because that's just 1g is equal to 180.
And I'm jumping around on the chalkboard on purpose to
intentionally confuse you.
OK, this is where I am.
So 2/7 plus 7/7g equals 180.
So we have 9/7g is equal to 180.
And then we just multiply both sides times the
reciprocal of 7/9.
Oops!
That's not a g.
That's a 9.
Once, again, an intentional device to confuse you.
These cancel out, and you get g equals 180 times 7/9.
Well, 180 divided by 9, this is just 20, right?
So g is equal to 140.
So if there's 140 girls in the room, how many boys
are there going to be?
Well, we know that the whole class has 180 people, and we
know b plus g is 180, so there's going to be-- the
boys are equal to 40.
And this is really about as difficult as I guess we could
say basic ratio problems get.
There's nothing really difficult about ratios.
They're just representing, for every amount of one thing, how
much do you have of the other thing?
And then you can use that ratio if you have some other
information in terms of how many total people there are, or
how many total objects there are, or how much of one object
there is, you can use that to figure out how much of the
other object there is or how many total objects there are.
I think you're now ready to try some of the ratio problems, and
I'm going to do another presentation on what I would
consider slightly more advanced ratio problems.
So have fun.