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 57.
And I would also tell you that the ratio of girls
to boys is equal to 4:15.
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 4:11.
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 57, right,
because there are 57 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 4/15 times the boys, right?
And then we can just substitute that back into this equation,
and then we get 4/15b plus b is equal to 57, is the same thing
as 19/15b is equal to 57.
Let me clean this up a little bit.
That's separate, and then let me go here.
And we say b is equal to 57-- oh, woops --it's actually 57
times 15 all of that over 19, right?
I just multiply both sides by 15/19.
So 57 divided by 19 is 3.
So b is equal to 45.
And we know there're a total of 57 kids in the class-- g plus
b is 57 --so we know that there are 12 girls, right?
57 minus 45.
Good.
So now we know that the current boys and girls
are 45 boys and 12 girls.
So let's write that down.
So there's 12 girls and 45 boys.
Now, the question says, how many boys need to leave for
the ratio of girls to boys equals 4 4:11?
So this is the number of girls right now, 12,
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 12 girls to
the 45 boys minus the x boys that leave, right?
If that confuses you, sit and look at that for a second.
We start off with 12 girls and 45 boys in the room.
And we're saying x boys are going to leave, so the new
ratio is going to be 12:45 minus x.
And we know from this part of the problem that that new
ratio is going to equal 4:11.
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 45 minus x, and that's going to
be equal to the new ratio.
So let's solve for x.
Well, 12 times 11 is what that's 132.
132 is equal to 4 times 45, 160, 180, minus 4x.
And then if you solve for x, I think you know how to do this
right now, and we can say minus 4x is equal to minus 48.
x is equal to 12.
There we solved it.
So we say that if 12 boys left the room, the new ratio of
girls to boys would be 4:11.
And does that make sense?
Well if 12 boys left the room, then the new ratio of girls to
boys would be 12:33, right?
Because 45 minus 12 is 33.
And that's the same thing as if you divide the
top and bottom by 3.
That's 4:11.
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 5:19.
And when we add 23 bananas the ratio of apples to bananas--
and actually let's write it right now, we now have 23
bananas more --is equal to 10:61.
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 5:19.
When I add 23 bananas, now the new ratio's going to be the
number of apples to b plus 23.
The new ratio is 10:61.
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 61a is equal to 10b plus 230 and if we
divide both sides by 61, we know that a is equal to
10/61b plus 230/61.
Right?
And we could take this equation and multiply both sides by b
and we could say that a is equal to 5/19b.
Right?
Well both of these are equal to a so we could set them
equal to each other.
And you get 5/19b is equal to 10/61b plus 230 times 61.
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 2 minutes
left in this youtube.
I'm just going to solve for b, and you get b is equal to 38.
if b is equal to 38, we know that the initial ratio is 5:19.
So that's pretty easy.
We just say a is equal to 5/19 times 38 is equal to 10.
So the initial number of apples was 10, the initial number
bananas is 38, right?
So initially we started off with 48 pieces of fruit, and
then we're going to add 23 more pieces of fruit, right?
And 43 plus 28, that's what, 71 pieces of fruit.
So, let me review real quick what we said.
We said the ratio of apples to bananas is 5:19.
That's a is the number of apples, b is the
number of bananas.
When I add 23 bananas, I now have b plus 23 bananas, the
new ratio of apples to total number bananas is 10:61.
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 23 pieces and I got 71 total pieces of fruit.
I think you're now ready to try some of the more
difficult ratio problems.
Have fun!