>> Julie Harland: Hi.
This is Julie Harland
and I'm your math gal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
Let's solve this
geometry problem.
The perimeter
of a rectangle is 144 feet.
The length is 3 times
the width.
Find the length and width
of the triangle.
I like to draw a picture.
Alright, so I'm going
to draw a picture
of a rectangle
and so I'm going to think
of the length as the long side
and the width
as the short side.
So, I read
through this carefully.
The perimeter is 144,
so I know all the way
around is 144, okay.
The length is 3 times
the width.
Now, I don't know the width,
but the length is 3
times that.
So, if I knew the width I
could multiply it by 3.
So, what I'm going
to do is let the width be W
and I could just do
that by putting the W
on the width
of this triangle I'm drawing.
Some people like to write it
out this way,
the width is W. That's okay
as well.
Now, the length is 3 times
the width.
So, you can find your
variables this way
and say well,
if the width is 3 then the
length is 3 times that.
That'll be 3W
or you can just put it
on your little diagram.
You can do both,
but you don't need to do both.
So, what we've done here is
define the width.
We've defined the length using
this variable W.
And if you're more comfortable
using an X that's fine.
I use W so when I'm done,
if I solve
for W I know that's what the
width is going to be.
Now I know this perimeter is
in feet, but I'm going
to do the algebra
without writing the feet.
That's what's cool
about algebra.
It sort of helps you get the
answer and in the final answer
we have to remember
to write the length
and width using the correct
unit and that's feet.
Okay, now what else do
we know.
We have to get an
equation here.
The perimeter
of a rectangle is 144 feet.
Now the perimeter means you
just add all the sides.
So, one thing you can do is
take and add the four sides
of a rectangle.
I'm going to show you the
three ways people might set
up an equation.
Somebody might say well,
over here that's a W
and up here is a 3W.
So, if I add all four sides
that gives me W plus 3W plus
another W plus 3W
and that equals 144;
that's certainly true.
Now, another formula
for perimeter
that you may have seen is
that it's twice the width plus
twice the length.
So, we can also use
this formula.
What would that look like?
Well, I would say twice
the width.
So what's the width?
In my diagram
over here it's W. So,
that's 2W plus twice
the length.
And the length is 3W
so it's 2 times 3W.
[ Writing ]
>> Julie Harland: Now,
if you look
at these two different
equations in the first
equation if you add the
like terms remember this 1W
and another 1W, you're going
to get 8W equals 144.
If you look
at the second equation you
would first multiply the 2
times 3W, which is 6W
and when you add
that to the 2W you're still
going to get 8W.
So, no matter
which of these equations you
happen to write
down you'll end
up with 8W equals 144.
So, let me show you the steps
had you written it this second
way; 2W plus 6W equals 144
or 8W equals 144.
And hopefully you'll notice
that's the same thing you
would get if you add the
like terms
from this first equation
I wrote.
So, there are a couple of ways
of doing it.
Let's go ahead
and finish this.
How do you solve for W now?
You need to divide both sides
by 8 and so now we're going
to just do our division
and that should give us I
think 18.
Now, notice that said 18,
not 18 feet.
Okay, we will take care
of that eventually.
So, we have 18 for our width.
So, if I go back up to here
if you had written it this way
I could just write the 18
right now and 3 times W would
be 3 times 18, which is 54.
And in fact, the shortcut
for feet is just putting this
little mark here.
I have 18 feet and 54 feet.
The other way to do it is
to look at this
little picture.
We said the width is 18,
so you could put 18 feet here.
And then I know the length is
3 time as long
from my original picture
I drew.
So, 3 times 18 is 54 feet.
And, of course,
that means this 18
over here right
and this is 54 here as well.
So, we can check
that the perimeter really
is 144.
Let's just check it.
[ Writing ]
>> Julie Harland:
If you add the sides
of the rectangle 18, 54, 18,
54, you do get 144, right.
So, you can check
that the sides add
up to the right amount here,
right.
So, that seems
to be our answer.
And here's a common mistake
students will say
that the width is 36,
because they say oh,
it's 18 plus 18.
No, no, no, no, no.
The width is just this length
here on the side.
So the width is 18
and the length is 54.
And so that's how we want
to write our answer.
So, we just write the width is
18 feet and the length is
54 feet.
Okay, I'm going to back
to my picture here
in the beginning.
Alright, remember how we set
it up.
I sometimes think a little
bit differently.
If the perimeter is 144 then
what would it be
if I just went half
way around?
Right so I just went along the
length and around the width.
That's only half way around.
That would only be half
the perimeter.
I like to do my problems
that way.
I get easier numbers.
Let me show you how
that looks.
So, here's the picture again
and here's a little bit
different reasoning.
So, let's just say we call it
the width.
No, let's see sorry.
We called that the width.
So, let's say we called this
the width and that the length
and that's all I filled in.
I know that if I add those two
numbers I should get the half
the perimeter.
If the perimeter is 144 what
is the-- what is half
of that, 72.
So, another way
of reasoning your answer here
is to write the length plus
the width is only half way
around so that would be 72.
I just get smaller numbers,
a little bit easier equation
to deal with
and I'll still get 18,
3W will still be 54
and my answer will be
the same.
The width is still 18 feet
and the length is still
54 feet.
This gives you multiple ways
to solve the same problem.
So, you have to,
when you're working
with word problems you have a
little bit of leeway just
like in most algebra problems
and you should do what makes
you know more sense to you.
And this is an alternative way
not usually see
in algebra books,
not usually explained this way
by most teachers,
but I like to
give alternatives.
So, this works as well.
As long as that makes sense
to you, you're only going half
way around.
But you could also go all the
way around.
You got all 4 sides
to get the 144
or use the formula,
twice the width plus twice the
length, which was right here.
And no matter how you do it,
we all come
up with the same answer
that the width is 18 feet
and the length is 54 feet.
[ Silence ]
>> Julie Harland:
Please visit my website
at yourmathgal.com
where you can view all
of my videos,
which are organized by topic.