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# Perimeter and unit conversion

Video transcript

The perimeter of a rectangular
fence measures 0.72 kilometers. The length of the fenced
area is 160 meters. What is it's width? Now, the first thing
that jumps out at me, and it might have
jumped out at you, is that they're giving
us different units. They're giving us the perimeter
in terms of kilometers, and they're giving us the
length in terms of meters. And I'm assuming that they want
the width in terms of meters, since that's what they're
giving us the length in. So what I want to do
right from the get go is just convert the
perimeter into meters. So we have the
perimeter, and I'll just represent that with a p, is
equal to 0.72 kilometers. Which I'll write km for short. Kilometers, 1,000 meters. That's what the
prefix kilo- means. And so we can say that
for every 1 kilometer we have 1,000 meters. Or we have 1,000 meters
per every 1 kilometer. And you might say
Sal, how do you know to multiply by 1,000,
instead of divide by 1,000? And one way to think about
it, and this is probably the best way to think
about it, is just look, a kilometer is a
bunch of meters. It's actually 1,000 meters. So if I'm converting
kilometers into meters, I should have a
much larger number. Whatever my number
is in kilometers, it must be a much
larger number of meters. So that's what tells
you to multiply. And also, if you care
about dimensional analysis, the dimensions
cancel out here too. We have kilometers in
the numerator, kilometers in the denominator. And so when you multiply it, you
have 0.72 times 1,000 meters. And to multiply
anything times 1,000, or really any power
of 10, you say, look, if I multiply it by 10,
I'll move the decimal to the right one space. That'd be multiplying it
by 10, it would be 7.2. Multiplying it by
100 would give us 72. If we're multiplying by
1,000, that would give us 720. And I'll put a trailing
0 here, just so that we have something
to move to the right of. So this is going to be
equal to 720 meters. So that is the perimeter. Now let's remind ourselves
what the perimeter even is. And then hopefully we
can figure out the width. So let me draw a
little box over here. And they tell us that
the length is 160 meters. So let's say that's this
dimension right over here. The length is 160 meters. So that would be this dimension
and this, it's a rectangle. So these sides are
both the same length. And our width is what
we need to solve for. So that's our width, and
this is also our width. And the perimeter is the
measure going around it. So the perimeter is going to be
this length, plus this width, plus that same length again,
plus that width over there. Or another way to think
about it-- and this might be a simpler
way-- so there's a couple of ways you
could think about it. You could say that
the perimeter is equal to the length
plus the width, plus the length plus the width. And then we know
what the length is. So the perimeter would be equal
to 160 meters plus the width. Actually, let me
write the units down. 160 meters plus the width,
plus 160 meters plus the width. And then we know what
know the perimeter is, that's actually 720 meters. So 720 meters is the perimeter. So you would get 720
meters is equal to 160 meters, plus the width, plus
160 meters plus the width. Now there's a bunch of different
ways to solve for the width. One way, you could
just say, look, if I just have the
width plus the length once, that's going to add
up to half of the perimeter. So if I just go halfway
around the rectangle, that's going to add up
to half the perimeter. So if I take my width, which
is w, plus my length, which is 160 meters, this should be
equal to 1/2 of the perimeter. This should be equal to 1/2
times our perimeter, which is 720 meters. Or you get width plus 160 meters
is equal to 1/2 times 720, or 720 divided by
2 is 360 meters. And so now you have
a situation where we have the width plus
160 meters is 360 meters. So we could now subtract
360 from both sides. Or sorry, we could subtract
160 from both sides to solve for it. Or you could even
do it in your head. If I say something plus 160 is
360, you could, in your head, say well, that
something must be 200. 200 plus 160 is 360. So you could just say
your width is 200 meters. Or if you want to do it a
little bit more formally, you could say look,
subtract 160 meters from both sides
of this equation. And you are left with the
width is equal to 200 meters. So that's one way to do it. We've solved the problem. The other way is,
you could actually go straight from this equation. So we get 720. I'm just going to assume
everything is in meters. 720. 160 meters plus
160 meters is 320. And that's going to
be the same thing as the width plus the width. Or 2 times the width. Anything plus itself is
just 2 times that anything. Now if this plus 320 is equal to
720, you could do in your head. You could say well,
what plus 320 is 720? Well, this thing
must be equal to 400. Or you could do
it more formally, and subtract 320 from
both sides of this. And you would get--
if you subtract 320 from here-- you would
get 400 is equal to, subtract 320 from this side,
you get 2 times the width. So if I have 2 times
something is equal to 400, that something must be 200. Or if we want to do
it more formally, you could divide both sides
of this equation by 2. Either way, you will get the
width is equal to 200 meters.