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Perimeter and unit conversion

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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.