Perimeter and unit conversion

the perimeter of a rectangular fence measures 0.72 kilometers the length of the fenced area is 160 meters what is its width now the first thing that jumps out at me 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 maybe 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 perimeters 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 killed the prefix kilo means and so we can say that for every 1 kilometer we have a thousand meters or we have 1,000 meters per every one kilometer and you might say Sal how do you know to multiply by a thousand instead of divided by a thousand and one way to think about it is probably the best way to think about it is this look a kilometer is a huge 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 - we have kilometers in the numerator kilometers in the denominator and so when you multiply it you have 0.72 times a thousand meters and to multiply anything times a thousand 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 would be multiplying it by 10 it would be 7.2 multiplying it by 100 would give us 72 if we're multiplying by a thousand that would give us 720 s and I'll put a trailing zero here just so that we have something to move to the right of so this is going to be equal to 720 720 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 so let me a little box and they tell us that the length is 160 meters so let's say that 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 ways you could think about you could say that the perimeter 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 and actually let me write the unit's down 160 meters plus the width plus 160 meters plus the width and then we then know we know what 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 half way around if I just go half way 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 plus 160 meters this should be equal to one-half of the perimeter this should be equal to 1/2 times our perimeter which is 720 meters or you get with plus 160 meters is equal to 1/2 times 720 or 720 divided by 2 is 360 360 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 so our sorry we can subtract 160 from both sides to solve for or you could even do it for 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 is 200 meters or if you want to do a little bit more formally you could say look subtract 160 meters from both sides from both sides of this equation and you are left with the width is equal to the width is equal to 200 meters so that's one way to do it we've solved the problem the other way is that you could actually go straight from this equation so we get 720 I'm just going to assume everything is in meters 720 is 160 meters plus 160 meters is 300 is 320 meters 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 if - if this plus 320 is equal to 720 you could do it in your head you could say well look what plus 320 is 720 well this thing must be equal to 400 or you can do it more formally and subtract 320 from both sides of this and you would get if you subtract 3 20 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 you want to do it more formally you can divide both sides of this equation by 2 either way you will get the width is equal to 200 meters