Unit Conversions (Practice)

Just wanted to do a little introduction or
some practice on unit conversion, since unit conversion is typically one of the simplest mistakes to make on a test problem or homework problem and it’s something you just need to practice
and get used to and comfortable with so that you don’t make those mistakes. So, I have three
examples I want to go through, and just kind of talk about a set-up. Once you get comfortable
doing this set-up, and more of it becomes automatic in your head, you don’t necessarily
need to use this strategy, but at least to start, and this is the way that’s suggested, and that is, if we are going, in this case, from say one cup
and we want to go to a certain amount of moles of water. We have to be able to figure out
what conversion factors we need, and what those are so we can convert from that one
cup of water to those moles of water. So what we typically do is draw something that looks
like this, with the line in the middle and units on both the top and the bottom. And
then, start putting our conversion factors in between. So, you have to have an approach,
and I just happen to know that we can go from a volume to a mass using density, and to go
from mass to moles using the molecular weight. So, I’ll go ahead and show you how to go about
that, and just in terms of the setup, how you would approach it. There are a couple
different conversion factors you could use for density. The one I can recall off the top
of my head is that there is 1 gram of water per 1 milliliter of water. So that makes it
easy to go from there. And then, you could memorize or look up some other volume conversions;
for instance, if you cook a lot, hopefully you’ll know that, in terms of cups and milliliters, there are 240 milliliters per cup. And so it’s important that you notice which side
the top or bottom that we’re putting these units. We want the “cup” on the bottom, so
that we can cross those off. And the same things here: we want the “milliliters of water”
on the top so we can cross those off. So now, we have “grams of water”. To go from grams
to moles, we can use the molecular weight. And since we want grams to cancel, we’re going
to put “grams” on the bottom. So how many grams of water are there per mole of water?
Well, you could look that up or calculate it, and it should be 18 grams. So now we can
cross out our “grams of water” and “grams of water” and we’re left with moles. The only thing you have left to
do is make these multiplications, where we have 240 over 18, and that is going to give
you the amount of moles of water. It comes out to about 13.3 moles of water. So let’s try another one. Here’s a question for
you: something I recall from when I was younger was whether or not you think you can out run
an alligator that could go up to 20 miles per hour if you can cover the 40 yard dash
in 5 seconds. Now, there are obviously a lot faster people out there, but relatively, five
seconds for forty yards isn’t too far off. So the way we would start this is write our
forty yards in five seconds, because that’s our velocity that we want to compare with
the velocity of an alligator. We’re going to continue this line across, and we want
to get to miles per hour. So now this is going to help us see what conversion factors we’re going to use. I’m going to go ahead and tell you to pause at this point
and try to fill this in and calculate what your speed in miles per hour is. Now, you could either choose to go with yards or seconds to start with. It doesn’t really matter. First
thing that comes to my mind is how to get rid of the seconds. We know that there’s 3600 seconds in one hour.
That’s going to cross those off. We have our hours and now we have to go from yards to
miles. You could probably look something up, or if you know the conversion between
yards to miles, you could use that. I do know that there are 3 feet per 1 yard, and there
are 5280 feet per one mile. So now you can see now that the feet are cancelling, the yards
are cancelling, and we have our miles per hour. So the only thing left to do is to multiply
our top numbers divided by our bottom numbers, and you should see that this comes out to
16.4 miles per hour, meaning that you’d probably be lunch. The final question, which is something that
you may come across in your travels, or if you’re curious. The question asks whether or not it’s cheaper to drive in the U.S. with an average 20 miles to the gallon right now, or in Germany, where
the average is about 6 Liters per 100 kilometers and that’s how they report it over there. So there’s
a couple different conversions you have to make, and some information that you’re not given
that you have to be resourceful enough to look up. So how do we go about this? Well we need to find some way to compare these
two to each other, so let’s do it two ways: let’s compare how much it costs to drive a
mile in each country, and also how much it would cost per gallon of gasoline. Alright, so let’s start with the U.S., since
it’s already in the units we were talking about. If we go 20 miles to 1 gallon of gasoline,
what unit conversion would get us towards miles to dollars? Well, we have to convert
the dollars to gallons–so we’ll say that one gallon probably costs us 3 dollars. So
this conversion should yield about 6.67 miles to the dollar, which is also going to be about
15 cents per mile. And we already have our price per gallon, since that’s what we’ve
looked up. Now we just have to convert Germany’s system
over. So let’s start with the six Liters per 100 kilometers. I’ll draw my single-dimensional
equation, and we want to get to dollars per mile. So let’s start with the kilometers. We know that
there’s 1.6 kilometers per 1 mile. That’ll give us our mile and cancel out kilometers.
Now all we have to do is get to our prices. we know that their price of gasoline, if you look it up
right now, is about 1 Euro per Liter, and there’s about 1.5 dollars per 1 Euro. These
may not be that accurate depending on when you look at this, but if we do this calculation,
then we’ll see it’s about 14 cents per mile. That’s really not that different from what
we see here. Well, per mile, it’s not–but if we wanted
to check to see what it was on a volume basis, 1 Euro over 1 Liter using our dimensional
equation, 3.78 Liters per 1 gallon and then again the price conversion $1.50 per Euro, if you calculate this out, you’ll see that the price of gas is about $5.70/gallon. Although
distance-wise it doesn’t cost less, the reason for that is that the cars in Germany are just getting
better gas mileage on average than the cars over here in the U.S.

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