Converting Units with Conversion Factors


In this video, we’re going to look at how
to convert units using conversion factors like this and canceling units. Some people
call this dimensional analysis, some people call this the factor label method but you
are going to call it easy by the end of this video because we’re going to go step by step
to show how to solve these kind of problems. So, here is our first one. We want to know,
what is 3.45 pounds expressed in grams? So the conversion that we’re going to be doing
is we’re going to be doing pounds to grams. We’re going to be starting with 3.45 pounds.
Okay, the next thing we got to do is we got to go and we got to find some kind of relationship
between pounds and grams. So how many pounds are there in how many grams? You can find
this information on the internet, you can find it in a textbook, you can probably find
it like in the back of a notebook where they’ll have a conversion table for the units. It’s
going to look something like this. You got a bunch of relationships between different
units and you want to find the equation that talks about pounds and grams. Okay? It’s going
to be this one down here, 1 pound (lb)=453.6 grams (g). So this is the statement that you’re
going to want, you can get it from a variety of places but it’s important. Now, it doesn’t
matter whether pounds or grams is first, it can be flipped just as long as it has both
pounds and grams. So now, we have this statement that tells us how pounds relate to grams.
We’re going to use this statement now to write two conversion factors. A conversion factor
is expressed as a fraction with a top and a bottom so here’s how we can take this and
write a conversion factor. We’re going to take this side of the equation, 1 pound and
put it on the top of the fraction and there’s a fraction line and then this part equation
of the equation… it’s going to be on the bottom so it’s going to be 453.6 grams. And
now that’s one of the two conversion factors. The other conversion factor that we’re going
to write is… we just take this and we flip it… so this 453.6 grams we put that now
on the top of the fraction and divide it by 1 pound. So two conversion factors that you
can write from the statement here, either one of them is correct but only one of them
is what we want to be using here. Okay? So we’re just going to use one of these. The
one that I’m going to use is this one because pounds is up here and pounds is down here.
Let me tell you what that means. So 3.45 pounds, that’s not a part of a fraction, right? There’s
no fraction line here and so if something does not have the bottom of a fraction you,
just assume that it is the top of a fraction it is the same as it being on the top of a
fraction if it doesn’t have a bottom, that’s what I mean. Now on the other hand, pounds
is down here on the bottom of the fraction, okay? And when we’re using conversion factors,
we want to get rid of pounds and want to be left with grams. And it turns out if the unit
is on top, on one side of the multiplication sign, and then it’s on the bottom, on the
other side of a multiplication sign, it cancels out. So pounds is on top here, pounds is on
bottom there, so they both cancel out and that leaves me with units of grams. So just
to review, I wanted to use this version of the conversion factor because pounds was on
the bottom and this way pounds will cancel out. Now I’ve cancelled out pounds, I’m
ready to do the math. What’s it going to be? I’m going to do 3.45 times 453.6 divided 1.
Or if you have a fancy scientific calculator, you can plug this whole thing in one expression.
You can do 3.45 times… and then write this conversion factor in parentheses (453.6 divided
by what?). You don’t really have to worry about 1 too much if you don’t want to. I’m
just putting it in there because sometimes this won’t be a 1. So I just want you to get
used to dividing by whatever is on the bottom of the fraction even if it happens to be 1
in this case. So however you decide to plug this in your calculator, when you crank through
it you’re going to get the same number and that’s going to be 1,560. What are the final
units here? Well I cancel out pounds so the final units I’m going to be left with are
grams. And not worrying too much about significant figures when I’m doing these unit conversions
just for this lesson because I don’t want to add another thing in here to confuse you.
There will be a lesson later about how to do significant figures with unit cancellation
but I don’t want you to worry about sig figs now, just worry about figuring out how to
do the unit conversions. So anyway, 1560 grams is our final answer, let’s do a couple more.
How many miles is 15,100 feet? The conversion we’re going to be doing in this problem is
for units of feet into miles. We will be starting with 15,100 feet. We got to go to our unit
conversion table or find information on the Internet to figure out what the relationship
is between miles and feet. We got it right here so this is going to be the statement
we’re going to use to write our conversion factors. Let me pull this off the table here.
Now don’t freak out that miles is on this side and feet is on this side. It doesn’t
matter what unit is on what side of the equation because you can just easily flip it, okay?
You don’t care what unit is on what side of the equation, all you care about is being
able to have this so you can write the conversion factor. So let’s write the two conversion
factors that we can get from this statement. I’ll take 1 mile and I’ll put it on top here
and I’ll take the other side, 5,280 feet, and I’ll put that on the bottom. And I’ll
write what we could call the reciprocal of this, where we take it and flip it up so that
5,280 feet is on the top and 1 mile is on the bottom. We’re going to be multiplying
our measurement in feet by one of these two conversion factors. Which one is it going
to be? We have feet not as part of a fraction, so you assume that feet is on the top of a
fraction, it’s the same as if it’s on the top of a fraction which means that we
are going to want a conversion factor that has feet on the bottom of a fraction so they
cancel out. So it is going to be this one here. And now, feet on the top and feet on
the bottom cancel out and they leave me units in miles which is what I’m looking for here.
Now how do I do the math? I’ll do 15,100 times 1 divided by 5,280 because it’s on
the bottom of the fraction or if you can plug larger expressions into your scientific calculator,
you can do 15,100 times and in parentheses you can do one divided by 5,280. Again you
might wonder why you have to keep doing the 1, you can leave the 1 out if you want to
but remember this isn’t always going to be a 1 so it is a good thing to get in to
the habit of multiplying by whatever is on the top of the fraction and then dividing
by whenever is on the bottom of the fraction even if it’s 1 for right now. You can do
either one of these expressions and you’re going to end up with an answer 2.86. Units
are in miles. Again, I’m not really paying attention to significant figures for these
calculations. That’s how you do this, let’s do two more problems so you really get the
hang of this. This problem is about units of money. On a certain day, the exchange rate
between the US dollars and Euros is 1 US dollar equals 0.78 Euros. On that day, how much is
125 Euros worth in US dollars? So we will be going from Euros to US dollars here, starting
with 125 Euros and the question gives us this relationship between dollars and euros which
I write it in bigger letters here. In the two previous problems, I took this statement
and I wrote two conversion factors top and the bottom but I flip the top and the bottom.
What I’m going to do here though is I’m going to look at what I’m starting with I’m
just going to write the one conversion factor that I need, okay? So I’m going to take
125 Euros and what do I want to multiply that by to cancel out Euros? I’m going to want
the version of the conversion factor that has Euros on the bottom so they’ll cancel
out. So I’m going to take this thing that has Euros, 0.78 Euros and that will be on
the bottom which means that then this 1 US dollar will be on top. See you don’t always
have to write out both of the conversion factors, you can figure out which of the two you need
based on what should be on the top and what should be on the bottom. Now, Euros up here,
Euros down there, they cancel out which leaves us with dollars which is good because that’s
what we’re looking for and the math is going to be 125 times 1 divided by 0.78 or 125 times
(1 divided by 0.78). This is going to give us 160 US dollars so the dollar is doing pretty
well compared to the Euro on this day. One more. How many liters is 23,500 milliliters?
These are both metric units and you know a lot of times people ask me this unit canceling
method, can I use it for metric units? Of course, you can use it for any type of units.
All you got to do is figure out what the relationship between your two units is. So we are going
from milliliters (mL) to liters (L) and you may already know this but there is 1000 milliliters
(mL) in 1 liter. That’s what I mean, this is all you need. You can convert any two units
that you want just as long as you know the relationship between them. So we have this
here for liters and milliliters. So 23,500 mL… which conversion factor am I going to
want to use here? Since I want to get rid of milliliters, I want to use a version of
this that puts milliliters on the bottom. So I’ll put 1000 mL down here so that they’ll
cancel out so that means that I’ll put 1 L on the top. Cancel, cancel, I’m left with
liters so this is going to be, I’m not even going to write it out because I think you’re
getting the hang of it, it’s going to be 23,500 times 1 divided by 1000 which is going to
be 23 .5, final units are in liters. So that’s how you can convert from one unit to another
by setting up your conversion factors and canceling your units. So where do you go from
here? There are two more videos that may be of interest to you. The first is to show how
to string multiple conversion factors together because you don’t just always have to use
one. Here I’m converting from days all the way to seconds by setting up a bunch of conversion
factors where all the units cancel. So I’ll show you how to do that in one of the next
videos and then another video you might want to watch is about understanding unit conversion
where I talk about the rationale, the reasoning behind why you set up conversion factors the
way you do, why the units cancel, and how this relates to things you might be able to
more easily understand.