The Factor-Label Method

Hi class. This is Mr. Andersen.
Today I’m going to show you how to use the factor-label method. Some science teachers
refer to this as dimensional analysis. And some people just call it common sense. And
so what is the factor-label method? The factor-label is the way that you solve a problem. And so
there’s a nice method you can use to do that. And so if I were to for example to ask you
how many hours are there in a day? That thought process you go through of remembering that
it’s 24 hours in a day is actually a simple for of the factor-label method. So what we
do with that is we take a value, let’s say 55 miles per hour. And we’re going to convert
that to a different unit, like meters per second. This becomes really important in chemistry,
physics, physical science, because you can solve these very complex problems. And as
long as you follow the methods that I lay out in this podcast you should be good to
go. Now an analogy or a good way to think about how this works is what’s called six
degrees of separation. So there’s a scientist back in the 1940s I think it was who said,
let’s say we have a person here who lives, we’ll say in New York City. And then we have
a person who lives way over here. Let’s say they live in Montana. He said that we could
take any two people and we could connect them with at least six degrees of separation. In
other words this guy might be friends with this guy. And this guy might have a sister
who is this person right here. Who might have a friend who is this person. Who also has
a friend who knows this person. And so the idea is that you’re connected to anybody on
the planet by no more than six degrees of separation. There’s a funny game with movies
and using Kevin Bacon. It’s called six degrees of Kevin Bacon that uses movie trivia to kind
of do the same thing. But again that’s just kind of an analogy. So what do we do in this?
Conceptually we’re taking a quantity. So let’s say that is miles per hour. And we’re going
to convert that to something like meters per second. And so all of these questions will
start with some kind of quantity. And then we’re going to end up with a desired quantity.
But you have to use your brain to figure out what kind of conversion factor we’re going
to use. In other words, what are some important things if we’re going from hours to seconds.
How are you actually going to convert that? Or miles to meters. We’re going to have to
know some kind of a conversion to make it from that given quantity to the desired quantity.
Okay. So this is my method. And there’s lots of different methods laid out to do the factor-label
method. But if you follow these steps you can solve pretty complex problems. So let’s
start with one that’s really really easy. And let’s say we say that you’ve got one day
and you want to convert that to hours. So what is the first step? You start with the
given quantity. And you always express it as a fraction. And so even though one day
doesn’t need to be written over one, let’s just do that. Because it’s going to all you
to solve the problems. Lot’s of times you’ll actually have units over units. And so it
makes it easier. Okay. Next we’re going to convert with a conversion factor. Okay. So
what does that mean? We’re here with days. But we want to eventually make it to hours.
And so what I’m going to do is I’m going to write days underneath and I going to write
hours on the top. So first we insert the conversion factor. Then we add our numbers. Well we know
that one day is 24 hours. So what’s next? We cancel the units. This is a day on the
top. So I’m going to cancel that out. And here’s a day on the bottom. And so I’m going
to cancel that out. And then the fourth step, what I do is I actually solve the math. And
so I’m going to multiple across the top. 1 times 24 hours is 24 hours. Now I’m going
to multiple across the bottom. 1 times 1, we lost the day, is 1. And so my answer equals
24 hours. Now you could have just done that in your head. But if you followed these steps
on all of the problems we work with on factor-label method, you’ll do fine. So let’s do a couple
of practice ones. So let’s say we start with this. I’ve got 12 days over here. So I’ve
got 12 days. So I write that over 1. I then figure out my conversion factor. Well, what
do I want to go to? I want to eventually make it to seconds. And you don’t even have to
know how many seconds there are in a day. So I do know that I could go from days to
hours. I also know that I could go from hours to minutes. And I also know that I could go
from minutes to seconds. Okay. So why was I doing that? Well if I’ve got days up here,
I could put days on the bottom. I know those are going to cancel. So now I just go back.
Once I have them all laid out, I now know that 1 day has 24 hours in it. Let’s go to
the next one. And that one hour has 60 minutes in it. And I know that 1 minute has 60 seconds
in it. So now the next step is to cross out and cancel out all of the units. So I’m going
to cancel out days. I’m going to cancel out hours. I’m going to cancel out minutes. And
now I’m left with seconds. And so now using my trusty calculator I’m going to take 12
times 24 times 60 times 60. And what do I get is, let’s write this down here, is 1,036,800
seconds. Okay. Now if you’ve watched my podcast on significant digits you know that this is
a silly answer to write because we only have 2 significant digits in this first one. This
answer can only have 2 significant digits as well. And so I would write this in scientific
notation. So that’s 1, 2 , 3, 4, 5, 6. And so this is going to be written as 1 point
0 times 10 to the 6th seconds. In other words that’s how many seconds are in 12 days. Let’s
try another one. Because that’s one had talked about earlier. Let me erase that. Let’s say
we want to go from 55 miles per hour. So I’m going to write 55 miles. And now look what
I’m going to do. I’m going to write that over 1 hour. So this is why we use fractions. Because
once we start having units over units it’s important that you’ve written it out that
way. So now what do I want to start with? Miles and I want to end us with meters. So
what I could do is I could put another conversion factor here, I know that 1 mile is exactly
1609 meters. So 1 mile is 1609 meters. I also know, since we’re going to seconds that I
could put hour up on the top. And I could go to minute on the bottom. And I could also
put the minute up on the top and I could put seconds on the bottom. So what do we do. We’ll
let’s cross them out. Oh, first I’ve got to come back here. So 1 hour has 60 minutes in
it. And then over here 1 minute has 60 seconds in it. So now I cross out all my values. I’m
going to cross out miles and miles. I’m going to cross out, what else? Hours right here.
And hours back here. And then I’m going to cross out minutes here and minutes here. So
what do I have left? Well I have meters on the top. That didn’t get cancelled out. And
then we have seconds on the bottom. And so now I’ve made it to meters per second. So
what’s that final step? I have to actually do the math. And so I’m going to go all the
way across the top. So using my trusty calculator I’m going to take 55 times 1609. And then
I’m going to take 60 times 60 which is 3600. And I’m going to divide that out. And so the
value I get is 24.5819 . . . . So it goes out like that. So how many significant digits
do we have? Well this had 2 significant digits. And so my answer can only have 2 significant
digits as well. So let me write my answer up here. My answer is going to be 25 meters
per second. That has 2 significant digits as well. Now one thing you might be wondering
is well this has two significant digits. But doesn’t this 1 here just have one significant
digit? And the right answer is no. And the reason why is that in a conversion we think
of these conversions actually having an infinite number of significant digits. And so we don’t
have to figure those in. Because we know that 1 mile is exactly 1609. And so we don’t have
to worry about ones like that. Okay. So that’s the factor-label method. And if you always
follow the steps, putting fractions to start. Then figuring out your conversion factors.
Finally crossing out the units. And then doing the math, you should make it there. Now there
are a few limitations. These work really well if we have a constant difference. In other
words there’s always 1609 meters in 1 mile. Or there’s a constant ratio between the two.
But we can’t do both of those at the same time. In other words, when you’re converting
from Fahrenheit degrees to Celsius degrees, remember you have to take that times 9 fifths
and then add 32. And so since you’re doing two things, the factor label method actually
falls apart at that point. And so factor-label method can solve a ton of things. But it does
have a few limitations. But if you always follow those four rules then you should be
good to go.

45 thoughts on “The Factor-Label Method

  1. I seriously hope you don't underestimate just how helpful your videos are. Thank you very much and I appreciate everything you do.

  2. Thank you so much, Mr. Anderson! Honors Chemistry was kicking my butt not even 2 weeks in, but your videos on Sig Figs and the Factor Label Method have helped me so much!

  3. I'm a student at NIU and because of my teachers accent I find lectures very hard to follow….Im so thankful for this video!!!!!

  4. i missed the day we got the lecture over this. and the next day i came in and the teacher toled me one on one how to do it and i was ok with the simple 1 conversion but i could not do the 2 or more but now i can. and all i can say now is thank you

  5. i missed two continuous days of class and even after watching multiple videos i couldn't figure this stuff out. thank you for your video, because this is the one where everything finally clicked

  6. With all the conversions from days into seconds, I couldn't help, but think of the RENT song. Thank you, Mr. Andersen!

  7. I have a question about significant digits:  Wouldn't the number of hours in a day be a constant, i.e. a real world value, and therefore have an infinite number of significant figures?  I am referring to 24 hours in a day used in the video.  You would need to defined the number of significant figures you are expecting in the response, correct?

  8. at 7:50 what did you multiply to get the answer? no matter what i multiply i cant do that, you also said something about dividing it. what am i doing wrong?

  9. your scientific notation was wrong you said 12 days=1036800 or 1.0*10^6 the 3,6 and 8 are significant numbers so it would be 1.0368*10^6

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