# The Factor-Label Method

—

Mr. Andersen shows you how to use the factor label method to solve complex conversions.

—

—

#### Transcript Provided by YouTube:

00:00

Hi class. This is Mr. Andersen. Today I’m going to show you how to use the

00:03

factor-label method. Some science teachers refer to this as dimensional analysis. And

00:08

some people just call it common sense. And so what is the factor-label method? The factor-label

00:13

is the way that you solve a problem. And so there’s a nice method you can use to do that.

00:19

And so if I were to for example to ask you how many hours are there in a day? That thought

00:25

process you go through of remembering that it’s 24 hours in a day is actually a simple

00:29

for of the factor-label method. So what we do with that is we take a value, let’s say

00:34

55 miles per hour. And we’re going to convert that to a different unit, like meters per

00:38

second. This becomes really important in chemistry, physics, physical science, because you can

00:43

solve these very complex problems. And as long as you follow the methods that I lay

00:48

out in this podcast you should be good to go. Now an analogy or a good way to think

00:54

about how this works is what’s called six degrees of separation. So there’s a scientist

00:59

back in the 1940s I think it was who said, let’s say we have a person here who lives,

01:04

we’ll say in New York City. And then we have a person who lives way over here. Let’s say

01:09

they live in Montana. He said that we could take any two people and we could connect them

01:14

with at least six degrees of separation. In other words this guy might be friends with

01:19

this guy. And this guy might have a sister who is this person right here. Who might have

01:26

a friend who is this person. Who also has a friend who knows this person. And so the

01:33

idea is that you’re connected to anybody on the planet by no more than six degrees of

01:38

separation. There’s a funny game with movies and using Kevin Bacon. It’s called six degrees

01:43

of Kevin Bacon that uses movie trivia to kind of do the same thing. But again that’s just

01:49

kind of an analogy. So what do we do in this? Conceptually we’re taking a quantity. So let’s

01:55

say that is miles per hour. And we’re going to convert that to something like meters per

02:06

second. And so all of these questions will start with some kind of quantity. And then

02:13

we’re going to end up with a desired quantity. But you have to use your brain to figure out

02:17

what kind of conversion factor we’re going to use. In other words, what are some important

02:21

things if we’re going from hours to seconds. How are you actually going to convert that?

02:25

Or miles to meters. We’re going to have to know some kind of a conversion to make it

02:30

from that given quantity to the desired quantity. Okay. So this is my method. And there’s lots

02:36

of different methods laid out to do the factor-label method. But if you follow these steps you

02:41

can solve pretty complex problems. So let’s start with one that’s really really easy.

02:46

And let’s say we say that you’ve got one day and you want to convert that to hours. So

02:54

what is the first step? You start with the given quantity. And you always express it

02:58

as a fraction. And so even though one day doesn’t need to be written over one, let’s

03:03

just do that. Because it’s going to all you to solve the problems. Lot’s of times you’ll

03:07

actually have units over units. And so it makes it easier. Okay. Next we’re going to

03:10

convert with a conversion factor. Okay. So what does that mean? We’re here with days.

03:16

But we want to eventually make it to hours. And so what I’m going to do is I’m going to

03:20

write days underneath and I going to write hours on the top. So first we insert the conversion

03:26

factor. Then we add our numbers. Well we know that one day is 24 hours. So what’s next?

03:34

We cancel the units. This is a day on the top. So I’m going to cancel that out. And

03:38

here’s a day on the bottom. And so I’m going to cancel that out. And then the fourth step,

03:42

what I do is I actually solve the math. And so I’m going to multiple across the top. 1

03:47

times 24 hours is 24 hours. Now I’m going to multiple across the bottom. 1 times 1,

03:54

we lost the day, is 1. And so my answer equals 24 hours. Now you could have just done that

04:01

in your head. But if you followed these steps on all of the problems we work with on factor-label

04:06

method, you’ll do fine. So let’s do a couple of practice ones. So let’s say we start with

04:11

this. I’ve got 12 days over here. So I’ve got 12 days. So I write that over 1. I then

04:19

figure out my conversion factor. Well, what do I want to go to? I want to eventually make

04:24

it to seconds. And you don’t even have to know how many seconds there are in a day.

04:29

So I do know that I could go from days to hours. I also know that I could go from hours

04:39

to minutes. And I also know that I could go from minutes to seconds. Okay. So why was

04:49

I doing that? Well if I’ve got days up here, I could put days on the bottom. I know those

04:53

are going to cancel. So now I just go back. Once I have them all laid out, I now know

04:57

that 1 day has 24 hours in it. Let’s go to the next one. And that one hour has 60 minutes

05:06

in it. And I know that 1 minute has 60 seconds in it. So now the next step is to cross out

05:13

and cancel out all of the units. So I’m going to cancel out days. I’m going to cancel out

05:19

hours. I’m going to cancel out minutes. And now I’m left with seconds. And so now using

05:23

my trusty calculator I’m going to take 12 times 24 times 60 times 60. And what do I

05:32

get is, let’s write this down here, is 1,036,800 seconds. Okay. Now if you’ve watched my podcast

05:46

on significant digits you know that this is a silly answer to write because we only have

05:50

2 significant digits in this first one. This answer can only have 2 significant digits

05:55

as well. And so I would write this in scientific notation. So that’s 1, 2 , 3, 4, 5, 6. And

06:02

so this is going to be written as 1 point 0 times 10 to the 6th seconds. In other words

06:12

that’s how many seconds are in 12 days. Let’s try another one. Because that’s one had talked

06:16

about earlier. Let me erase that. Let’s say we want to go from 55 miles per hour. So I’m

06:23

going to write 55 miles. And now look what I’m going to do. I’m going to write that over

06:30

1 hour. So this is why we use fractions. Because once we start having units over units it’s

06:35

important that you’ve written it out that way. So now what do I want to start with?

06:40

Miles and I want to end us with meters. So what I could do is I could put another conversion

06:44

factor here, I know that 1 mile is exactly 1609 meters. So 1 mile is 1609 meters. I also

06:57

know, since we’re going to seconds that I could put hour up on the top. And I could

07:02

go to minute on the bottom. And I could also put the minute up on the top and I could put

07:10

seconds on the bottom. So what do we do. We’ll let’s cross them out. Oh, first I’ve got to

07:14

come back here. So 1 hour has 60 minutes in it. And then over here 1 minute has 60 seconds

07:23

in it. So now I cross out all my values. I’m going to cross out miles and miles. I’m going

07:28

to cross out, what else? Hours right here. And hours back here. And then I’m going to

07:34

cross out minutes here and minutes here. So what do I have left? Well I have meters on

07:40

the top. That didn’t get cancelled out. And then we have seconds on the bottom. And so

07:45

now I’ve made it to meters per second. So what’s that final step? I have to actually

07:48

do the math. And so I’m going to go all the way across the top. So using my trusty calculator

07:53

I’m going to take 55 times 1609. And then I’m going to take 60 times 60 which is 3600.

08:02

And I’m going to divide that out. And so the value I get is 24.5819 . . . . So it goes

08:15

out like that. So how many significant digits do we have? Well this had 2 significant digits.

08:21

And so my answer can only have 2 significant digits as well. So let me write my answer

08:24

up here. My answer is going to be 25 meters per second. That has 2 significant digits

08:33

as well. Now one thing you might be wondering is well this has two significant digits. But

08:38

doesn’t this 1 here just have one significant digit? And the right answer is no. And the

08:43

reason why is that in a conversion we think of these conversions actually having an infinite

08:49

number of significant digits. And so we don’t have to figure those in. Because we know that

08:53

1 mile is exactly 1609. And so we don’t have to worry about ones like that. Okay. So that’s

09:00

the factor-label method. And if you always follow the steps, putting fractions to start.

09:05

Then figuring out your conversion factors. Finally crossing out the units. And then doing

09:09

the math, you should make it there. Now there are a few limitations. These work really well

09:15

if we have a constant difference. In other words there’s always 1609 meters in 1 mile.

09:21

Or there’s a constant ratio between the two. But we can’t do both of those at the same

09:26

time. In other words, when you’re converting from Fahrenheit degrees to Celsius degrees,

09:33

remember you have to take that times 9 fifths and then add 32. And so since you’re doing

09:37

two things, the factor label method actually falls apart at that point. And so factor-label

09:42

method can solve a ton of things. But it does have a few limitations. But if you always

09:47

follow those four rules then you should be good to go.

—

This post was previously published on YouTube.

—

Photo credit: Screenshot from video.