In this section, we want to start our work with conversions, and we're going to use a method called unit analysis.
I've written over here on the board some of the common U.S. units of conversion: 12 inches is 1 foot, 3 feet is 1 yard, and 5,280 feet is 1 mile.
Now, if I look at this 3 feet and 1 yard right here, since 3 feet is equal to 1 yard,
that means that 1 yard divided by 3 feet, and 3 feet divided by 1 yard are both equal to the number 1. So, this is the number 1,
and this is the number 1 because 3 feet and 1 yard are exactly the same thing. Now, we call each of these a conversion factor.
In the metric system, I have 1,000 millimeters is the same as 1 meter. 100 centimeters is 1 meter; and 1,000 meters is 1 kilometer, or kilometer.
So, these are the conversion for the metric system; these are the conversions for the U.S. system.
Each one of these conversions right here will give us a conversion factor, which is a fraction that's equal to the number 1.
Now, we want to use these to convert different lengths into other lengths with other units. Let's look at our first example:
we have 27 inches, and we want to convert this to feet. Now, the conversion factor I'm going to use is this:
12 inches = 1 foot. Now, what that means is that 12 inches over 1 foot is the same as 1 foot over 12 inches,
and both of these are equal to the number 1. So, this is a conversion factor; it's equal to the number 1.
This is a conversion factor; it's equal to the number 1. I'm going to use one of these conversion factors to convert 12 inches into feet.
So, I'm sorry, 27 inches; so, I have 27 inches, and I'm going to multiply by a conversion factor
that will allow these inches to divide out and leave me with feet. Now remember, this is 27 inches divided by the number 1.
So, I'm going to put inches in the denominator, and then feet in the numerator. So, I'll pick the conversion factor that has that:
so, here's 12 inches and 1 foot; this is the number 1. So, multiplying 1 times 27 inches won't change its value, but notice that my inches will divide out.
I end up with 27/12 feet. Now, I'll just reduce this down to the lowest terms,
or let's see; both these are divisible by 3. So, I'll divide 3 in there, I get 9; 3 in there, 4 feet.
So, I end up with 9/4 feet as a improper fraction. So, converting 27 inches into feet, I get 9/4 feet. Now let's see, that would be 2 1/4 feet.
So, 2 1/4 feet is exactly the same as 27 inches. Now, this method of doing this is called unit analysis
because I analyzed the units right here to decide what kind of conversion factor I want to multiply by.
Convince yourself that that conversion factor is really the number 1. Let's look at our next example:
I want to convert 2 yards to feet. Now, the conversion factor I'm going to use here is this: there are 3 feet in every 1 yard.
That means that 3 feet divided by 1 yard is the same as 1 yard divided by 3 feet; and both of these are equal to the number 1.
So, I'll start over here with 2 yards, remember that's 2 yards over 1. And I want to multiply by one of these conversion factors
that will make my yards divide out. So, yards is in the numerator here; I'm going to put it in the denominator over here;
and I'll put feet in the numerator. So, I have 3 feet divided by 1 yard. Now, the yards divide out. I multiply 2(3);
I end up with 6 feet in the numerator; 1 in the denominator. Now intuitively, that makes sense also, right? Because if there's 3 feet in 1 yard,
then in 2 yards, there must be twice that amount, or 6 feet. So, you can see this method that we're using right here agrees with our common sense.
Again, it's called unit analysis because I analyzed the units to decide on what conversion factor to use. Here's our next problem:
4.8 kilometers to meters. Well, let's see. There's 1,000 meters in every kilometer,
so, this is what I'm going to use to convert 4.8 kilometers to meters. So, I'll start with my 4.8 kilometers, and that's divided by 1;
I want to multiply by a conversion factor that's going to allow those kilometers to divide out, and meters to stay.
So I want to make this equal to the number 1, and this is what I know: 1,000 meters in every 1 kilometer.
Now, those divide out; 4.8 times 1,000 will be equal to 4, one, two, three, so 4,800 meters. So, I look at the equality that I have right here:
the relationship between meters and kilometers. I use that to write the correct conversion factor
knowing that I'm starting with kilometers; I want to end up with meters, so I want the kilometers to divide out.
So, that's why that goes in the denominator because it's in the numerator over here. Here's another problem.
Problem number 4: 19 centimeters to meters. Now, 19 centimeters, that's about the length of a regular pencil.
So, a regular pencil is about 19 centimeters long. I want to convert that into meters. So, this is what I know: there's 100 centimeters in every 1 meter.
So, 1 meter is the same as 100 centimeters. So, I'm going to start with my 19 centimeters,
remember that's over 1; you don't have to write this over 1, but I'm doing it just to remind you that it is in the numerator,
times a conversion factor that will have centimeters in the denominator, so they will divide out.
I want to end up with meters, and I want this to be equal to 1. So, there will be 100 centimeters in every 1 meter, this is the same as the number 1.
1 meter over 1 meter; centimeters divide out; and I have 19(1) ÷ (1)(100), that would be 0.19 meters.
So, I like this method of converting between units because it's a method that you can use if you take chemistry,
or if you take physics or physical science. This unit analysis is a method that's used in a lot of other classes. Let's look at our next example.
Here's one that's not as familiar to us: a furlong--something that's used in horse racing--is 220 yards. How many feet are in 12 furlongs?
So, even if we have unfamiliar units like a furlong, if we have one kind of conversion right here,
then we'll be able to convert from furlongs into anything that we want. Now, I know this: 1 furlong is equal to 220 yards.
Now, I'm going to be asked for the number of feet in 12 furlongs. So, I need a conversion between feet and yards also. And this is what I know:
3 feet is equal to 1 yard. So, I'm going to have to use these two items right here to convert 12 furlongs into feet. So, I'll start with 12 furlongs,
and remember, that's over 1. Now, let's convert furlongs to yards. So, I know that there is 220 yards in every 1 furlong.
Now, that will allow my furlongs to divide out; I end up with yards; and what I want to end up with for the answer to this problem is feet.
So, I need to multiply by another conversion factor that has yards in the denominator and feet in the numerator.
So, I know that there's 3 feet in every 1 yard. So, those will divide out. Now, what I end up with is 12(220)(3).
And I did that multiplication already, that's 7,920. Furlongs have divided out; yards have divided out; now, I'm left with feet, exactly what I wanted.
So, sometimes the conversion process requires that we use more than one of these conversion factors.
This is the number 1 and this is the number 1. So, I multiplied by the number 1 twice. That's not going to change the value of my 12 furlongs.
It simply changes what it's equal to; in this case, feet. Let's look at one that involves miles per hour.
Let's convert 55 miles per hour into feet per second. So, I've got, I want to go between miles and feet;
so, I know there's 5,280 feet in every 1 mile. And I also want to go from hours to seconds;
so, I know that there is 60 minutes in 1 hour, and that there is 60 seconds in 1 minute.
So, let's see if I can put these three items together with the appropriate conversion factors to convert 55 miles per hour to feet per second.
So, first of all, 55 miles per hour is the same as 55 miles divided by 1 hour.
So, 55 miles per hour is the same as 55 miles in the numerator and 1 hour in the denominator. So, I need to convert miles into feet;
and I need to convert hours into seconds. So, I have two things to do. We'll do them one at a time. First of all, to convert into miles into feet:
I need the miles to divide out, and I want to be left with feet. I know that there's 5,280 feet in every 1 mile. So now, my miles have divided out.
Next, I want hours to divide out. Let's go from hours to minutes. Hours is in the denominator;
so I need hours in the numerator over here for it to divide out; so, minutes will go in the denominator.
And what I have is 60 minutes in every 1 hour. Now, my hours have divided out, and I'm left with minutes.
I want minutes to divide out and give me seconds. So, minutes are in the denominator down here. For them to divide out,
I need minutes in the numerator over here; I want to be left with seconds, and what I know is that there is 60 seconds in every 1 minute.
Now, minutes will divide out, and I'll be left with, on top, feet; and on the bottom, seconds. So, I end up with here,
and I hope that you'll be able to see this, 55(5,280) ÷ 60(60).
And what I have on top is feet; and what I have on the bottom is seconds. Now, I've done that arithmetic,
and that turns out to be 80.7 feet per second to the nearest tenth. So, I've rounded that to the nearest tenth.
I think it comes out 80.666 or something like that. So, what we have is 80.7 to the nearest tenth.
(So, what we want to do is go to that wide camera for a second.)
Okay, and so this conversion process, see it takes one, two, three conversion factors to do this. This is the number 1;
this is the number 1; and this is the number 1. So, I've multiplied by the number 1, three times. This is a more complicated problem;
and a lot of these more complicated problems, you won't be able to look at the initial thing that you're given
and see how to get to the result right away. You just take it piece by piece. I know I have miles per hour and I want to convert into feet per second;
maybe I can't see how to go from hours to seconds directly, but I can go from hours to minutes, and then from minutes to seconds.
So, you can't go wrong this way because you continually multiply by the number 1. So, even if you make a mistake
and don't end up with the units that you want, still you haven't changed what you started with because you've multiplied by the number 1.
The better you get at these problems, the better off you're going to be in classes like chemistry, physics, nursing classes,
where you have to do drug dosages and things like that. This method of unit analysis is used in a lot of other places.