We're going to continue our work with conversions now by looking at units of area and units of volume.
In the U.S. system for units of area, for instance, we have 144 square inches is the same as 1 square foot; 1 acre is 43,560 square feet;
640 acres is 1 square mile. So, these are all written in the book, so you have access to them by looking in the book.
In the metric system, I have 1 square meter is the same as a 100 decimeters squared; and 1 square centimeter is 100 millimeters squared,
things like that. When it comes to volume, volume measures capacity: how much it takes to fill something up,
so, one cubic foot is the same as 1,728 cubic inches; 1 pint is 16 fluid ounces; 1 quart is 2 pints; and 1 gallon is the same as 4 quarts.
So, these are the different conversions for units of volume in the U.S. system of measure.
In the metric system, I have 1 liter is the same as 1,000 milliliters; and 100 liters is 1 hectoliter; and a 1,000 liters is 1 kiloliter.
So, what we want to do is use these units of measure to convert from one kind of unit to another;
and the units are going to involve both area and volume. Let's look at our first example: I want to convert 288 square inches into square feet.
So, the conversion factor that I'm going to use is this: there's 144 square inches in 1 square foot. So, I start with my 288 square inches;
and I want to multiply by a conversion factor that's going to have my square inches divide out and leave me with square feet. So, I see
that I must put the 144 on the bottom and the 1 on the top; the inches will divide out; and I'm left with 288(1) ÷ 144 square feet.
Now, when I divide 288 ÷ 144, I end up with 2 square feet. So, converting this unit of area into this unit of area is really pretty simple,
as long as I know my basic conversion factor here between square inches and square feet: 144 square inches is 1 square foot.
So, I know this is the conversion factor to multiply by. And again, remember this is the number 1:
144 square inches is exactly the same as 1 square foot. So, I have 1 square foot over 1 square foot, this is just the number 1.
Let's look at our next example. For problem 2, we want to convert 1,920 acres into square miles,
so, again, a conversion of units of area in the U.S. system. This is what I know: there are 640 acres in 1 square mile.
So, this I should be able to use to do my conversion. So, I'll start with my 1,920 acres;
and I'm going to multiply by a conversion factor that will allow acres to divide out.
So, I want to divide out acres, and I want to be left with square miles. So, to do that, I need the 640 in the denominator and the 1 in the numerator
because that's what this tells me. Now, acres divide out, and I end up with 1,920(1) ÷ 640, and that is square miles.
Now, if I do this division, I think it divides out evenly. It does. I get 3 square miles.
So, 1,920 acres is exactly the same as 3 square miles; and I know that because there's 640 acres in every 1 square mile;
that means that 1 square mile divided by 640 acres is simply the number 1. So, 1 times 1,920 acres isn't going to change its value,
but it is going to change the units that it's written in terms of, so, 3 square miles. Let's do another one. I want to convert 12 pints to quarts.
Now, these are units of capacity, or volume. And if I look at my list of units, I see that 1 quart is the same as 2 pints.
So, 1 quart is equal to 2 pints. Now, that should be enough to allow me to convert from pints into quarts. So, I start here with 12 pints;
and I want to multiply by a conversion factor that's going to allow pints to divide out and leave me with quarts.
So, since there are 2 pints in every 1 quart, I'm going to have 2 pints, 1 quart, like this. Pints divide out,
and I'm left with 12(1), which is 12; ÷ 2, which is 6 quarts. So, as long as we have a list of conversion factors like this,
we can really do any kind of conversion, whatsoever. As a matter of fact,
we don't even need to have a good intuitive idea of what 1 pint or 1 quart is,
as long as we know that 1 quart and 2 pints are exactly the same thing. Then, it's easy to convert between pints and quarts.
Given any number of pints or quarts, we can convert into any number of quarts or pints. Here's another example:
in the metric system, I have 127 milliliters that I want to convert into liters. Now, this is what I know: there's 1,000 milliliters in every 1 liter,
so, here's my conversion factor; and I'm going to take 127 milliliters and I'm going to multiply by a conversion factor
that has milliliters in the denominator and liters in the numerator. So, I see that it must have 1,000 here, and 1 up here; milliliters divide out.
127(1) ÷ 1,000 would be 0.127 liters. So, 127 thousandths liters is the same as 127 milliliters. So, converting from milliliters to liters,
I multiply by this conversion factor: 1 liter divided by 1,000 milliliters. So, in the metric system, it seems to be a little bit easier
because all the conversion factors involve powers of 10, at least, people seem to think it's a little bit easier.
We're not as familiar with the metric system, maybe, as we are with the U.S. system, but still the conversions proceed in exactly the same way.
Here's our next example: "How many 8-fluid-ounce glasses of water will it take to fill a 3-gallon aquarium?"
So, what I'm going to do here is find out the number of fluid ounces there are in 3 gallons;
and then, I'll take that total number of fluid ounces and divide by 8; and that will be the number of those glasses.
So, I need to convert from gallons into fluid ounces. Now, I can't go directly there, so let's see what we've got.
I do know that 1 gallon is equal to 4 quarts; and I also know that 1 quart is equal to 2 pints, so, 1 quart is equal to 2 pints.
And then, I also know that 1 pint is 16 fluid ounces, so, 1 pint is equal to 16 fluid ounces.
Well, this should be enough to allow me to change gallons into fluid ounces. So, let's start here with my 3 gallons.
So, I start with my 3 gallons; and I'm going to multiply by a conversion factor that will allow gallons to divide out.
So, I use this one right here: I'll go from gallons to quarts, so, gallons in the denominator; quarts in the numerator;
and what I have is 1 gallon is the same as 4 quarts. Now, my gallons have divided out. I'll go from quarts to pints.
So, I have my quarts divide out, and I'll be left with pints; and I want this to be the number 1. So, here I have 2 pints and 1 quart.
So, I'll put a 2 here, and a 1 here. My quarts have divided out. The last thing is to convert from pints to fluid ounces.
So, I'll multiply over here. Let's see, I want pints in the denominator, fluid ounces in the numerator.
So, what I have, 16 fluid ounces is the same as 1 pint; 16 divided by 1. Okay, so now, I multiply (3)(4)(2)(16).
The pints divide out; that will give me the total number of fluid ounces in 3 gallons.
So, I end up with (3)(4)(2)(16) fluid ounces. Now, that comes out to be 384 fluid ounces.
So, there's the total number of fluid ounces in 3 gallons. Now, I'm going to take this number of fluid ounces,
and see how many 8-fluid-ounce glasses there are in this by taking 384 and dividing by 8. Fluid ounces over fluid ounces will divide to 1.
So, I divide this by 8, and I end up with 48 glasses. So, if you want to fill up a 3-gallon aquarium and you have an 8-fluid-ounce glass to do it with,
it's going to take 48 of those glasses to fill up this aquarium. Notice that I started with gallons, and I ended up with fluid ounces.
In the meantime, I went from gallons to quarts, from quarts to pints, and from pints to fluid ounces.
I just went over to my list and found as many things as I needed to use. Each of these conversion factors, again, is the number 1.
And that's a look at using these conversion factors with area and volume.