Over here, I've written the fraction a over b.
The number on top, a, is called the numerator of the fraction; the number b that's on the bottom is called the denominator of the fraction.
Both a and b are called terms of the fraction. When the numerator is less than the denominator, we have what's called a proper fraction;
and when the numerator is greater than or equal to the denominator, then we have what's called an improper fraction.
We can visualize fractions a number of different ways. One way is to visualize them is on the number line.
So, if I take the section of the number line that goes from 0 to 1 and divide it up into three equal segments, this one, this one, and this one,
each of those segments is 1/3 of the distance from 0 to 1. So, this point right here on the number line would have a coordinate of 1/3;
this would be two of those line segments, so that point would have a coordinate of 2/3; and of course, this point right here, which is 1,
the number 1, would have a coordinate in terms of fractions of 3/3. So, let's go to the board now and look at our first problem.
For examples here, I've written the fraction 2/5; 2 is the numerator, and 5 is the denominator.
Since the numerator 2 is less than the denominator 5, this is an example of a proper fraction.
Now next, I have problem number 2 here: the fraction 7/3.
As you can see, the numerator 7 is larger than the denominator 3, so this is an example of an improper fraction. So, it's improper.
That doesn't mean that there's something wrong with it, it's just called an improper fraction; the numerator 7 is larger than the denominator 3.
And again, 7 and 3 are called terms of the fraction, as are 2 and 5, in the first example. So, here's a proper fraction;
here's an improper fraction. Now as far as fractions go, we have two properties of fractions that we're going to use throughout this chapter.
So, I'm want to walk over here and take a look at those properties.
Our first property of fractions says that the fraction a over b is equivalent to the fraction a times c over b times c. Or,
we can multiply the numerator and denominator of our fraction by any nonzero number, and it won't change the value of the fraction.
So, here c represents any nonzero number.
So, as long as I multiply the numerator by the same thing I multiply the denominator by, I know I won't change the value of the fraction.
The same thing holds for division. I can divide the numerator and denominator by the same nonzero number,
and always produce an equivalent fraction. Now as far as the properties of fractions go, that's it. There's only two properties of fractions.
So, they're easy numbers to work with. We can multiply the numerator and denominator by any nonzero number we want,
and we can divide the numerator and denominator by any nonzero number we want; we'll always be sure that we've produced an equivalent fraction.
As far as addition or subtraction goes, those won't work. If we add the same number to the numerator and denominator of a fraction,
we'll end up changing the value of the fraction most times. So, let's go to the board now and see how we can use these two properties.
Here, I've written: write each fraction as an equivalent fraction with denominator 6. So, here I have my fraction 2/3.
I want to rewrite it as a fraction with a denominator of 6. So, in order to go from 3 to 6, I need to multiply by 2.
So, multiplying the denominator by 2 will produce a denominator of 6. If I do that to the denominator,
I have to do the exact same thing to the numerator. So, 2/3 is equivalent to the fraction 4/6;
and I've produced 4/6 by multiplying the numerator and denominator of this fraction by 2.
Let's look at our second example here: 55 over 66. I'm going to write this as an equivalent fraction with the denominator of 6.
To do that, I have to divide this denominator by 11: 66 divided by 11 produces 6.
Since I did that to the denominator, I have to do the exact same thing to the numerator. So, I'll divide the numerator by 11;
and that produces a 5 in this numerator. So, the fraction 5/6 is equivalent to the fraction 55 over 66.
And I found that by using property two, where I divide the numerator and denominator by the same nonzero number.
Okay, let's go look at our next problem: find an equivalent fraction with a denominator of 12x.
Now, if you are taking this class out of the basic math book, you'll notice that we haven't worked with variables yet in basic math.
It's still a good idea to watch these examples because the principles that we're using are the same.
But this kind of problem is the kind of problem that you'll only find in the pre-algebra book. So, here I have 3/4.
I want to rewrite that as a fraction with the denominator 12x. To go from 4 to 12x with multiplication or division, I have to multiply by 3x:
3x(4) gives me 12x. If I do that to the denominator here, I have to do the same thing to the numerator.
So, I'll multiply by 3x; that produces 9x in the numerator. So, the fraction 9x over 12x is equivalent to the fraction 3/4: that is,
for any value of x over here other than 0, this fraction and this fraction have the exact same value.
The next problem that we're going to look at we'll have to use a calculator on.
So, it says: fill in the correct denominator; and I have 7/11 is equal to 5,761 over something. So, to see how I go from 7 to 5,761,
I'm going to use my calculator, and I'm going to take 5,761 and divide it by 7 to see what it is I multiplied by here. So, 5,761 ÷ 7 = 823.
So, that 823 that I multiply 7 by to get this, is the same number I'll multiply 11 by to get my new denominator over here.
So, I'll just simply take that and multiply times 11, and I end up with 9,053. So, even with large numbers like this
and a calculator, it's still easy to work these problems. 5,761 divided by 9,053 is exactly the same as the fraction 7/11.