Chapter 5
This chapter is devoted entirely to understanding what fractions are and how they work from the ground up. Begin by reviewing the two parts of a fraction: the numerator and the denominator.
In the picture above, each circle represents a whole unit. One full circle means the number 1, two full circles represent the number 2, and so on. Fractions essentially divide units into parts. The units shown here have been divided into four equal parts, because the denominator of our fraction is 4. In any fraction, the denominator tells you how many equal pieces a unit has been broken into.
The circle at the top has three of the pieces shaded in, and one piece unshaded. That’s because the top of the fraction is 3. For any fraction, the numerator tells you how many of the equal pieces you have.
Take a look at how changes to the numerator and denominator change a fraction. First, consider how changes affect the denominator. You’ve already seen what
looks like; here is what
looks like.
The numerator hasn’t changed (it’s still 3), so you still have three shaded pieces, but now the circle has been divided into five pieces instead of four. One effect is that each piece is now smaller. The fraction 
is smaller than 
. Rule: as the denominator of a number gets bigger, the value of the fraction gets smaller. The fraction 
is smaller than 
, because each fraction has three pieces, but when the circle (or number) is divided into five equal portions, each portion is smaller, so three 
portions are less than three 
portions.
As you split the circle into more and more pieces, each piece gets smaller and smaller:
Conversely, as the denominator gets smaller, each piece becomes bigger and bigger.
Now look at what happens as you change the numerator. The numerator tells you how many pieces you have, so if you make the numerator smaller, we get fewer pieces:
Conversely, if you make the numerator larger, you get more pieces. Look more closely at what happens as you get more pieces. In particular, you want to know what happens when the numerator becomes equal to or greater than the denominator. First, notice what happens when you have the same numerator and denominator. If you have 
pieces, this is what the circle looks like:
Remember, the circle represents one whole unit. So when all four parts are filled, you have one full unit, or 1. So 
is equal to 1. Rule: If the numerator and denominator of a fraction are the same, that fraction equals 1.
Following is what happens as the numerator becomes larger than the denominator. What does 
look like?
Each circle is only capable of holding four pieces, so when you fill up one circle, you have to move on to a second circle and begin filling it up, too. So one way of looking at
is that you have one complete circle, which you know is equivalent to 1, and you have an additional
. So another way to write 
is 1 +
. This can be shortened to
(“one and one-fourth”).
In the last example, the numerator was only a little larger than the denominator. However, that will not always be the case. The same logic applies to any situation. Look at the fraction 
. Once again, this means that each circle (i.e., each whole number) is divided into four pieces, and you have 15 pieces.
In this case, you have three circles completely filled. To fill three circles, you needed 12 pieces. (Note: 3 circles × 4 pieces per circle = 12 pieces.) In addition to the three full circles, you have three additional pieces. So you have:
.
Whenever you have both an integer and a fraction in the same number, you have a mixed number. Any fraction in which the numerator is larger than the denominator (e.g.,
) is known as an improper fraction. Improper fractions and mixed numbers express the same thing. How to convert from improper fractions to mixed numbers and vice-versa will be discussed later in the chapter.
Take a moment to review what you’ve learned about fractions so far. Every fraction has two components: the numerator and the denominator.
The denominator tells you how many equal pieces each unit circle has. Assuming that the numerator stays the same, as the denominator gets bigger, each piece gets smaller, so the fraction gets smaller as well.
The numerator tells you how many equal pieces you have. Assuming that the denominator stays the same, as the numerator gets bigger, you have more pieces, so the fraction gets bigger.
When the numerator is smaller than the denominator, the fraction will be less than 1. When the numerator equals the denominator, the fraction equals 1. When the numerator is larger than the denominator, the fraction is greater than 1.
For each of the following sets of fractions, decide which fraction is larger:
