Find practice problems and solutions for these concepts at Defining Fractions Practice Problems.
What's Around The Bend
 Defining a Fraction
 Parts of a Fraction
 Writing Fractions
 Using Fractions to Represent Real Situations
 Proper and Improper Fractions
 Like and Unlike Fractions
 Comparing Fractions
 Equivalent Fractions
 Simplifying Fractions
Defining a Fraction
What is a fraction, and why do we need them? A fraction is a type of number. We use fractions to represent a part of a whole. Let's say a whole pie is made up of eight slices. If you eat two slices, what part of the whole pie have you eaten? Fractions help us answer that question. You have eaten two out of eight slices, or twoeighths of the pie.
The following circle represents our pie. You can see that the circle is divided into eight equal slices. Two of those slices are shaded. The shaded area is twoeighths of the pie, which can be written as .
Every fraction is made up of two numbers with a horizontal line between them. The top number is called the numerator. The bottom number is called the denominator. In the fraction , the numerator is 2 and the denominator is 8. The numerator of a fraction can be any number, including zero, but the denominator of a fraction can never be zero. A fraction with a denominator of zero is undefined.
Fuel for Thought
A fraction represents a part of a whole. A fraction itself is a division statement. The top number of the fraction, the numerator, is divided by the bottom number of the fraction, the denominator.

The denominator is the number of parts that make the whole. Look again at our pie. There are 8 parts, or slices, that make up the whole. If the pie had only 4 slices, then the denominator of our fraction would be 4. The numerator is the number of parts that are shaded. Because our pie has 2 slices shaded, the numerator of our fraction is 2. The numerator always tells us how many parts of the whole we have.
Look at the following fraction. What fraction of this circle is shaded?
There are 3 parts of the circle that are shaded. The numerator of our fraction will be 3. This circle is divided into only 6 equal parts, so the denominator of our fraction will be 6. Because there are 3 parts that are shaded out of 6 total parts, we can say that of the circle is shaded.
We just saw how to turn pictures (shaded circles) into fractions. Now let's go in the opposite direction: Let's draw pictures to represent fractions.
When you look at a fraction, read it as "the numerator out of the denominator." The fraction is "2 out of 4." To represent the fraction with a circle, we need to show 2 out of 4 parts shaded. First, draw a circle divided into 4 equal parts. Then, shade 2 of those parts:
Pace Yourself
Fractions help us describe everyday situations. If you have 15 math problems for homework, and you've completed 5 of them, what fraction of your math homework is complete? The answer: 5 out of 15, or .
If of your class has brown hair, and there are 25 students in your class, how many have brown hair? Because the fraction means "12 out of 25," there are 12 students in your class with brown hair.
Find three other reallife scenarios that could be described with fractions.

Types of Fractions
So far, every fraction we have looked at has been a proper fraction. A proper fraction has a value that is between –1 and 1. In other words, the part of the whole is always less than the whole.
Inside Track
How can you determine if a fraction is proper? First, ignore any positive or negative signs. Then, compare the numerator to the denominator. If the numerator is less than the denominator, the fraction is proper.

What do you call a fraction whose numerator is greater than or equal to its denominator? An improper fraction. The fractions , , and are all improper fractions. Later, we'll see how to turn improper fractions into mixed numbers.
Every fraction can be described as either proper or improper. We also have a way of describing pairs of fractions. If two fractions have the same denominator, then they are like fractions. If two fractions have different denominators, then they are unlike fractions.
The fractions and are like fractions. Both have a denominator of 4. Like fractions are easy to compare: The fraction with the greater numerator is the greater fraction. The fractions and are unlike fractions, because they do not have the same denominator. Unlike fractions are harder to compare. In fact, you should always convert unlike fractions to like fractions before comparing.
Comparing Fractions
Often, you will want to compare two fractions to determine which fraction is larger. Let's begin by comparing like fractions. Which fraction is bigger, or ? The fractions are like, so all we need to do is compare the numerators. Because 4 is greater than 2, is greater than .What about and ? Because 7 is greater than 6, is greater than .
Comparing unlike fractions is a bit trickier. Think of comparing unlike fractions as comparing two different units of measure. Which length is longer, 28 centimeters or 11 inches? It is very difficult to say—but it would be much easier to figure out if both lengths were given in centimeters, or if both lengths were given in inches. That's why we always turn unlike fractions into like fractions before comparing them.
How do we turn unlike fractions into like fractions? We find a common denominator, and rewrite both fractions with that new denominator. The best way to find a common denominator for two unlike fractions is to find the least common multiple of those two denominators.
Fuel for Thought
A common denominator for two fractions is a number that is a multiple of each of the denominators of those fractions. For instance, 15 is a common denominator for the fractions and because 15 is a multiple of 5 and a multiple of 3. 15 isn't the only multiple that 5 and 3 have in common—they also have 30, 45, 60, and many others in common—but 15 is the least common multiple. Of all the multiples that 3 and 5 have in common, 15 is the smallest.

That may sound hard, but it's as simple as remembering your times tables. Take the unlike fractions and . We need to find a common denominator for these fractions before we can compare them. To find the least common multiple of 4 and 6, think of the 4 times table and the 6 times table:
4 times table: 4, 8, 12, 16, 20, 24, …
6 times table: 6, 12, 18, 24, 30, 36, …
By listing the multiples of 4 and 6, we can find the numbers that are multiples of both 4 and 6. Notice the numbers that appear in both times tables: 12 and 24. These numbers are common multiples of 4 and 6. 12 is the smallest multiple that is common to 4 and 6. We say that 12 is the "least common multiple" of 4 and 6.
Now that we've found the least common multiple, we can convert both fractions into twelfths—fractions with denominators of 12. To do this, we must change both the numerator and the denominator of each fraction.
Caution!
There are infinitely many ways to write the value of a single fraction. When converting a fraction to a new fraction with a different denominator, be sure that the value of the fraction does not change. Whatever you do to the denominator, you must also do to the numerator. If you create a new fraction with a denominator that is 10 times greater than the denominator of your original fraction, then you must multiply the numerator of your original fraction by 10 to find the numerator of your new fraction.

We know that the denominators of our new fractions will be 12. But what will the numerators be? Let's start with . The denominator of this fraction is 4. The new denominator will be 12. That means that the new denominator is 3 times larger than the old denominator. How did we figure that out? By dividing the new denominator, 12, by the old denominator, 4. Because the new denominator is 3 times larger than the old denominator, the new numerator must be 3 times larger than the old numerator. Because 2 × 3 = 6, the numerator of our new fraction is 6. We had to change the numerator in the same way we changed the denominator: = . These fractions may look different, but they have the exact same value. In fact, we could have multiplied the numerator and the denominator by 6, creating the fraction , and that would have been equal to (and ) too. As long as you multiply the numerator by the same number by which you multiply the denominator, the value of the fraction will not change.
Now let's convert to a fraction with a denominator of 12. Follow the same steps. Divide the new denominator, 12, by the old denominator, 6: = 2. The new denominator is 2 times bigger than the old denominator. Therefore, the new numerator must be 2 times bigger than the old numerator: 4 × 2 = 8. The numerator of our new fraction is 8. = .
Now that we have rewritten our two unlike fractions, and , as the like fractions and , we are ready to compare them. Because 8 is greater than 6, is greater than , and is greater than .
Let's look at another example. Which is bigger, or ? We follow the same steps as in the previous example. First, we find the least common multiple of 5 and 10:
5 times table: 5, 10, 15, 20, 25, 30, …
10 times table: 10, 20, 30, 40, 50, 60, …
Every multiple of 10, including 10 itself, is a multiple of 5. Because 10 is the least common multiple of 5 and 10, we don't have to change the fraction at all. We just need to convert to tenths. Divide the new denominator, 10, by 5: = 2. Because the new denominator is 2 times bigger than the old denominator, the new numerator must be 2 times bigger than the old numerator: 3 × 2 = 6. = . Now we are ready to compare. Because 6 is greater than 5, is greater than , so is greater than .
Inside Track
Here are some tips for comparing fractions:
 If two fractions have the same numerator, but different denominators, the fraction with the smaller denominator is the bigger fraction. For instance, > . Check by finding common denominators: = , and = . is greater than . is also greater than , , and
 If the denominator of one or both of the fractions is a prime number and the other denominator is not a multiple of that prime number, the least common denominator will be the product of the two denominators. For instance, if you are comparing and , the least common denominator will be 40 (5 × 8), because 5 is a prime number and 8 is not a multiple of 5. Check by listing the multiples of each:
5 times table: 5, 10, 15, 20, 25, 30, 35, 40, …
8 times table: 8, 16, 32, 40, …
 If two fractions are positive and one of them is improper while the other is proper, the improper fraction is greater—you don't even have to find common denominators. All positive improper fractions are equal to 1 or more, while all positive proper fractions are equal to less than 1.

Simplifying Fractions
We've seen how to take fractions and convert them to new fractions with larger denominators. Now, let's go in the other direction. We can make the numbers in the fractions smaller, or simpler, by reducing them without changing the value of the fraction.
We can reduce a fraction by dividing its numerator and denominator by the same number. Remember, whatever we do to the numerator, we must also do to the denominator. In order to reduce a fraction to its simplest form, we must divide the numerator and the denominator by the largest number that is a factor of both the numerator and the denominator. This number is called the greatest common factor.
Fuel for Thought
The greatest common factor of two numbers is the largest number that divides evenly into both numbers. For instance, the greatest common factor of 12 and 18 is 6. Both 12 and 18 can be divided evenly by 6. Other numbers (1, 2, and 3) are also factors of both 12 and 18, but 6 is the greatest common factor.

Example
Let's simplify the fraction . We begin by listing the factors of each number:
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
The factors that 8 and 12 have in common are 1, 2, and 4. Therefore, 4 is the greatest common factor. We can reduce to its simplest form if we divide its numerator and denominator by 4: = 2 and = 3. Therefore, = . is in its simplest form.
Example
Now let's simplify . Again, begin by listing the factors of each number:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
12 and 36 have many common factors, but 12 is the greatest common factor. We can reduce to its simplest form if we divide its numerator and denominator by 12: = 1 and = 3. Therefore, = . is in its simplest form.
If a proper fraction has a prime number in its denominator, then it is already in simplest form. However, a proper fraction could have a composite number in its denominator, and it may also be in simplest form. For instance, the fraction is in simplest form even though 8 is not a prime number, but the fraction must be in simplest form, because 41 is a prime number.
Caution!
Finding the greatest common factor of two numbers is not the same as finding the least common multiple of two numbers. When you are simplifying a fraction, you are looking for the greatest common factor between a numerator and denominator of a single fraction. When you are finding common denominators for a pair of fractions, you are looking for the least common multiple of the denominators of the fractions.

We've seen how to write fractions and how to express reallife situations as fractions. We've also learned to classify fractions as proper or improper and how to classify pairs of fractions as like or unlike. By converting unlike fractions to like fractions, we saw how to compare them. In the next chapter, we'll use these skills to help us add and subtract fractions. Many tests ask for answers to be put in simplest form. Now that we know how to reduce fractions, we can always write our answers in simplest form. We're ready to start REALLY working with fractions.
Find practice problems and solutions for these concepts at Defining Fractions Practice Problems.