Introduction to Three Kinds of Fractions
If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.
—Galileo Galilei, mathematician and astronomer (1564–1642)
This first fraction lesson will familiarize you with fractions, teaching you ways to think about them that will let you work with them more easily. This lesson introduces the three kinds of fractions and teaches you how to change from one kind of fraction to another, a useful skill for making fraction arithmetic more efficient. The remaining fraction lessons focus on arithmetic.
Fractions are one of the most important building blocks of mathematics. You come into contact with fractions every day: in recipes (cup of milk), driving ( of a mile), measurements (acres), money (half a dollar), and so forth. Most arithmetic problems involve fractions in one way or another. Decimals, percents, ratios, and proportions, which are covered in Lessons 6–12, are also fractions. To understand them, you have to be very comfortable with fractions, which is what this lesson and the next four are all about.
What is a Fraction?
A fraction is a part of a whole.
 A minute is a fraction of an hour. It is 1 of the 60 equal parts of an hour, or (onesixtieth) of an hour.
 The weekend days are a fraction of a week. The weekend days are 2 of the 7 equal parts of the week, or (twosevenths) of the week.
 Money is expressed in fractions. A nickel is (onetwentieth) of a dollar, because there are 20 nickels in one dollar. A dime is (onetenth) of a dollar, because there are 10 dimes in a dollar.
 Measurements are expressed in fractions. There are four quarts in a gallon. One quart is of a gallon. Three quarts are of a gallon.
Tip
It is important to know what "0" means in a fraction! = 0, because there are zero of five parts. But is undefined, because it is impossible to have five parts of zero. Zero is never allowed to be the denominator of a fraction!

The two numbers that compose a fraction are called the:
For example, in the fraction , the numerator is 3, and the denominator is 8. An easy way to remember which is which is to associate the word denominator with the word down. The numerator indicates the number of parts you are considering, and the denominator indicates the number of equal parts contained in the whole. You can represent any fraction graphically by shading the number of parts being considered (numerator) out of the whole (denominator).
Example:
Let's say that a pizza was cut into 8 equal slices, and you ate 3 of them. The fraction tells you what part of the pizza you ate. The pizza below shows this: It's divided into 8 equal slices, and 3 of the 8 slices (the ones you ate) are shaded. Since the whole pizza was cut into 8 equal slices, 8 is the denominator. The part you ate was 3 slices, making 3 the numerator.
If you have difficulty conceptualizing a particular fraction, think in terms of pizza fractions. Just picture yourself eating the top number of slices from a pizza that's cut into the bottom number of slices. This may sound silly, but most of us relate much better to visual images than to abstract ideas. Incidentally, this little trick comes in handy for comparing fractions to determine which one is bigger and for adding fractions to approximate an answer.
Sometimes the whole isn't a single object like a pizza, but a group of objects. However, the shading idea works the same way. Four out of the following five triangles are shaded. Thus, of the triangles are shaded.
Three Kinds of Fractions
There are three kinds of fractions  proper, improper, and mixed  explained in this Stud Guide.
Proper Fractions
In a proper fraction, the top number is less than the bottom number:
The value of a proper fraction is less than 1.
Example:
Suppose you eat 3 slices of a pizza that's cut into 8 slices. Each slice is of the pizza. You've eaten of the pizza.
Improper Fractions
In an improper fraction, the top number is greater than or equal to the bottom number:
The value of an improper fraction is 1 or more.
 When the top and bottom numbers are the same, the value of the fraction is 1. For example, all of these fractions are equal to , etc.
 Any whole number can be written as an improper fraction by writing that number as the top number of a fraction whose bottom number is 1, for example, .
Example:
Suppose you're very hungry and eat all 8 slices of that pizza. You could say you ate of the pizza, or 1 entire pizza. If you were still hungry and then ate 1 slice of your best friend's pizza, which was also cut into 8 slices, you'd have eaten of a pizza. However, you would probably use a mixed number, rather than an improper fraction, to tell someone how much pizza you ate. (If you dare!)
Tip
If a shape is divided into pieces of different sizes, you cannot just add up all the sections. Break the shape up into equal sections of the smaller pieces and use the total number of smaller pieces as your denominator. For example, break this box into 16 of the smaller squares instead of counting this as just six sections. The fraction that represents the shaded area would then be .

Mixed Numbers
When a fraction is written to the right of a whole number, the whole number and fraction together constitute a mixed number:
The value of a mixed number is greater than 1: It is the sum of the whole number plus the fraction.
Example:
Remember those 9 slices you ate? You could also say that you ate pizzas because you ate one entire pizza and one out of eight slices of your best friend's pizza.
Changing Improper Fractions into Mixed or Whole Numbers
Fractions are easier to add and subtract as mixed numbers rather than as improper fractions. To change an improper fraction into a mixed number or a whole number:
 Divide the bottom number into the top number.
 If there is a remainder, change it into a fraction by writing it as the top number over the bottom number of the improper fraction. Write it next to the whole number.
Example: Change into a mixed number.
1. 
Divide the bottom number (2) into the top number (13) to get the whole number portion (6) of the mixed number: 

2. 
Write the remainder of the division (1) over the original bottom number (2): 

3. 
Write the two numbers together: 

4. 
Check: Change the mixed number back into an improper fraction (see steps starting on page 20). If you get the original improper fraction, your answer is correct. 
Example: Change into a mixed number.
1. 
Divide the bottom number (4) into the top number (12) to get the whole number portion (3) of the mixed number: 

2. 
Since the remainder of the division is zero, you're done. The improper fraction is actually a whole number: 
3 
3. 
Check: Multiply 3 by the original bottom number (4) to make sure you get the original top number (12) as the answer. 
Changing Mixed Numbers Into Improper Fractions
Fractions are easier to multiply and divide as improper fractions rather than as mixed numbers. To change a mixed number into an improper fraction:
 Multiply the whole number by the bottom number.
 Add the top number to the product from step 1.
 Write the total as the top number of a fraction over the original bottom number.
Example: Changeinto an improper fraction.
1. 
Multiply the whole number (2) by the bottom number (4): 
2 × 4 = 8 
2. 
Add the result (8) to the top number (3): 
8 + 3 = 11 
3. 
Put the total (11) over the bottom number (4): 

4. 
Check: Reverse the process by changing the improper fraction into a mixed number. Since you get back , your answer is right. 
Example: Change into an improper fraction.
1. 
Multiply the whole number (3) by the bottom number (8): 
3 × 8 = 24 
2. 
Add the result (24) to the top number (5): 
24 + 5 = 29 
3. 
Put the total (29) over the bottom number (8): 

4. 
Check: Change the improper fraction into a mixed number. Since you get back , your answer is right. 
Tip
Reach into your pocket or coin purse and pull out all your change. You need more than a dollar's worth of change for this exercise, so if you don't have enough, borrow some loose change and add that to the mix. Add up the change you collected and write the total amount as an improper fraction. Then convert it to a mixed number.

Three Kinds of Fractions Sample Questions
 Change into a mixed number.
 Change into an improper fraction.
Solutions
Question 1
1.

Divide the bottom number (3) into the top number (14) to get the whole number portion (4) of the mixed number:


2.

Write the remainder of the division (2) over the original bottom number (3):


3.

Write the two numbers together:


4.

Check: Change the mixed number back into an improper fraction to make sure you get the original

Question 2
1.

Multiply the whole number (3) by the bottom number (5):

3 × 5 =15

2.

Add the result (15) to the top number (2):

15 + 2 = 17

3.

Put the total (17) over the bottom number (5):


4.

Check: Change the improper fraction back to a mixed number.



Dividing 17 by 5 gives an answer of 3 with a remainder of 2:



Put the remainder (2) over the original bottom number (5):



Write the two numbers together to get back the original mixed number:


Find practice problems and solutions for these concepts at Three Kinds of Fractions Practice Questions.