**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 (one-*sixtieth*) 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 (two-*sevenths*) of the week.**Money is expressed in fractions**. A nickel is (one-*twentieth*) of a dollar, because there are 20 nickels in one dollar. A dime is (one-*tenth*) 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

*. 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).*

**d**own*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.

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