Fractions

Fractions

Fractions are used when we speak of portions of some whole. “How much of the whole” is named by the fraction.

A fraction has two numbers, a numerator and a denominator, separated by a horizontal bar, this horizontal bar is called a vinculum.

<-- vinculum

A reduced fraction is one where the numerator and the denominator are relatively prime, that is, they have no common factor. A proper fraction is one whose numerator is less than its denominator. An improper fraction is one whose numerator is greater than its denominator.

Examples:

proper and reduced

proper (not reduced)

improper and reduced

improper (not reduced)

So, back to the meaning of a fraction. We're talking about equal parts of some whole.

We need some whole. Let's say we have a bag of 10 pennies. The bag is the whole. This bag of pennies can be divided into either

There is no other way to divide this bag of pennies into equal parts.

Consider the 2 equal parts:

One of the 2 equal parts contains the same number of pennies as does the other of the 2 equal parts, namely 5 pennies.

We write this fraction of the bag like this: Note the denominator, 2, is the number of equal parts this fraction is naming. The numerator, 1, is the number of these equal parts this fraction names. Since this fraction of the bag is also one half the pennies in the bag, we actually read this fraction as “one half.”

Now consider the 5 equal parts:

Each part has 2 pennies. Each part would be named using the fraction , again the 5 is stating that we're talking about 5 equal parts and in this case the 1 means we're talking about one of these 5 equal parts. This fraction is spoken “one fifth.”

If we want to talk about two of these equal parts we'd write and speak “two fifths.” of this bag would contain 2 pennies plus 2 pennies, that is 4 pennies.

Three of these equal parts would be written , spoken “three fifths,” and would represent 2 + 2 + 2 = 6 pennies.

Four of these equal parts would be written , spoken “four fifths,” and would represent 2 + 2 + 2 + 2 = 8 pennies.

Finally, all five of these equal parts would be written , spoken “five fifths” and would represent 1 whole.

2 pennies is less than 4 pennies, 4 pennies is less than 6 pennies, 6 pennies is less than 8 pennies, and 8 pennies is less than 10 pennies. In other words,

< < < <

Fractions can be compared as long as the denominators are equal. With this in mind how does fit in with these fifths? Well, represents 5 pennies. 5 pennies is greater than 4 pennies but less than 6 pennies. So,

< <

Now consider the 10 equal parts:

Each of these ten equal parts represents 1 penny and as a fraction would be written , and is spoken “one tenth.”

So we have one tenth, two tenths, three tenths, four tenths, 5 tenths, 6 tenths, 7 tenths, 8 tenths, 9 tenths and finally ten tenths, the whole, all written

, , , , , , , , ,

each representing, 1 penny, 2 pennies, 3 pennies, ... 9 pennies, and 10 pennies, the whole bag, respectively.

The following illustrates how all of these fractions are related (fractions in a column are equal, and each column represents a fraction larger than the fraction in the preceeding column):

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