## Fractions

This section deals with situations where we want to represent part of a whole. In an Introduction to Numbers, we suggested that we do not want to represent a part cat but would want to represent a piece of a pie by some number or numerical expression. We discuss how to do this below.

**What Is a Fraction?**

A fraction is not a number. Rather, it is an arithmetic expression and more explicitly a division expression. The fraction representing three quarters of a pie is sometimes written as a 3 above a horizontal line with a 4 underneath the line. This may be impossible to express in this way using most software programs. Consequently, we tip the fraction over so it fits on one line of text producing the expression 3/4. This is now an expression of integer division. The top number of the fraction which is also the first operand in the division expression, is called the **numerator**. The bottom number in the fraction, or second operand, is called the **denominator**.

**Proper and Mixed Fractions**

A proper fraction has both numerator and denominator as integers (whole numbers). Since most fractions that we work with are proper fraction we will drop the “proper” in the name and just call them fractions.

Every integer can be expressed as an equivalent fraction composed of the integer as the numerator and 1 as the denominator. For example 3 = 3/1.

A mixed fraction has two parts, an integer part and a proper fraction part. Example: 2 2/3 is composed of the integer 2 and the fraction 2/3. We sometimes use mixed fractions as a way of expressing fractions that have a numerator larger than the denominator. For example, the fraction 413/17 can be expressed as the mixed fraction 24 5/17.

Another way of looking at the fraction 413/17 is as the division problem 413 รท 17 which has the answer of 24 with a remainder of 5 or 5/17.

To convert a mixed fraction into a proper fraction, multiply the integer part by the denominator of the fractional part and add the fractional part to the result. For example:

3 5/6 ? 3/1 + 5/6

= (3*6)/(1*6) + 5/6

= 18/6 + 5/6

= 23/6

Next we will learn how to add and subtract fractions.

**Addition and Subtraction of Fractions**

Procedure:

- Change any mixed fraction into its equivalent proper fraction (example express 1 1/4 as 5/4).
- Find a common denominator for all fractions:
- For two fractions, multiply the denominators together to get a common denominator.
- For more than two fractions, multiply the first two denominators together

- Express each fraction as an equivalent fraction with the common denominator.
- Add and subtract the numerators to give a single number that will be the numerator of the result with the common denominator as the denominator of the result.

Example:

#### Multiplication and Division of Fractions

#### Ratios and Percentages