Introduction to Fractions

Proper and Improper Fractions

In a fraction, the number above the fraction bar is called the numerator. The number below is called the denominator.

If the numerator is less than the denominator, the fraction is called a proper fraction. If the numerator is greater than the denominator, the fraction is called an improper fraction.

Examples of proper fractions:

Examples of improper fractions:

Just a note about the position of the negative sign in a fraction. For example, the fractions, , from above can be also written as follows:

The negative in front of a fraction bar is the same as a negative in the numerator or a negative in the denominator BUT NOT BOTH.

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Reducing Fractions

Find a number that divides into the numerator and denominator evenly. Divide this number into the numerator and denominator and write down the answer to these divisions and that is the answer.

Example: Reduce .

Solution: Since 3 goes into 12 and 18, we will do this division. 3 into 12 is 4 and 3 into 18 is 6. Thus . But wait, reduces by 2 (2 into 4 is 2 and 2 into 6 is 3). Thus, .

Note: You may have noticed from the beginning that 6 went into 12 and 18 and this would have led to right away. Sometimes there are many ways to reduce a fraction but either way you decide to reduce, it will always lead to the same answer. (Assuming you do all of the division correctly!)

You try the following problems.

Reduce the following fractions:

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Mixed Numbers

A mixed number is a number that is the sum of a whole number and a proper fraction.

Example of a mixed number:

Converting a Mixed Number to an Improper Fraction

To change a mixed number to an improper fraction:

Multiply the denominator and the whole number, add the numerator and put this new value in the numerator. The denominator of the mixed number is the denominator of the improper fraction.

Example: Convert to an improper fraction:

Solution:

You try the following problems:

Convert the following mixed numbers to improper fractions:

Note: You will use this skill with all of the operations of fractions.

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Converting an Improper Fraction to a Mixed Number

To change an improper fraction to a mixed number:

Divide the denominator into the numerator - this answer is the whole number part. The remainder part of the division is your new numerator. And again, the denominator of the improper fraction is still the denominator of the mixed number.

Example: Convert to a mixed number:

Solution:

Note: Writing an improper fraction as is NOT REDUCING A FRACTION! (All you are doing is changing the fractions form.)

You try the following!

Convert the following Improper Fractions to Mixed Numbers.

Note: This skill is less useful. You are not required to change an answer that is improper to a mixed number (despite what you learned in elementary school). Just be sure your any fraction is reduced.

Converting an improper fraction to a mixed number is necessary if you need to graph an improper fraction on a real number line or the Cartesian Coordinate System. You need to convert the improper fraction to a mixed number to know what two numbers your improper fraction is between.

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