Fractions and Mixed Numbers Study Guide

Updated on Aug 24, 2011

Find practice problems and solutions for these concepts at Fractions and Mixed Numbers Practice Problems.

What's Around The Bend

  • What's a Fraction?
  • Changing a Fraction without Changing Its Value
  • Addition and Subtraction with Fractions
  • Multiplying and Dividing Fractions
  • Reciprocals
  • Reciprocals on the Number Line
  • Introducing Mixed Numbers
  • Converting to a Fraction
  • Converting a Fraction to a Mixed Number
  • Adding and Subtracting Mixed Numbers
  • Multiplying and Dividing Mixed Numbers

Fractions are used to represent parts of a whole. You can think of the fraction bar as meaning "out of." You can also think of the fraction bar as meaning "divided by."

Although most people call the top part of the fraction the "top" and the bottom part of a fraction the "bottom," the technical names are numerator and denominator.


You are never allowed to have a zero in the denominator. A fraction whose denominator is zero is undefined.

A proper fraction has a numerator that is smaller than its denominator. Examples are , , and . Improper fractions have numerators that are bigger than their denominators. Examples include , , and .

What if the numerator and denominator are equal (making the fraction equal to 1), as is the case with , , and ? Are these proper or improper fractions? The rule is that these fractions must be called improper fractions.

A fraction that represents a particular part of the whole is sometimes referred to as a fractional part. For example, let's say that a family has 4 cats and 2 dogs. What fractional part of their pets are cats? Because 4 out of the total 6 animals are cats, the fractional part of their pets that are cats is equal to . You can reduce this fraction to .

Fuel for Thought

A proper fraction (with positive numerator and denominator) has a value that is less than 1. If the value is 1 or greater, the fraction is called an improper fraction.

You can identify an improper fraction easily: Its numerator is as big as or bigger than its denominator. Is that a no-no? No, it is not a no-no. There is really nothing improper about an improper fraction.

There's one more term you need to know, and then we can stop talking about fractions and start using them. That term is mixed number, which consists of a whole number paired with a proper fraction. Examples would include , , and .

Working with Fractions

What is it that doesn't get smaller when you reduce it? A fraction! There are two operations you can do to change the look and feel of a fraction without changing its value: You can reduce it (as in converting to ) or you can augment it (as in converting to ).

Reducing a Fraction

When you reduce to , you don't change the value of the fraction. How is it possible to change the fraction without changing its value?

Well, both and reside at the same point on the number line. That means is the same number as . A piece of lumber that is of an inch thick has the same thickness as a piece of lumber that is of an inch thick. The principle used to preserve the value of the fraction was to divide the numerator and the denominator by the same number (in this case, 2).

When you reduce a fraction, you preserve its value.

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