Fractions Math 173 DF Fall 2008 R. Raina

Overview ► Fraction Basics ► Types of Fractions ► Simplifying Fractions ► Multiplying and Dividing ► Adding and Subtracting ► Prime Factorization Technique ► Converting from Fractions to Decimals

The Basics: A fraction has a numerator, a denominator, and a fraction line. If a whole quantity is divided into parts, each of those parts is called a fraction of the whole. Since division by zero is not permitted, it should be understood in our work with fractions that the denominator cannot be zero. Numerator Denominator Fraction Line Example: There are 5 people in a room, two men and three women. Thus of the people are men, and are women.

Common vs. Algebraic Fractions ► Common Fraction:  A fraction whose numerator and denominator are both integers ► Algebraic Fraction  A fraction whose numerator and/or denominator contain literal quantities. Examples:

Proper, Improper and Mixed Fractions ► A proper common fraction is one whose numerator is smaller than its denominator. ► A mixed number is the sum of an integer and a fraction. ► A improper common fraction is one whose numerator is larger than its denominator.

Changing Improper Fractions to Mixed Form ► To change an improper fraction to a mixed number: - Divide the denominator into the numerator. - Write the remainder over the denominator.

Changing Mixed Numbers to Improper Fractions ► To change a mixed number to an improper fraction: - Multiply the whole number by the denominator and add this number to the numerator. - Write the sum over the denominator.

Simplifying a Fraction ► Reducing to lowest terms:  dividing both numerator and denominator by any factor that is contained in both. ► Changing signs:  any two of the three signs of a fraction may be changed without changing the value of a fraction.

Simplify the following fractions:

Multiplication and Division What are we doing?  

Multiplication ► Fraction Multiplication (Steps): 1) Change mixed numbers to improper fractions. 2) Change whole numbers to fractions by dividing by one. 2) Change whole numbers to fractions by dividing by one. 3) Multiply all numerators together – this number is the numerator of your answer. 4) Multiply all denominators together – this number is the denominator of your answer. 5) Simplify you answer. How do we do it?

Division ► Dividing Fractions (Steps) 1) Change mixed numbers to improper fractions. 2)Change whole numbers to fractions by dividing by one. 3) Invert and Multiply (Take the reciprocal of the fraction after the sign, Change the division sign to a multiplication sign.) 5) Multiply the two fractions together How do we do it?

Adding and Subtracting += + = +=

► If the fractions have a common denominator: - Add the numerators together. - Keep the common denominator. - Simplify if required. ► If the fractions do not have a common denominator: - Find the lowest common denominator (LCD) – use prime factorization. - Find the equivalent fraction with the chosen denominator. - Add the fractions. Simplify if required. Adding

Subtracting ► If the fractions do not have a common denominator: - Find the lowest common denominator (LCD) – use prime factorization. - Find the equivalent fraction with the chosen denominator. - Subtract the fractions. Simplify if required. ► If the fractions have a common denominator: - Subtract the numerators together. - Keep the common denominator. - Simplify if required.

A Useful Technique – Prime Factorization Ex) Find the LCM of 18 and 24. ► Find the prime factors of each number, and line them up vertically. If a number appears in a column more than once, move only one down.

Decimals and Fractions ► To change a fraction to an equivalent decimal, divide the numerator by the denominator. ► To change a decimal number to a fraction:  Write a fraction with the decimal the numerator and 1 in the denominator  Multiply by a multiple of 10 that will make the numerator a whole number.  Reduce to lowest terms.

► To express a repeating decimal as a fraction: __ __ Example: Express as a fraction