HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 2.1 Introduction to Fractions and Mixed Numbers

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Understand the basic concepts of fractions. o Multiply fractions. o Find equivalent fractions. o Reduce fractions to lowest terms. o Change mixed numbers to improper fractions. o Change improper fractions to mixed numbers.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Understanding Fractions a.If a whole pizza is cut into 3 equal pieces, then 2 of these pieces represent of the pizza (see pizza portion of the figure). The remaining piece (missing portion of the pizza) represents of the pizza.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Understanding Fractions (cont.) b.In the rectangle, 3 of the 4 equal parts are shaded. Thus of the rectangle is shaded and is not shaded.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Proper Fractions and Improper Fractions A proper fraction is a fraction in which the numerator is less than the denominator. Proper fractions have values less than 1. Examples of proper fractions: Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Proper Fractions and Improper Fractions (cont.) An improper fraction is a fraction in which the numerator is greater than or equal to the denominator. Improper fractions have values greater than or equal to 1. Examples of improper fractions: Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Proper Fractions indicates 5 of 6 equal parts.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Improper Fractions Each whole square is separated into 3 equal parts. The shading here indicates 5 of these equal parts and can be represented by the improper fraction.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Number 0 in Fractions For any nonzero value of b, For any value of a, is undefined. Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: The Number 0 in Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Multiply Fractions 1.Multiply the numerators. 2.Multiply the denominators. Multiplication with Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Multiplication with Fractions Find Solution

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Multiplication with Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Find an Equivalent Fraction Multiply the numerator and denominator by the same nonzero whole number. Finding Equivalent Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Finding Equivalent Fractions Find the missing numerator that will make the fractions equal. Solution

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Finding Equivalent Fractions Find the missing numerator that will make the fractions equal. Solution

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 9: Finding Equivalent Fractions Find the missing numerator that will make the fractions equal. Solution

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Application of Multiplication with Fractions In a certain voting district, of the eligible voters are actually registered to vote. Of those registered voters, are independents (have no party affiliation). What fraction of the eligible voters are registered independents?

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Application of Multiplication with Fractions (cont.) Solution Since the independents are a fraction of the eligible voters, we multiply. Thus of the eligible voters are registered as independents.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Reduce a Fraction to Lowest Terms 1.Factor the numerator and denominator into prime factors. 2.Use the fact that and “divide out” all common factors. Note: Reduced fractions can be improper fractions. Reducing Fractions to Lowest Terms

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Reducing Fractions to Lowest Terms

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12: Reducing Fractions to Lowest Terms Reduce to lowest terms. Solution We can divide out common factors (prime or not) with the understanding that a number divided by itself equals 1.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 13: Reducing Fractions to Lowest Terms Reduce to lowest terms. Solution Finding a common factor could be difficult here. Prime factoring helps.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Change a Mixed Number to an Improper Fraction 1.Multiply the whole number by the denominator of the proper fraction. 2.Add the numerator of the proper fraction to this product. 3.Write this sum over the denominator of the fraction. Changing Mixed Numbers to Improper Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Change a Mixed Number to an Improper Fraction (cont.) For example: Changing Mixed Numbers to Improper Fractions

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solution Step 1:Multiply the whole number by the denominator: 8 ∙ 10 = 80. Step 2:Add the numerator: = 89. Step 3:Write this sum over the denominator: Example 14: Changing a Mixed Number to an Improper Fraction Change to an improper fraction.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solution Step 1: Multiply 11 ∙ 3 = ____. Step 2: Add the numerator: _____ + _____ = ____. Step 3: Write this sum, _______, over the denominator _______. Therefore, ______. Completion Example 15: Changing a Mixed Number to an Improper Fraction Change to an improper fraction.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Change an Improper Fraction to a Mixed Number 1.Divide the numerator by the denominator. The quotient is the whole number part of the mixed number. 2.Write the remainder over the denominator as the fraction part of the mixed number. Changing Improper Fractions to Mixed Numbers

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solution Divide 67 by 5. Example 16: Changing an Improper Fraction to a Mixed Number Change to a mixed number. whole number part numerator

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems 1.Find the products. a. b. c. d. 2.Reduce to lowest terms. a. b. c. 3.Change to an improper fraction.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems (cont.) 4.Change to a mixed number. 5.Write as a mixed number.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.a.b.c. d. 2.a. b.c