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Chapter 2 - Fractions Section 2.1 Factors and Prime Numbers 1-2 Chapter 2 – Slide 2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. What Factors Mean and Why They are Important In a multiplication problem, the whole numbers that we are multiplying are factors. 23 6 The 2 and 3 are both factors of 6. 1-3 To identify the factors of a whole number, we divide the whole number by the numbers 1, 2, 3, 4, 5, 6, and so on, looking for remainders of 0. Chapter 2 – Slide 3 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Divisibility Tests 1-4 Chapter 2 – Slide 4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example What are the factors of 54? Is 54 divisible by Answer 1 2 3 4 5 6 7 8 9 10 1-5 Chapter 2 – Slide 5 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Identifying Prime and Composite Numbers A prime number is a whole number that has exactly two different factors: Itself and 1. A composite number is a whole number that has more than two factors. That is, whole numbers that are not prime. 1-6 Chapter 2 – Slide 6 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Indicate whether each number is prime or composite. a. 3 b. 96 c. 47 d. 39 1-7 Chapter 2 – Slide 7 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Finding the Prime Factorization of a Number The prime factorization of a whole number is the number written as the product of its prime factors. A good way to find the prime factorization of a number is by making a factor tree. 1-8 Chapter 2 – Slide 8 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples What is the prime factorization of the following. Create a factorization tree for each. (a) 90 (b) 720 1-9 Chapter 2 – Slide 9 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Finding the Least Common Multiple The least common multiple (LCM) of two or more whole numbers is the smallest nonzero whole number that is a multiple of each number. To Computer The Least Common Multiple (LCM) 1. Find the prime factorization of each number. 2. Identify the prime factors that appear in each factorization. 3. Multiply these prime factors, using each factor the greatest number of times that it occurs in any factorization. 1-10 Chapter 2 – Slide 10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 1. 2. 3. 1-11 Find the LCM of 9 and 15. Find the LCM of 4, 6 and 18. The prime factors are 3 and 5, 3 occurs twice and 5 occurs once. What is the LCM. Chapter 2 – Slide 11 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example You walk dogs to make some extra money. You walk a poodle every 4th day and a retriever every 6th day. You walked both dogs today. In how many days will you walk both dogs again? 1-12 Chapter 2 – Slide 12 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Section 2.2 Introduction to Fractions 1-13 Chapter 2 – Slide 13 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. What Fractions Are and Why They are Important 1-14 A fraction can mean part of a whole. A fraction is any number that can be written in the form a / b, where a and b are whole numbers and b is not zero. The denominator (on the bottom) stands for the number of parts into which the whole is divided. The numerator (on top) tell us how many parts of the whole the fraction contains. The fraction line separates the numerator from the denominator and stands for “out of” or “divided by” Chapter 2 – Slide 14 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Fraction Diagrams and Proper Fractions 3 4 The number is an example of a proper fraction because its numerator is smaller than its denominator. Example: In the diagram what does the shaded portion represent? The whole is divided into ____ parts. There are ___ shaded parts. The shaded portion represents _____. 1-15 Chapter 2 – Slide 15 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example The college of education accepted 115 out of 175 applicants for admission. What fraction of the applicants were accepted? 1-16 Chapter 2 – Slide 16 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Mixed Numbers and Improper Fractions A number with a whole number part and a proper fraction, is called a mixed number. A mixed number can also be expressed as an improper fraction. An improper fraction has a numerator greater than or equal to its denominator. 1-17 Chapter 2 – Slide 17 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Draw a diagram to show that 1 13 3 . 4 4 1 3 4 13 4 1-18 Chapter 2 – Slide 18 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Changing Mixed Numbers to Improper Fractions 1. 2. 3. 1-19 Multiply the denominator of the fraction by the whole-number part of the mixed number. Add the numerator of the fraction to this product. Write this sum over the denominator to form the improper fraction. Chapter 2 – Slide 19 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Write the following mixed number as an improper fraction. 1. 4 2 2. 7 3. 1-20 2 2 9 4. 3 5 4 3 7 5 Chapter 2 – Slide 20 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Changing Improper Fractions to Mixed Numbers 1. 2. Divide the numerator by the denominator to get the whole part. If there is a remainder, write that remainder over the denominator Example Write each improper fraction as a mixed number. 29 95 a. b. 4 9 1-21 Chapter 2 – Slide 21 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Finding Equivalent Fractions Equivalent fractions are fractions that have the same value as the original fraction. For example, the whole number 2 can be written as 4 6 40 or or 2 3 20 To find an equivalent fraction, multiply the numerator and a denominator of b by the same number n, a an b bn Where both b and n are nonzero. 1-22 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Chapter 2 – Slide 22 Example Find two fractions equivalent to: a. 3 5 b. 7 9 1-23 Chapter 2 – Slide 23 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Reducing Fractions A fraction is said to be in simplest form (or reduced to lowest terms) when the only common factor of its numerator and denominator is 1. 1-24 Chapter 2 – Slide 24 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Express each in simplest form (or reduce to lowest terms). a. 4 16 b. 42 35 1-25 Chapter 2 – Slide 25 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Like and Unlike Fractions Definitions Like fractions are fractions with the same denominator. Unlike fractions are fractions with different denominators. To Compare Fractions If the fractions are like, compare their numerators. If the fractions are unlike, write them as equivalent fractions with the same denominator and then compare their numerators. 1-26 Chapter 2 – Slide 26 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example 8 7 Compare and . Which fraction is greater? 15 9 1-27 Chapter 2 – Slide 27 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Least Common Denominator (LCD) For two or more fractions, their least common denominator (LCD) is the least common multiple of their denominators. 1-28 Chapter 2 – Slide 28 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example 1. Write as equivalent fractions with a common denominator of 18. 5 6 1 2 2. Order from smallest to largest: 1-29 13 18 5 1 13 , , 6 2 18 Chapter 2 – Slide 29 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Section 2.3 Adding and Subtracting Fractions 1-30 Chapter 2 – Slide 30 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Adding and Subtracting Like Fractions To add (or subtract) like fractions, add (or subtract) the numerators. Use the given denominator. Write answer in simplest form. 1-31 Chapter 2 – Slide 31 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 1. 4 8 Add: . 15 15 2. 5 12 7 Find the sum of , and . 21 21 21 3. Find the difference between 1-32 15 2 and . 9 9 Chapter 2 – Slide 32 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Adding and Subtracting Unlike Fractions To Add (or Subtract) Unlike Fractions 1-33 Write the fractions as equivalent fractions with the same denominator, usually the LCD. Add (or subtract) the numerators, keeping the same denominator. Writhe the answer in simplest form. Chapter 2 – Slide 33 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 1. 6 3 Add: 16 24 2. Subtract: 7 3 12 8 3. Combine: 1-34 2 5 1 3 6 10 Chapter 2 – Slide 34 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Adding Mixed Numbers To Add Mixed Numbers 1-35 Write the fraction as equivalent fractions with the same denominator, usually the LCD. Add the fractions. Add the whole numbers. Write the answer in simplest form. Chapter 2 – Slide 35 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 1. 2 1 Add: 4 3 5 5 2. Find the sum of 3 3 , 4 7 , and 7. 8 8 3. Find the sum of 1-36 1 5 3 1 4 and 2 . 4 6 5 Chapter 2 – Slide 36 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Subtracting Mixed Numbers To Subtract Mixed Numbers 1-37 Write the fractions as equivalent fractions with the same denominator, usually the LCD. Rename (or borrow from) the whole number on top if the fraction on the bottom is larger than the fraction on top. Subtract the fraction. Subtract the whole number. Write the answer in simplest form. Chapter 2 – Slide 37 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 1. 2. 7 1 Subtract: 4 1 9 9 Subtract: 15 1 9 11 12 12 3. Subtract: 4. Subtract 1-38 2 8 1 9 3 1 10 4 8 16 Chapter 2 – Slide 38 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example 5 5 1. Add: 5 7 6 8 Sometimes it is easier to add or subtract mixed numbers by first converting them to improper fractions. 1 7 1. Subtract: 13 11 4 8 1 4 3 2. Combine and check. 6 1 2 3 5 10 1-39 Chapter 2 – Slide 39 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Section 2.4 Multiplying and Dividing Fractions 2-40 Chapter 2 – Slide 40 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Multiplying Fractions To Multiply Fractions • Multiply the numerators. • Multiply the denominators • Write the answer in simplest form. 2-41 Chapter 2 – Slide 41 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 2 8 1. 3 9 4 2. 9 2 3 3. What is of 18? 8 3 7 8 4. 4 9 3 5. A recipe calls for ¼ cup of sugar. If Betty is making ½ the recipe, how much sugar should she use? 1-42 Chapter 2 – Slide 42 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Multiplying Mixed Numbers To Multiply Mixed Numbers • Convert the mixed numbers to improper fractions. • Multiply the fractions. • Write the answer in simplest form. 2-43 Chapter 2 – Slide 43 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 1. 1 3 2 1 2 4 4 1 2. 5 3 2 9 7 8 ¼ ft 3. An area rug is being put down as depicted in the following drawing. How many square feet of flooring is not covered by the rug? 12 ½ ft 12 ft 15 ½ ft 1-44 Chapter 2 – Slide 44 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Dividing Fractions To Divide Fractions and Mixed Numbers • Convert mixed numbers to improper fractions. • Change the divisor to it’s reciprocal (flip over) and multiply. • Write your answer in simplist form. 2-45 Chapter 2 – Slide 45 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Examples 3 7 1. 4 8 3 2. 20 3 5 7 3. What is divided by 8? 15 7 2 4. 1 1 8 3 1-46 3 1 1 5. 4 2 3 8 7 4 Chapter 2 – Slide 46 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example 1 Mrs. Henderson prepares a 5 pound roast for a 2 celebration. How many servings have been prepared if each person receives 1/3 lb of roast? 1-47 Chapter 2 – Slide 47 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Checking Answers As with adding and subtracting mixed numbers, it is important to check our answers when multiplying or dividing. We can check a product or a quotient of mixed numbers by estimating the answer and then confirming that our estimate answers are reasonably close. 2-48 Chapter 2 – Slide 48 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example 3 1 1 1. Simplify and check: 4 2 3 8 7 4 1-49 Chapter 2 – Slide 49 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.