1. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 1

2. 2 Chapter 2 Fractions Prepared by Dr. Elena Skliarenko

3. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 3 Recognize the three types of fractions Convert improper fractions to whole or mixed numbers and mixed numbers to improper fractions Convert fractions to lowest and highest terms Fractions #2 Learning Unit Objectives Types of Fractions and Conversion Procedures LU2.1

4. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 4 Add like and unlike fractions Find the least common denominator (LCD) by inspection and prime numbers Subtract like and unlike fractions Add and subtract mixed numbers with the same or different denominators Fractions #2 Learning Unit Objectives Adding and Subtracting of Fractions LU2.2

5. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 5 Multiply and divide proper fractions and mixed numbers Use the cancellation method in the multiplication and division of fractions Fractions #2 Learning Unit Objectives Multiplication and Division of Fractions LU2.3

6. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 6 Use a calculator in operations with fractions and mixed numbers Convert improper fractions into mixed numbers Add and subtract like and unlike fractions and mixed numbers Multiply and divide fractions and mixed numbersFractions #2 Learning Unit Objectives Using a Calculator LU2.4

7. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 7 Types of Fractions 3, 4, 12, 11 15 8 26 35 19, 9, 13, 42 19 4 10 29 2 9 Proper Mixed Numbers Improper Numerator Denominator 2 1, 3 7 5, 8 6 8, 9 5 6 10

8. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 8 Converting Improper Fractions to Whole or Mixed Numbers 2 Steps 1 Divide the numerator by the denominator 2a. If you have no remainder, the quotient is a whole number 2b. If you have a remainder, the quotient is a mixed number 24 21 1 4 4 5 R 1 4 21 20 1 = 5 = 1

9. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 9 Reducing Fractions to Lowest Terms by Inspection Find the lowest whole number that will divide evenly into the numerator and denominator 24 24 ÷ 6 4 30 30 ÷ 6 5 = =

10. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 10 Finding the Greatest Common Divisor Step 1. Divide the numerator into the denominator - 12 30 2 12 30 24 6 2 6 12 12 0 Step 2. Divide the remainder in Step 1 into the divisor of Step 1 12 ÷ 6 2 30 ÷ 6 5 = Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0

11. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 11 Divisibility Tests = = = = Last digit is 0,2,4,6,8 Sum of the digits is divisible by 3 Last two digits can be divided by 4 Last digit is 0 or 5 The number is even and 3 will divide into the sum of the digits The last digit is 0 12 6 14 7 36 12 69 23 140 1(40) 160 1(60) 15 3 20 4 12 2 18 3 90 9 100 10 3 + 6 = 9 ÷ 3 = 3 6 + 9 = 15 ÷ 3 = 5 35 7 40 8 = = = =

12. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 12 Least Common Denominator (LCD) The smallest nonzero whole number into which all denominators will divide evenly. 5 16 20 10 40 60 20 + What is the least common denominator?

13. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 13 Adding and Subtracting Fractions with the Same Denominator Add the numerators and place the total over the denominator Subtract the numerators and place the total over the denominator 2 3 5 9 9 9 = - = + 7 1 6 ÷ 2 3 12 12 12 ÷ 2 6 - ==

14. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 14 Adding and Subtracting Proper Fractions with Different Denominators Find the LCD Change each fraction to a like fraction Add the numerators Find the LCD Raise the fraction to its equivalent Subtract the numerators 1 1 3 8 9 12 5 2 8 64 - + 24 + 9 + 8 + 6 = 47 72 72 72 72 72 40 64 - 2 64 38 = 19 64 32 ++

15. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 15 Multiplying Proper Fractions 2 Steps Multiply the numerator and the denominators Reduce the answer to lowest terms 5 2 4 40 20 2 5 7 70 35 == x x

16. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 16 Multiplying Mixed Numbers Convert the mixed numbers to improper fractions Multiply the numerator and denominators 1 1 7 3 7 1 3 2 3 2 2 2 2 3 X X 1 ===

17. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 17 Dividing Proper Fractions 1 2 1 3 3 8 3 8 2 16 Invert (turn upside down) the divisor (the second fraction ) Multiply the fractions ==X ÷

18. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 18 Using a Calculator How to use the ab/c button? How to use the ab/c button? You can use the “magic” ab/c button You can use the “magic” ab/c button for conversion of improper fractions for conversion of improper fractions into a mixed number. into a mixed number. Example 1: 1 + 1 + 1 + 1 3 8 9 12 Enter given fractions and add them in sequence. 1 a b/c 3 + 1 a b/c 8 + a b/c + 1 a b/c 9 + 1 a b/c + 12 = 47 r 72

19. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. 19 Proper use of the calculator