2. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Fraction Notation: Multiplication and Division 3
3.1
Multiples and Divisibility
3.2
Factorizations
3.3
Fractions and Fraction Notation
3.4
Multiplication and Applications
3.5
Simplifying
3.6
Multiplying, Simplifying, and More with Area
3.7
Reciprocals and Division
3.8
Solving Equations: The Multiplication Principle
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 2

3. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying a
Multiply by 1 to find an equivalent expression using a different denominator. b
Simplify fraction notation. c
Test to determine whether two fractions are equivalent. b
Simplify fraction notation like n/n to 1, 0/n to 0, and
n /1 to n.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 3

4. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying
MULTIPLICATIVE IDENTITY FOR FRACTIONS
When we multiply a number by 1, we get the same number: a
Multiply by 1 to find an equivalent expression using a different denominator.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 4

5. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying a
Multiply by 1 to find an equivalent expression using a different denominator.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 5

6. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying a
Multiply by 1 to find an equivalent expression using a different denominator. 1
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 6

7. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying a
Multiply by 1 to find an equivalent expression using a different denominator. 3
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 7

8. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 8

9. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation.
To simplify fraction notation, we reverse the process of multiplying by 1:
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 9

10. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying
SIMPLIFYING FRACTION NOTATION
1. Factor the numerator and factor the denominator.
2. Identify any common factors in the numerator and denominator.
3. Use the common factors to remove a factor equal to 1. b
Simplify fraction notation.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 10

11. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation.
Simplify.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 11

12. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation. 10
Since both 105 and 135 end in 5, we know that 5 is a factor of both the numerator and the denominator:
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 12

13. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation. 10
A fraction is not “simplified” if common factors of the numerator and the denominator remain. Because 21 and 27 are both divisible by 3, we must simplify further:
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 13

14. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation.
Canceling is a shortcut that you may have used for removing a factor that equals 1 when working with fraction notation. With concern, we mention it as a possibility for speeding up your work. Canceling may be done only when removing common factors in numerators and denominators. Each common factor allows us to remove a factor equal to 1 in a product.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 14

15. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation.
Generally, slashes are used to indicate factors equal to 1 that have been removed.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 15

16. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying b
Simplify fraction notation.
• If you cannot factor, do not cancel! If in doubt, do not cancel!
• Only factors can be canceled, and factors are never separated by + or – signs.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 16

17. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying c
Test to determine whether two fractions are equivalent.
When denominators are the same, we say that fractions have a common denominator. One way to compare fractions is to find a common denominator and compare numerators. We can multiply each fraction by 1,
using the other denominator to write 1.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 17

18. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying c
Test to determine whether two fractions are equivalent.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 18

19. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying
A TEST FOR EQUALITY c
Test to determine whether two fractions are equivalent.
We call and cross products. Since the cross products are the same we know that
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 19

20. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying c
Test to determine whether two fractions are equivalent. 12
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 20

21. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying c
Test to determine whether two fractions are equivalent. 12
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 21

22. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying c
Test to determine whether two fractions are equivalent. 14
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 22

23. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
3.5
Simplifying c
Test to determine whether two fractions are equivalent. 14
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 23