1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 2.6 Order of Operations with Fractions and Mixed Numbers

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Compare fractions by finding a common denominator and comparing the numerators. o Evaluate expressions with fractions. o Simplify complex fractions. o Evaluate expressions with mixed numbers. o Find the average of a group of fractions or mixed numbers.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Comparing Two or More Fractions To Compare Two Fractions (to Find which is Larger or Smaller) 1.Find the least common denominator (LCD). 2.Change each fraction to an equivalent fraction with that denominator. 3.Compare the numerators.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Which is larger: How much larger? Solution Step 1: Find the LCD. Example 1: Comparing Fractions Step 2: Find equal fractions with denominator 24.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Comparing Fractions (cont.) Step 3: is larger than since 21 is larger than 20.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Which is larger: How much larger? Solution Step 1: Step 2: Example 2: Comparing Fractions

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Comparing Fractions (cont.) Step 3: is larger than since 33 is larger than 32.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Arrange in order from smallest to largest. Then find the difference between the smallest and the largest. Solution Step 1: LCD = 30 Step 2: Example 3: Comparing Fractions

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Comparing Fractions (cont.) Step 3: Smallest to largest:

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Evaluating Expressions with Fractions Rules for Order of Operations 1.Simplify within grouping symbols, such as parentheses ( ), brackets [ ], and braces { }. Start with the innermost grouping. 2.Evaluate any numbers or expressions indicated by exponents.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Evaluating Expressions with Fractions Rules for Order of Operations (cont.) 3.Moving from left to right, perform any multiplications or divisions in the order in which they appear. 4.Moving from left to right, perform any additions or subtractions in the order in which they appear.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Order of Operations with Fractions Evaluate the expression: Solution

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Order of Operations with Fractions (cont.)

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Order of Operations with Fractions Evaluate the expression: Solution

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Order of Operations with Fractions Evaluate the expression: Solution

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Order of Operations with Fractions (cont.)

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 7: Order of Operations with Fractions Evaluate the expression: Solution

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Simplifying Complex Fractions To Simplify Complex Fractions 1.Simplify the numerator so that it is a single fraction, possibly an improper fraction. 2.Simplify the denominator so that it also is a single fraction, possibly an improper fraction. 3.Divide the numerator by the denominator, and reduce if possible.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Simplifying Complex Fractions Simplify the complex fraction: Solution

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Simplifying Complex Fractions (cont.) So,

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Notes The complex fraction in Example 8 could have been written as Thus the fraction bar in a complex fraction serves the same purpose as two sets of parentheses, one surrounding the numerator and the other surrounding the denominator. Simplifying Complex Fractions

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Order of Operations with Mixed Numbers Evaluate the expression: Solution

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Order of Operations with Mixed Numbers (cont.) Now divide.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Order of Operations with Mixed Numbers Evaluate the expression: Solution Work inside each set of parentheses:

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Order of Operations with Mixed Numbers (cont.) Now divide.

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Order of Operations with Mixed Numbers Evaluate the expression: Solution

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Order of Operations with Mixed Numbers (cont.)

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Order of Operations with Mixed Numbers (cont.) Alternate Solution Or, working with separate parts, we can write:

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Order of Operations with Mixed Numbers (cont.)

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the average of Solution Finding the average is the same as evaluating the expression Example 12: Average of Mixed Numbers

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12: Average of Mixed Numbers (cont.) Find the sum first and then divide by 3.

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12: Average of Mixed Numbers (cont.) The average is

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Evaluate each of the following expressions by using the rules for order of operations. 1.2.3. 4.Simplify the complex fraction: 5.Find the average of Practice Problems

34. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.2.3. 4.5.