1. Introduction to Fractions and Mixed Numbers
Section 2.1
Introduction to Fractions and Mixed Numbers

2. Objectives Understand the basic concepts of fractions.
Multiply fractions.
Find equivalent fractions.
Reduce fractions to lowest terms.
Change mixed numbers to improper fractions.
Change improper fractions to mixed numbers.

3. 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.

4. 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.

5. Fractions
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:

6. Fractions
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:

7. Example 2: Proper Fractions
indicates 5 of 6 equal parts.

8. 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 .

9. Fractions
The Number 0 in Fractions For any nonzero value of b, For any value of a, is undefined.

10. Example 4: The Number 0 in Fractions

11. Multiplication with Fractions
To Multiply Fractions
1. Multiply the numerators.
2. Multiply the denominators.

12. Example 5: Multiplication with Fractions
Find Solution

13. Example 6: Multiplication with Fractions

14. Finding Equivalent Fractions
To Find an Equivalent Fraction Multiply the numerator and denominator by the same nonzero whole number.

15. Example 7: Finding Equivalent Fractions
Find the missing numerator that will make the fractions equal. Solution

16. Example 8: Finding Equivalent Fractions
Find the missing numerator that will make the fractions equal. Solution

17. Completion Example 9: Finding Equivalent Fractions
Find the missing numerator that will make the fractions equal. Solution

18. 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?

19. 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.

20. Reducing Fractions to Lowest Terms
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.

21. Example 11: Reducing Fractions to Lowest Terms

22. 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.

23. Completion Example 13: Reducing Fractions to Lowest Terms
Reduce to lowest terms.
Solution
Finding a common factor could be difficult here. Prime factoring helps.

24. Changing Mixed Numbers to Improper Fractions
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.
Write this sum over the denominator of the fraction.

25. Changing Mixed Numbers to Improper Fractions
To Change a Mixed Number to an Improper Fraction (cont.)
For example:

26. Example 14: Changing a Mixed Number to an Improper Fraction
Change to an improper fraction.
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:

27. Completion Example 15: Changing a Mixed Number to an Improper Fraction
Change to an improper fraction.
Solution
Step 1: Multiply 11 ∙ 3 = ____.
Step 2: Add the numerator: _____ + _____ = ____.
Step 3: Write this sum, _______, over the denominator _______. Therefore, ______.

28. Changing Improper Fractions to Mixed Numbers
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.

29. Example 16: Changing an Improper Fraction to a Mixed Number
Change to a mixed number.
Solution Divide 67 by 5.
whole number part
numerator

30. Practice Problems
1. Find the products. a. b. c. d. 2. Reduce to lowest terms. a. b. c. 3. Change to an improper fraction.

31. Practice Problems (cont.)
4. Change to a mixed number. 5. Write as a mixed number.

32. Practice Problem Answers
1. a. b. c. d. 2. a. b. c