Copyright © Cengage Learning. All rights reserved. 3 Fractions and Mixed Numbers

Copyright © Cengage Learning. All rights reserved. S E C T I O N 3.1 An Introduction to Fractions

33 Objectives 1. Identify the numerator and denominator of a fraction. 2. Simplify special fraction forms. 3. Define equivalent fractions. 4. Build equivalent fractions. 5. Simplify fractions.

44 An Introduction to Fractions Whole numbers are used to count objects, such as CDs, stamps, eggs, and magazines. When we need to describe a part of a whole, such as one-half of a pie, three-quarters of an hour, or a one-third-pound burger, we can use fractions. One-half of a cherry pie Three-quarters of an hour One-third pound burger

55 Identify the numerator and denominator of a fraction 1

66 A fraction describes the number of equal parts of a whole. For example, consider the figure below with 5 of the 6 equal parts colored red. We say that (five-sixths) of the figure is shaded. In a fraction, the number above the fraction bar is called the numerator, and the number below is called the denominator. Fraction bar numerator denominator

77 Example 1 Identify the numerator and denominator of each fraction: a. b. Strategy: We will find the number above the fraction bar and the number below it. Solution: a. b. numerator denominator numerator denominator

88 Identify the numerator and denominator of a fraction If the numerator of a fraction is less than its denominator, the fraction is called a proper fraction. A proper fraction is less than 1. If the numerator of a fraction is greater than or equal to its denominator, the fraction is called an improper fraction. An improper fraction is greater than or equal to 1. Proper fractions Improper fractions

99 Identify the numerator and denominator of a fraction There are times when a negative fraction is needed to describe a quantity. For example, if an earthquake causes a road to sink seven-eighths of an inch, the amount of downward movement can be represented by

10 Identify the numerator and denominator of a fraction Negative fractions can be written in three ways. The negative sign can appear in the numerator, in the denominator, or in front of the fraction. Notice that the examples above agree with the rule for dividing integers with different (unlike) signs: the quotient of a negative integer and a positive integer is negative.

11 Simplify special fraction forms 2

12 Simplify special fraction forms Recall that a fraction bar indicates division. This fact helps us simplify four special fraction forms. Fractions that have the same numerator and denominator: In this case, we have a number divided by itself. The result is 1 (provided the numerator and denominator are not 0). We call each of the following fractions a form of 1.

13 Simplify special fraction forms Fractions that have a denominator of 1: In this case, we have a number divided by 1. The result is simply the numerator. Fractions that have a numerator of 0: In this case, we have division of 0. The result is 0 (provided the denominator is not 0).

14 Simplify special fraction forms Fractions that have a denominator of 0: In this case, we have division by 0. The division is undefined. is undefined

15 Example 3 Simplify, if possible: Strategy: To simplify each fraction, we will divide the numerator by the denominator, if possible. Solution: This corresponds to dividing a quantity into 12 equal parts, and then considering all 12 of them. We would get 1 whole quantity. This corresponds to dividing a quantity into 24 equal parts, and then considering 0 (none) of them. We would get 0.

16 Example 3 – Solution cont’d This corresponds to dividing a quantity into 0 equal parts, and then considering 18 of them. That is not possible. This corresponds to "dividing" a quantity into 1 equal part, and then considering 9 of them. We would get 9 of those quantities. is undefined

17 Define equivalent fractions 3

18 Define equivalent fractions Fractions can look different but still represent the same part of a whole. To illustrate this, consider the identical rectangular regions as shown below. The figure is divided into 10 equal parts. Since 6 of those parts are red, of the figure is shaded.

19 Define equivalent fractions The figure shown below is divided into 5 equal parts. Since 3 of those parts are red, of the figure is shaded. We can conclude that because and represent the same shaded portion of the figure. We say that and are equivalent fractions.

20 Build equivalent fractions 4

21 Build equivalent fractions Writing a fraction as an equivalent fraction with a larger denominator is called building the fraction. To build a fraction, we use a familiar property that is also true for fractions: We also use the following rule for multiplying fractions.

22 Build equivalent fractions To build an equivalent fraction for with a denominator of 8, we first ask, “What number times 2 equals 8?” To answer that question we divide 8 by 2 to get 4. Since we need to multiply the denominator of by 4 to obtain a denominator of 8, it follows that should be the form of 1 that is used to build an equivalent fraction for. Multiply by 1 in the form of Note the form of 1 highlighted in pink. Use the rule for multiplying two fractions. Multiply the numerators. Multiply the denominators.

23 Build equivalent fractions We have found that is equivalent to To build an equivalent fraction for with a denominator of 8, we multiplied by a factor equal to 1 in the form of Multiplying by changes its appearance but does not change its value, because we are multiplying it by 1.

24 Example 4 Write as an equivalent fraction with a denominator of 35. Strategy: We will compare the given denominator to the required denominator and ask, “What number times 5 equals 35?” Solution: To answer the question “What number times 5 equals 35?” we divide 35 by 5 to get 7. Since we need to multiply the denominator of by 7 to obtain a denominator of 35, it follows that should be the form of 1 that is used to build an equivalent fraction for.

25 Example 4 – Solution cont’d Multiply by a form of We have found that is equivalent to Multiply the numerators. Multiply the denominators.

26 Build equivalent fractions

27 Simplify fractions 5

28 Simplify fractions Every fraction can be written in infinitely many equivalent forms. For example, some equivalent forms of are: Of all of the equivalent forms in which we can write a fraction, we often need to determine the one that is in simplest form.

29 Simplify fractions To simplify a fraction, we write it in simplest form by removing a factor equal to 1. For example, to simplify, we note that the greatest factor common to the numerator and denominator is 5 and proceed as follows: Factor 10 and 15. Note the form of 1 highlighted in pink. Use the rule for multiplying fractions in reverse: write as the product of two fractions, A number divided by itself is equal to Use the multiplication property of 1: the product of any fraction and 1 is that fraction.

30 Simplify fractions We have found that the simplified form of is To simplify, we removed a factor equal to 1 in the form of The result,, is equivalent to. To streamline the simplifying process, we can replace pairs of factors common to the numerator and denominator with the equivalent fraction.

31 Example 7 Simplify each fraction: Strategy: We will factor the numerator and denominator. Then we will look for any factors common to the numerator and denominator and remove them. Solution: To prepare to simplify, factor 6 and 10. Note the form of 1 highlighted in pink.

32 Example 7 – Solution Since 3 and 5 have no common factors (other than 1), is in simplest form. cont’d Simplify by removing the common factor of 2 from the numerator and denominator. A slash / and the 1’s are used to show that is replaced by the equivalent fraction. A factor equal to 1 in the form of was removed. Multiply the remaining factors in the numerator: 1  3 = 3. Multiply the remaining factors in the denominator: 1  5 = 5.

33 Example 7 – Solution cont’d To prepare to simplify, factor 21. Simplify by removing the common factor of 7 from the numerator and denominator. Multiply the remaining factors in the denominator: 1  3 = 3.

34 Simplify fractions We can easily identify common factors of the numerator and the denominator of a fraction if we write them in prime-factored form.

35 Simplify fractions Use the following steps to simplify a fraction. Negative fractions are simplified in the same way as positive fractions. Just remember to write a negative sign – in front of each step of the solution.

36 Simplify fractions For example, to simplify we proceed as follows: