fractions

Identifying Fractions So far in your course, you’ve been working with whole numbers. As you learned in the previous lesson, a whole number is any counting number, starting with the number 0. Whole numbers can’t express all the amounts we need to express, however. Sometimes, fractions, or parts of numbers, are needed. The word fraction means “broken.” So, a fraction is a part of a whole thing that’s been broken into pieces. For example, if you break a 16-ounce chocolate bar into two equal parts, each part is a fraction of the whole bar. (See the following figure.) Each broken section is one-half (½) of the candy. If you continue breaking the chocolate bar apart, the pieces, or fractions, will get smaller and smaller. No matter how many pieces you break, however, you can never have more than 16 ounces of chocolate altogether. Dividing a Chocolate Bar Fractions have three common uses in everyday mathematics. Use 1: Fractions can stand for part of one whole thing. Use 2: Fractions can stand for part of a group. Use 3: Fractions can show division. Example of Use 1: At a restaurant, a waiter divides a cheesecake into eight equal pieces. By the end of the evening, five of the pieces have been sold. What part of the cheesecake was sold? Answer: 5⁄8 of the cheesecake was sold. 5⁄8 stands for the part of one whole cheesecake that was sold. Cheesecake Example Example of Use 2: Tom has five children—four boys and one girl. What fraction of Tom’s children are boys? Answer: 4⁄5 of Tom’s children are boys. 4⁄5 stands for the part of the group that are boys. Children Example Example of Use 3: Write the fraction as a division problem. Each of these three uses of fractions will be covered in detail in this lesson. Reading and Writing Fractions A fraction is always written as two numerals separated by a dividing line. The number above the line is called the numerator. The number below the line is called the denominator. Name the numerator and the denominator in each fraction. Fraction Numerator Denominator You’ll sometimes see fractions written in different ways. All of these ways are correct. It doesn’t matter which way you write your fractions—that’s just a matter of preference. As long as you write the numerator, then a line, then the denominator, you’ll always be correct. In this study lesson, you’ll see fractions written in all of these ways. As you learned in the previous study lesson, whole numbers can be represented by both word names and numerals. Fractions can also be given in words. Sometimes, you may be trying to read a fraction with very large numbers in it, such as 95⁄167. Instead of reading “ninety-five one hundred sixty-sevenths,” it’s usually easier and clearer to read “ninety-five over one hundred sixty-seven.” The following chart shows you the word names for some of the most often used fractions. one half one eighth one sixteenth one third one ninth one seventeenth two thirds one tenth one eighteenth one fourth one eleventh one nineteenth three fourths one twelfth one twentieth one fifth one thirteenth one thirtieth one sixth one fourteenth one hundredth one seventh one fifteenth one thousandth Understanding the Uses of Fractions We’ve already mentioned the three uses of fractions—to show part of a whole, to show part of a group, and to show division. When you’re using fractions to show part of a whole thing, the denominator of the fraction will tell you how many pieces the whole quantity was broken into. The numerator of the fraction will tell you how many pieces of the whole quantity you have. For example, if you have 3/8 of a pizza, the denominator (8) tells you that the pizza was cut into eight pieces, and the numerator (3) tells you that you have three of those eight pieces of pizza. Pizza ExampleFractions are often used to represent part of a whole thing. The numerator of a fraction can be any number, including 0. If you have 0⁄8 of a pizza, you have none of the eight pieces the pizza was cut into. Any fraction with zero in the numerator simply equals zero. If the numerator of the fraction is the same as the denominator, then the fraction equals 1. If you have 8⁄8 of the pizza, you have all eight of the eight pieces, or 1 whole pizza. There are four quarters in a dollar. You have three quarters. What fraction of a dollar do you have? Solution:  One dollar is broken up into four quarters, so the denominator of the fraction is 4. You have three quarters, so the numerator of the fraction is 3. Answer: Three quarters equals of a dollar. What fraction of the following shape is shaded? A rectangle is divided into twelve equal-sized rectangles of which seven are shaded. Solution: The shape is divided into twelve parts. The denominator of the fraction is 12. The figure has seven parts shaded, so the numerator of the fraction is 7. Answer: of the shape is shaded. When you’re using fractions to show part of a group, you can read the line between the numerator and denominator of a fraction as “out of.” For example, 1⁄10 of the people in the United States don’t speak English as their main language. The fraction 1⁄10 refers to the part of the group that doesn’t speak English. This means that “one out of ten” people don’t speak English as their main language. Look at the following example problems. What fraction of this group of symbols are suns? Illustration of weather symbols showing four sun symbols, one snowflake, one moon, and one cloud with thunder. Solution: There are seven symbols altogether. 7 is the denominator of the fraction. 4 of the symbols are suns, so 4 is the numerator. Answer: of the symbols are suns. Julia planted tulips and daffodils in her garden. In the spring, 50 plants bloomed. Twenty-one of these plants were tulips, and 29 plants were daffodils. What fraction of the plants were tulips? What fraction of the plants were daffodils? Solution: The total number of plants was 50. 50 will be the denominator of the two fractions. Twenty-one out of the 50 plants were tulips. The numerator of the first fraction is 21. Twenty-nine out of the fifty plants were daffodils. The numerator of the second fraction is 29. Answer: of the plants were tulips, and were daffodils. The third common use of fractions is to show division. This makes sense, since when you break or cut something into pieces, you’re dividing it. In this situation, the line between the numerator and denominator of a fraction can be read “divided by.” For example, look at the fraction 4⁄4. If you have 4⁄4 of a cake, you have all of the pieces the cake was divided into, or one whole cake. By reading this fraction as “four divided by four,” you also get 1, since 4 ÷ 4 = 1. Remember from your previous studies that it’s impossible to divide by zero, so it’s impossible to use zero in the denominator of a fraction. The denominator tells you how many pieces the whole was broken into, and it doesn’t make sense to speak of something broken up into zero pieces. Write the following division problems in four ways. Way #1 Way #2 Way #3 Way #4 2 divided by 3 2 ÷ 3 50 divided by 10 50 ÷ 10 16 divided by 23 16 ÷ 23 Comparing Fractions Now that you understand how to use fractions, let’s take a closer look at how to compare them. Suppose you have 1/2 of a cake. It’s fairly easy to picture the amount of cake that 1/2 stands for. But given two fractions of cake, 2/5 and 3/8, which would you expect to be larger? In the previous study lesson, you learned about the number line. The number line can contain fractions as well as whole numbers. Look at the following example. Number Line with Fractions This illustration shows you where certain fractions will fall on the number line. As you move to the left on the number line, the fractions get smaller, the same as whole numbers do. As you move to the right, the fractions get larger. The number line can help you picture the values of simple fractions, like ½, ⅓, and ¼. However, when you’re trying to compare more difficult fractions, such as 17⁄23 and 49⁄92, keep the following rules in mind: Rule 1: If the denominators of two fractions are the same, the numerators will tell you which fraction is larger. The larger the numerator, the larger the fraction. Rule 2: If the numerators of two fractions are the same, the denominators will tell you which fraction is larger. The larger the denominator, the smaller the fraction. Which fraction is larger, or ? Solution: This problem is an illustration of Rule 1. The denominators in these two fractions are the same, so the numerators will tell you which fraction is larger. Since 15 is greater than 11, is larger than . A rectangle is made up of sixteen equal-sized rectangles of which eleven are shaded. A rectangle is made up of sixteen equal-sized rectangles of which fifteen are shaded. is shaded is shaded These drawings show this concept clearly. Which fraction is larger, or ? Solution: This problem demonstrates Rule 2. The numerators in these two fractions are the same, so the denominators will tell you which fraction is larger. Since 5 is smaller than 9, is greater than . A rectangle is divided into five smaller rectangles of which two are shaded. A rectangle is divided into nine smaller rectangles of which two are shaded. is shaded is shaded These illustrations confirm your answer. Proper and Improper Fractions If the numerator of a fraction is less than its denominator, then the fraction is less than 1. Any fraction that has a numerator that’s smaller than the denominator is called a proper fraction. Any fraction that has a numerator equal to or greater than its denominator is called an improper fraction. If the numerator of a fraction equals its denominator, the fraction equals 1. If the numerator is greater than the denominator, the fraction represents an amount greater than 1. Proper and improper fractions are illustrated in the following drawings. One bar split into three equal parts of which two are shaded. One bar split into three equal parts which are all shaded. One bar split into three equal parts of which two are shaded. One bar split into three equal parts of which all are shaded. is shaded is shaded is shaded proper fraction improper fraction improper fraction Tell whether each fraction is proper or improper, and whether it’s less than, equal to, or greater than 1. Fraction Type of Fraction Comparison with 1 Proper Improper Improper Proper Mixed Numbers Defining Mixed Numbers Previously, you learned that an improper fraction is a fraction equal to or greater than 1. Very often, improper fractions are written as mixed numbers. A mixed number is a whole number plus a proper fraction, written side by side. Let’s take a closer look at what this means by reviewing the following illustration. Representative of Eight Over Six This drawing represents the improper fraction 8⁄6. In the drawing, all six parts of the first shape are shaded, plus two parts of the second shape. In other words, one whole shape is shaded, plus 2⁄6 of a second shape. Thus, the improper fraction 8⁄6 can also be written as 1 plus 2⁄6, or, in a shorter form, 12⁄6. The number 12⁄6 is called a mixed number because it contains a “mix” of the whole number 1 and the proper fraction 2⁄6. The mixed number 12⁄6 is read as “one and two sixths.” You can also think of a mixed number as a sum. Thus, 12⁄6 is the sum of 1 plus 2⁄6. Write each sum as a mixed number, and name the whole number part and the proper fraction part. Sum Mixed Number Whole Number Proper Fraction 3 1 4 Doug drank two glasses of water. He then poured himself ½ of another glass and drank that too. How much water did Doug drink? Write your answer as a mixed number. Solution: Make a drawing to illustrate the problem. Answer: Doug drank 2 + glasses, or glasses of water. Changing Improper Fractions to Mixed Numbers Whenever you need to change an improper fraction into a mixed number, you can “draw” the solution by shading areas as you did in the previous examples, or you can treat the improper fraction just like a division problem. Remember, the line between the numerator and denominator of a fraction can be read “divided by.” To change an improper fraction to a mixed number, follow these steps. Step 1: Divide the denominator into the numerator of the fraction. The quotient is the whole number part of your answer. Step 2: The remainder is the numerator of the fraction part of your answer. Step 3: The divisor is the denominator of the fraction part of your answer. The following example illustrates this process. Change the improper fraction to a mixed number. Solution: Set up the fraction as a division problem. The fraction can be read “64 divided by 7.” 64 ÷ 7 = 9r1 Divide the denominator into the numerator. The quotient (9) is the whole number part of your answer. The remainder (1) is the numerator of the fraction part of your answer. The divisor (7) is the denominator of the fraction part. Answer: Change the improper fraction to a mixed number. Solution: Set up the fraction as a division problem. The fraction can be read as “15 divided by 2.” 15 ÷ 2 = 7r1 Divide 2 into 15. The quotient (7) is the whole number part of your answer. The remainder (1) is the numerator of the fraction part of your answer, and the divisor (2) is the denominator. Answer: Changing Mixed Numbers to Improper Fractions You’ve just learned how to change an improper fraction into a mixed number by dividing. Now, you’ll learn how to do just the opposite—change a mixed number to an improper fraction by multiplying. This makes sense, since multiplication is the inverse, or opposite, operation of division. To change a mixed number to an improper fraction, follow these three steps. Step 1: Multiply the whole number part of the mixed number by the denominator of the fraction part. Step 2: Add the product to the numerator of the fraction part. Step 3: Write the sum obtained in Step 2 over the denominator of the fraction part. This procedure may seem a little complicated at first, but the following two examples should help make things clear. Change to an improper fraction. Solution: Look at the mixed number as a multiplication problem. Multiply the denominator of the fraction part (7) times the whole number (5). 7 × 5 = 35 35 + 4 = 39 Add the numerator of the fraction part (4). Place the sum (39) over the denominator of the fraction part (7). Answer: Change to an improper fraction. Solution: Follow the previous steps to solve the problem. Multiply the denominator (5) times the whole number (14). 5 × 14 = 70 70 + 3 = 73 Add the numerator (3). Write the sum (73) over the denominator of the fraction part (5). Answer:

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