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Unit 2: Fractions and Mixed NumbersLESSON 1: INTRODUCTION TO FRACTIONSThis lesson covers the following information: Understanding fractions Finding equal fractions Simplifying and expanding fractions Identifying improper fractions and converting it to a mixed number Identifying a mixed number and converting it to an improper fractionHighlights include the following: Fractions are a part of a whole. The denominator is the number of parts the whole unit is divided into. The numerator is the number of those parts that are of interest. A proper fraction is a fraction in which the numerator is smaller than the denominator.Proper fractions represent quantities less than 1.An improper fraction is a fraction in which the numerator is larger than or equal to the denominator.Any number over itself equals 1.Any number over 1 equals the number.Fractions are called equivalent fractions if they represent the same quantity. Multiply or divide the numerator and denominator of a fraction by the same nonzero number.A fraction is in simplest form, or reduced form, when the numerator and the denominator have nocommon factors other than 1. To write a fraction in its simplest form (lowest terms), divide both the numerator anddenominator by the greatest common factor (GCF) that divides evenly into both numbers.In some circumstances, we will need to write fractions so that they have a particular denominator. Whenwe do this, we are said to expand fractions.To make an improper fraction a mixed number, divide the numerator by the denominator.To make a mixed number from an improper fraction, multiply the denominator times the whole number.Then, add the numerator to the product. Keep the denominator. 4ReflectionFractions represent parts of a whole and do not always have to be less than 1 (a whole). While proper fractionsrepresent quantities smaller than 1, improper fractions represent quantities that are 1 or larger. If we multiply ordivide both the numerator and denominator of a fraction by the same number (factor), the resulting fraction isequivalent to the original fraction. This property is used to both simplify (reduce) fractions and to expandfractions. 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.1

Unit 2: Fractions and Mixed NumbersNotes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.2

Unit 2: Fractions and Mixed NumbersCrossword PuzzleUse the clues to solve the puzzle.Across6. The number in the bottom of a fraction.Down1. A fraction in which the numerator is smaller than the denominator. Its value is less than 1.2. A fraction in which the numerator is larger than or equal to the denominator. Its value is 1 or larger.3. The bar that divides the numerator from the denominator in every fraction. The fraction bar means“divided by”4. The number in the top of a fraction.5. A number used to represent part of a whole unit. 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.3

Unit 2: Fractions and Mixed NumbersPractice Problems1. Eight children went to the baseball game. Three children had popcorn and five children had candy. Whatfraction of the students had candy?2. Becca made 12 cupcakes for her coworkers. She put frosting on seven of them. What fraction of thecupcakes did not have frosting?3. Kendall had 20 plats of wood. His wife wanted to use five of the plats to complete a craft. What fractionof plats did Kendall have left?4. Elliot was doing a research for his science class. He was watching a heard of eight deer. Three deer ranaway. What fraction of deer ran?5. Coryn received an arrangement of flowers. There were a dozen flowers and three were roses. Whatfraction of the flowers were roses?6. Bob bought bagels. Three were plain bagels, two were blueberry, and one was raisin. What fraction of thebagels were blueberry?7. The city has 100 restaurants. 50 of the restaurants serves pizza. Of these restaurants, only 30 are open onMonday nights. What fraction of restaurants that serves pizza is open on Monday?8. The local community center served lunch for school children during the summer months. There were 75children who attend the lunch. 32 children drink milk and 43 children prefer juice. What fraction ofchildren drink juice with lunch?9. An apple tree had 33 apples on the lowest branches. 12 apples fell to the ground. What fraction of applesremained on the tree?10. Alley joined a local theater group. There were 25 men and 23 women. What fraction of the group werewomen? 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.4

Unit 2: Fractions and Mixed NumbersLESSON 2: MULTIPLICATION WITH FRACTIONSThis lesson covers the following information: Multiplying fractions Using strategies to make multiplication easierHighlights include the following: The rule for multiplying fractions is if a, b, c, and d are numbers and b and d are not 0, thena c aici b d bid Multiply the numerators and multiply the denominators.When numerators and denominators have common factors when multiplying fractions, cancel thecommon factors from the numerator and denominator before multiplying.When multiplying fractions, if common factors are present between numbers in the numerator andnumbers in the denominator cancel them prior to multiplying.When multiplying more than two fractions at a time, it is possible to cancel any numerator with anydenominator. The fractions do not need to be next to each other to cancel.Since any numerator can be canceled with any denominator, any fraction that can be reduced to lowestterms can be reduced before canceling.ReflectionWhen multiplying fractions, simply multiply the numerators and then multiply the denominators. However, whennumerators and denominators have common factors, it is easier to cancel those common factors beforemultiplying.Notes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.5

Unit 2: Fractions and Mixed NumbersWord SearchFind all the words in the list. Words can be found in any LSIMPLIFIEDCOMMONFACTORSNUMERATORREDUCED 2015 ICCB and CAITi-pathways.org The GED Mark is a registered trademark of the American Council on Education.6

Unit 2: Fractions and Mixed NumbersPractice Problems1.4 4i 5 52.2 6i 3 83.2 7i 7 104.3 6i 10 85.2 4i 5 76.8 4i 9 107.2 6i 4 98.1 2i 7 89.4 2i 7 710.3 3i 9 4 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.7

Unit 2: Fractions and Mixed NumbersLESSON 3: DIVIDING FRACTIONSThis lesson covers the following information: Dividing fractions Identifying strategies for dividing fractionsHighlights include the following: Any two numbers are reciprocals of one another if their product is 1. So, 4 and ¼ are reciprocals because 1 (4) 145 5.1 Any number over 1 equals the number When you are multiplying reciprocals, each numerator cancels with the denominator of the other fraction,leaving you with 7 7 49i 1.7 7 49To find the reciprocal of a number, simply have to flip it over. So, if a and b are both real numbers otherthan 0, then the reciprocal of a b .b aTo divide by a fraction, multiply by the reciprocal. To divide two fractions, flip the second fraction over and multiply the two fractions usingcanceling and reducing when possible. This also works when dividing a whole number by a fraction (create a fractionwhole number,1flip the second fraction and multiply). When dividing a fraction by a whole number, create a fraction with thewhole number, flip it, and1multiply.ReflectionYou learned that you divide a fraction by multiplying by its reciprocal. You find the reciprocal of a fraction bysimply flipping it over.Notes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.8

Unit 2: Fractions and Mixed NumbersFallen PhrasesUnscramble the letters and solve the puzzle.Practice Problems1.1 2 32.6 9 11 113.5 11 8 124.2 3 55.1 1 2 46. 13 3 47.4 13 5 108.5 2 16 3 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.9

Unit 2: Fractions and Mixed Numbers9. You have2of a bag of candy and you want to divide it equally between 3 people. How much will each3person get?10. Angie is making homemade salsa. Each batch calls for1teaspoon of red pepper flakes. How many4batches can she make if she has 6 teaspoons of red pepper flakes? 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.10

Unit 2: Fractions and Mixed NumbersLESSON 4: ADDING FRACTIONSThis lesson covers the following information: Determining the least common denominator (LCD) Adding fractionsHighlights include the following: A basic principle of addition is that we can only add two quantities if they are alike. The least common denominator (LCD) of a list of fractions is the smallest positive number divisible by allthe denominators in the list. There are several strategies that will help in finding the LCD of any size numbers. Determine if one of the denominators is a multiple of the other. If it is, then o Convert the second fraction to the denominator of the multiple.o The LCD cannot be smaller than the largest denominator. Find the multiples of the largest denominator until you find a multiple that the other denominator(s) willdivide evenly into. Convert the fractions to the LCD by multiplying both the numerator and denominator of each by the samefactor. Another method for finding the LCD involves factoring each denominator. Write the prime factorization of each denominator. List all the factors that the two denominators have in common. Add to the list the remaining factors from each denominator. Multiply out the combined list of factors. The result is the least common denominator. If the two fractions to be added already have a common denominator, then simply add the numerators andwrite the result over the common denominator. Simplify/reduce the resulting fraction, If the fractions to be added do not have a common denominator, we must find the LCD, then convert eachfraction to its equivalent fraction that has the common denominator. Add the fractions (with the newcommon denominator), and simplify the results.ReflectionIn this lesson, you learned that, in order to add two or more fractions, they must first have a commondenominator. Therefore, if the fractions being added do not have a common denominator, then you must find theleast common denominator (LCD), write the fractions as equivalent fractions with a denominator of the LCD, addthe numerators and place the sum over the LCD, and simplify if possible.Notes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.11

Unit 2: Fractions and Mixed NumbersLetter TilesUnscramble the tiles to reveal a hidden message.Practice Problems1. Find the LCD of75and 126 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.12

Unit 2: Fractions and Mixed Numbers2. Find the LCD of15and 693. Find the LCD of52and 734. Find the LCD of21and 765. Find the LCD of79and 4116. Find the LCD of53and 1287. Sally wants to start walking every night. The first night she walkswalks3of a mile. How far did Sally walk?83of a mile. The second night, she48. A group of community members want to create a garden and picnic area in a square grass lot. If they wantto use13of the land for the picnic area andof the area for the garden, how much space are they using?489.25of Rosie’s patrons tipped at the end of their meal.of Shelly’s patrons tipped her. Which waitress36received the most tips?10. Richard walked12of a mile to work every morning. Then, he walkedof a mile to the library. If he25walked round trip each day, how far did Richard walk? 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.13

Unit 2: Fractions and Mixed NumbersLESSON 5: SUBTRACTION WITH FRACTIONSThis lesson covers the following information: Subtracting fractionsHighlights include the following: If the two fractions to be subtracted already have a common denominator, then simply subtract thenumerators and write the result over the common denominator. Simplify the resulting fraction, if possible. When the fractions to be subtracted do not have a common denominator, then just as when addingfractions, you must first find the LCD. Convert each fraction to its equivalent fraction that has the common denominator. Subtract thenumerators of the like fractions, and simplify the result. A common error when subtracting fractions is to write the difference of the numerators over thedifference of the denominators. This will not give correct results.ReflectionAs with addition of fractions, in order to subtract two fractions, they must first have a common denominator.Therefore, if the fractions being subtracted do not have a common denominator, then you must find the leastcommon denominator (LCD), then write the fractions as equivalent fractions with a denominator of the LCD,subtract the numerators and place the difference over the LCD, and simplify if possible.Notes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.14

Unit 2: Fractions and Mixed NumbersCryptogramSolve the puzzle to view a hidden message.Practice Problems1.11 7 15 102.8 5 9 123.15 3 16 44.2 3 3 10 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.15

Unit 2: Fractions and Mixed Numbers5.4 2 5 36.5 1 6 47.2 1 3 68. There was25inch of rain last month. This month, there wasinches of rain. How much more rain was36there in the second month?9. Kris was studying his cell phone bill and he spent12of his minutes talking to his mother, andof the153minutes talking to his girlfriend. How much time did he spend talking to his best friend?10. Three children were sharing a box of cookies. The first student ateof the cookies. How much was left for the third child? 2015 ICCB and CAIT 12the box. The second student ate25i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.16

Unit 2: Fractions and Mixed NumbersLESSON 6: MIXED NUMBERSThis lesson covers the following information: Using mixed numbers to represent figures Converting mixed numbers to improper fractions Convert improper fractions to mixed numbersHighlights include the following: A mixed number consists of a whole number and a fraction part. An improper fraction is a fraction in which the numerator is larger than or equal to the denominator andrepresents a quantity greater than or equal to 1. Mixed numbers can always be converted into improper fractions, and improper fractions can always beconverted into mixed numbers. To convert a mixed number to an improper fraction, multiply the whole number by the denominator andadd the numerator To convert an improper fraction into a mixed number, divide. The quotient will be the whole number of the mixed number and the remainder over the denominator ofthe original fraction will provide the fraction part of the mixed number.ReflectionTo count whole items or units, we use whole numbers. To describe part of a whole unit, use fractions. In manyapplications, however, a quantity may be expressed as a combination of the two. When we combine a wholenumber with a fraction, we obtain a mixed number.Notes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.17

Unit 2: Fractions and Mixed NumbersWord SearchFind all the words in the list. Words can be found in any direction.DENOMINATORFRACTION BARLARGER THANSIMPLIFYEQUAL TOIMPROPERMIXED NUMBERWHOLE UNIT 2015 ICCB and CAITi-pathways.org The GED Mark is a registered trademark of the American Council on Education.18

Unit 2: Fractions and Mixed NumbersPractice Problems1. Convert to a mixed number:6192. Convert to a mixed number:7383. Convert to a mixed number:76104. Convert to a mixed number:5055. Convert to a mixed number:4756. Convert to an improper fraction: 92137. Convert to an improper fraction: 39108. Convert to an improper fraction: 12359. Convert to an improper fraction: 841110. Convert to an improper fraction: 6715 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.19

Unit 2: Fractions and Mixed NumbersLESSON 7: MULTIPLYING AND DIVIDING MIXED NUMBERSThis lesson covers the following information: Multiplying and dividing mixed numbersHighlights include the following: Multiply mixed numbers in the same way that we multiply fractions except we first have to convert themixed numbers to improper fractions. Express the final answer as a mixed number when possible. Convert each mixed number to an improper fraction. Cancel numerators with denominators. Multiply all numerators. Multiply all denominators. Reduce/simplify if possible. Divide fractions by multiplying the fraction in front of the division symbol by the reciprocal of thefraction (flipped over) behind the division symbol. To divide mixed numbers, first convert each mixed number into an improper fraction, then applythe property for dividing fractions.ReflectionYou have seen that multiplying and dividing mixed numbers is very similar to multiplying and dividing fractions.The only difference is that the mixed numbers must first be converted into improper fractions before theproperties for multiplying and dividing fractions can be applied.Notes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.20

Unit 2: Fractions and Mixed NumbersLetter TilesEach block of letters is considered a tile. Unscramble the tiles to solve the phrase.Practice Problems1. Theresa has started her job at Hoyle farms. She was told to feed the cattle 4She needs to feed the horses 1horses fed?2times more hay than the cattle receive. How many bails of hay are the3 2015 ICCB and CAIT 1bales of hay every day.2i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.21

Unit 2: Fractions and Mixed Numbers2. How many bales of hay with Theresa need for 51days?23. Last year, Suzanne was in charge of a salad lunch at her church. She ordered 3year, she needed 11batches of rolls. This32times the original order. How many batches of rolls did order?34. There are two new trees in the yard. Ken wants to plant them about 12tall. The walnut tree is 5maple tree?15feet tall and the maple tree is 2 times as tall as the oak tree. How tall is the385. Jon and Carly started training for a race. Jon ran 2miles as Jon. How many miles did Carly run?11miles in 6 minutes. Carly ran 1 time as many236.144 2 257.164 3 278.312 5 429.813 3 9210. 571 1 10 2 2015 ICCB and CAIT 21feet apart. The oak tree is 333i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.22

Unit 2: Fractions and Mixed NumbersLESSON 8: ADDING AND SUBTRACTING MIXED NUMBERSThis lesson covers the following information: Add mixed numbers Subtract mixed numbersHighlights include the following: When adding mixed numbers, it is easier to write the problem vertically rather than horizontally. Weconvert the fraction parts to common denominators. First add the fraction parts and then the whole-number parts. In some cases, when we add the fraction parts of two mixed numbers, the result is an improper fraction. When this occurs, we convert this improper-fraction part of the sum into a mixed number and carry thewhole-number part of the result to be combined with the whole number parts of the original mixednumbers. To subtract two mixed numbers, first subtract the fraction parts and then subtract whole parts together. As with addition of mixed numbers, we subtract mixed numbers by subtracting the similar parts.In other words, when subtracting mixed numbers, we first subtract the fraction parts and then thewhole-number parts. We must subtract the fraction parts first in case it is necessary to borrow.ReflectionAs with multiplying and dividing, it is possible to add two or more mixed numbers by first converting each mixednumber into an improper fraction. We can then proceed to follow the rules for adding fractions. However, theequivalent improper fractions can often involve very large numerators that are difficult to handle. Therefore, it isusually easier to add the similar parts. In other words, when adding and subtracting mixed numbers, it is usuallyeasier to add the fraction parts together and the whole-number parts. You may have the following questions:Notes: 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.23

Unit 2: Fractions and Mixed NumbersCryptogramSolve the puzzle to reveal the hidden phrase.Practice Problems93 2 10101.42.317 5 443. 124.21 5 33154 9 66 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.24

Unit 2: Fractions and Mixed Numbers5.212 4 556.734 2 887.113 2 448. 139.97 7 10102 12 1 3 310. 77 1 10 10 2015 ICCB and CAIT i-pathways.orgThe GED Mark is a registered trademark of the American Council on Education.25

Proper fractions represent quantities less than 1. An improper fraction is a fraction in which the numerator is larger than or equal to the denominator. Any number over itself equals 1. Any number over 1 equals the number. Fractions are called equivalent