# Fractions Packet. Contents

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1 Fractions Packet Contents Intro to Fractions.. page Reducing Fractions.. page Ordering Fractions page Multiplication and Division of Fractions page Addition and Subtraction of Fractions.. page Answer Keys.. page Note to the Student: This packet is a supplement to your textbook Created by 00 page of

2 Intro to Fractions Reading Fractions Fractions are parts. We use them to write and work with amounts that are less than a whole number (one) but more than zero. The form of a fraction is one number over another separated by a fraction (divide) line. i.e. and These are fractions. Each of the two numbers tells certain information about the fraction (partial number). The bottom number (denominator) tells how many parts the whole (one) was divided into. The top number (numerator) tells how many of the parts to count. says Count one of two equal ports. says Count three of four equal parts. says Count five of nine equal parts. Fractions can be used to stand for information about wholes and their parts: EX. A class of 0 students had people absent one day. absentees are part of a whole class of 0 people. 0 represents the fraction of people absent. EX. A Goodbar candy breaks up into small sections. If someone ate of those sections that person ate of the Goodbar. Created by 00 page of

3 Exercise Write fractions that tell the following information: (answers on page ). Count two of five equal parts. Count one of four equal parts. Count eleven of twelve equal parts. Count three of five equal parts. Count twenty of fifty equal parts. It s miles to Gramma s. We have already driven miles. What fraction of the way have we driven?. A pizza was cut into twelve slices. Seven were eaten. What fraction of the pizza was eaten?. There are students in a class. have passed the fractions test. What fraction of the students have passed fractions? The Fraction Form of One Because fractions show how many parts the whole has been divided into and how many of the parts to count the form also hints at the number of parts needed to make up the whole thing. If the bottom number (denominator) is five we need parts to make a whole:. If the denominator is we need parts to make a whole of parts:. Any fraction whose top and bottom numbers are the same is equal to. 00 Example: 00 Created by 00 page of

4 Complementary Fractions Fractions tell us how many parts are in a whole and how many parts to count. The form also tells us how many parts have not been counted (the complement). The complement completes the whole and gives opposite information that can be very useful. says Count of equal parts. That means of the was not counted and is somehow different from the original. implies another (its complement). Together and make the whole thing. says Count of equal parts. That means of the parts have not been counted which implies another the complement. Together and make which is equal to one. Complementary Situations It s miles to town We have driven miles. That s of the way but we still have miles to go to get there or of the way. + = = ( is all the way to town). A pizza was cut into pieces. were eaten. That means there are slices left or of the pizza. + = = (the whole pizza). Mary had 0 dollars. She spent dollars on gas dollar on parking and dollars on lunch. In fraction form how much money does she have left? Gas = parking = lunch = = ; is the complement (the leftover money) Altogether it totals or all of the money. 0 Created by 00 page of

5 Exercise (answers on page ) Write the complements to answer the following questions:. A cake had slices. were eaten. What fraction of the cake was left?. There are 0 people in our class. are women. What part of the class are men?. It is miles to grandma s house. We have driven miles already. What fraction of the way do we have left to go?. There are cookies in the jar. 0 are Oreos. What fraction of the cookies are not Oreos? Reducing Fractions If I had 0 dollars and spent 0 dollars on a CD it s easy to see I ve spent half 0 of my money. It must be that. Whenever the number of the part (top) 0 and the number of the whole (bottom) have the same relationship between them that a pair of smaller numbers have you should always give the smaller pair answer. is half of. is half of 0. is the reduced form of and 0 0 and and many other fractions. 0 A fraction should be reduced any time both the top and bottom number can be divided by the same smaller number. This way you can be sure the fraction is as simple as it can be. both and 0 can be divided by 0 0 describes the same number relationship that 0 did but with smaller numbers. is the reduced form of 0. 0 both and can be divided by. Created by 00 page of

6 is the reduced form of. When you divide both the top and bottom numbers of a fraction by the same number you are dividing by a form of one so the value of the fraction doesn t change only the size of the numbers used to express it. These numbers are smaller but they can go lower because both and can be divided by again. or Exercise (answers on page ) Try these. Keep dividing until you can t divide anymore.. =. =. =. =. =. = 0 Good knowledge of times tables will help you see the dividers you need to reduce fractions. Here are some hints you can use that will help too. Hint If the top and bottom numbers are both even use. Hint If the sum of the digits is divisible by then use. looks impossible but note that (++) adds up to three and (++) adds up to. Both and divide by and so will both these numbers: The new fraction doesn t look too simple but it is smaller than when we first started. Created by 00 page of

7 Hint If the numbers of the fraction end in 0 and/or you can divide by. 0 0 Hint If both numbers end in zeros you can cancel the zeros in pairs one from the 0 top and one from the bottom. This is the same as dividing them by for each 0 cancelled pair Hint If you have tried to cut the fraction by and gotten nowhere you should try to see if the top number divides into the bottom one evenly. For none of the other hints help here but =. This means you can reduce by. For more help on reducing fractions see page Exercise (answers on page ) Directions: Reduce these fractions to lowest terms Created by 00 page of

8 Higher Equivalents There are good reasons for knowing how to build fractions up to a larger form. It is exactly the opposite of what we do in reducing. If reducing is done by division it makes sense that building up should be done by multiplication. A fraction can be built up to an equivalent form by multiplying by any form of one any number over itself. 0 All are forms of ; all will reduce to Comparing Fractions Sometimes it is necessary to compare the size of fractions to see which is larger or smaller or if the two are equal. Sometimes several fractions must be placed in order of size. Unless fractions have the same bottom number (denominator) and thus parts of the same size you can t know for certain which is larger or if they are equal. Which is larger or? Who knows? A ruler might help but rulers aren t usually graduated in thirds or sixths. Did you notice that if were doubled it would be? Created by 00 page of

9 So build up by ; Then it s easy to see that is larger because it counts more sixth parts than so < means Which is larger or? Build up by.. so Exercise (answers on page ) Use < > or = to compare these fractions Created by 00 page of

10 Mixed Numbers A mixed number is one that is part whole number and part fraction. are samples of mixed numbers. Mixed numbers have to be written as fractions only if you re going to multiply or divide them or use them as multipliers or divisors in fraction problems. This change of form is easy to do. Think about. That s whole things and half another. Each of the wholes has halves ( ). The number is ++ or. That s and with the original there s a total of. You don t have to think of every one this way; just figure the whole number times the denominator (bottom) and add the numerator (top). 0 Exercise (answers on page ) Change these mixed numbers to top heavy fractions: These top heavy forms are work forms but they are not usually acceptable answers. If the answer to a calculation comes out a top heavy fraction it will have to be changed to a mixed number. This can be done by reversing the times and plus to divide and minus. became by. can go back to by dividing and. Created by 00 page 0 of

11 The answer is the whole number. The remainder is the top number of the fraction and the divider is the denominator (bottom fraction number). Exercise (answers on page ) Reduce these top heavy fractions to mixed numbers Top heavy fractions may contain common factors as well. They will need to be divided out either before or after the top heavy fraction is changed to a mixed number. but can be divided by. Then = If you had noticed that both and are even you could divide out right away and then go for the mixed number. Either way the mixed number is the same. Exercise (answers on page ) 0 0. =. =. =. =. = 0 Created by 00 page of

12 Estimating Fractions One of the most important uses of estimation in mathematics is in the calculation of problems involving fractions. People find it easier to detect significant errors when working with whole numbers. However the extra steps involved in the calculation with fractions and mixed numbers often distract our attention from an error that we should have detected. Students should ask the following questions as motivation for estimating: ) Would estimates help in the calculation? ) Is the answer I get reasonable? ) Does the answer seem realistic? Try to make every fraction you work with into a whole number. 0 and should be your targets with fractions. Mixed numbers should be estimated to the nearest whole number (except Ex.). Here are some examples of problems using estimation: Ex. note: ⅔ is closer to (than 0) and ½ should be considered This symbol means approximately equal to Ex. 0 note: ⅓ is closer to 0 (than ) Ex. - note: ⅓ is closer to (than ) and ½ should be considered closer to (than ) Ex. note: ⅔ is closer to (than ) Ex. Ex. see Ex. above see Ex. above Ex. see Ex. above Ex. note: ⅓ is made into a because it is easier to divide by Exercise Estimate the answers to the following fractions operations (answers on page ) ) ) ) ) ) ) Basic College Mathematics th Ed. Tobey & Slater p. Created by 00 page of

13 Reducing Fractions Divide by if The top AND bottom numbers are EVEN numbers Like: Divide by if The sum of the top numbers can be divided by AND the sum of the bottom numbers can be divided by Like: can be divided evenly by can be divided evenly by Divide by if The top AND bottom numbers end in 0 or Like: 0 0 Divide by 0 if The top AND bottom numbers end in 0. Like: Divide by if The top AND bottom numbers end in or 0 or or 00 Like: Created by 00 page of

14 Divisibility RULES! Dividing by Add up the digits: if the sum is divisible by three then the number divides by three. Ex therefore divides by 0 Dividing by Look at the last two digits. If they are divisible by four then the number divides by four. Ex. therefore divides by Dividing by If the digits can be divided by two and three then the number divides by six Ex And therefore divides by Dividing by Take the last digit double it and then subtract it from the other numbers. If the answer is divisible by seven then the number divides by seven. Ex therefore divides by Dividing by If the last three digits are divisible by eight then the number divides by eight. Ex therefore divides by 0 Created by 00 page of

15 Dividing by If the number divides by both and then the number will divide by Ex. 0 0 And therefore divides by 0 0 Dividing by Delete the last digit. Subtract nine times the deleted digit from the remaining number. If what is left is divisible by thirteen then the number divides by thirteen. Ex. Forget it! This is too much work! Remember to try to reduce with any number that makes the reduction simple and easy for you. Good Luck! Created by 00 page of

16 ORDERING Fractions Being able to place numbers in order (smallest to largest or largest to smallest) is fundamental to the understanding of mathematics. In these exercises we will learn how to order fractions. Ordering Fractions There are several ways to order fractions. One way is to use common sense. This method can be simple but requires some general knowledge. If nothing else it can be used as a starting point to finding the necessary order. Take a look at the following examples: Ex. Place the following fractions from smallest to largest order The larger the number on the bottom of a fraction (assuming the numerator is the same for all the fractions) the smaller the fraction is. In the above example is the smallest fraction because the is the largest denominator. Next in order would be the because the is the next largest denominator. This leaves the which has the smallest denominator. Therefore the order for these fractions is: Ex. Place the following fractions from smallest to largest The larger bottom number here is the in. But the student should ask Is this the smallest fraction? By inspection it does not seem to be. But with fractions of this sort (different numerators) students run into the most problems when ordering. Created by 00 page of

17 Another way to order fractions is to find common denominators for all the fractions; build up the fractions; then compare the top numbers (numerators) of all the fractions. Look at the following example: Ex. Order the following fractions from smallest to largest The fractions will be rewritten with common denominators. This process is called building. Once the denominators change then the numerators will change by the same amount By looking at the top numbers the order of these fractions is: Exercise A (answers on page ) Order these fractions from SMALLEST to largest... Exercise B (answers on page ) Order these fractions from LARGEST to smallest... Created by 00 page of

18 Multiplication and Division of Fractions Worksheets When multiplying fractions simply multiply the numerators (top number of the fractions) together and multiply the denominators (bottom number of the fractions) together. It is good practice to check to see if any of the numbers can cancel. Canceling is done when the numerator and denominator can be divided evenly by the same number. Note: canceling can happen top-to-bottom and/or diagonally but never across. Ex. : fraction by to get the canceled answer this product can be canceled. Divide the numbers in the The fractions in Ex. can cancel before they are multiplied. Ex. : The s cancel by dividing by. Cross them out and place s close by. Now multiply the top numbers together then the bottom numbers. The product is the final answer. Ex. : can be rewritten as 0 Cancel by dividing by. Then cancel by dividing by 00. Multiply and get the product. Ex. : can be written like. Cancel by dividing by. Finally multiply to find the product Created by 00 page of

19 Exercise (answers on page 0) Multiply these fractions. Cancel and simplify if possible Created by 00 page of

20 Multiplying Mixed Numbers Change mixed numbers into improper fractions then multiply as before. 0 Ex. : Change the mixed numbers to improper fractions by: ) multiplying the bottom number by the whole number ) add the top number ) keep the bottom number. Cancel top and bottom. Multiply. Improper fractions simplify by dividing. Ex.: Change the mixed number into an improper fraction. Change the whole number into an improper fraction. Cancel. Multiply. Simplify to get the quotient. Exercise (answers on page 0) Multiply these fractions. Cancel and simplify if necessary Created by 00 page 0 of

21 Dividing Fractions When dividing fractions invert (turn over) the fraction to the right of the ( divide by ) symbol. Cancel (if possible) then multiply. Ex. : Ex. : Exercise (answers on page 0) Divide these fractions. Cancel if necessary and simplify Created by 00 page of

22 Dividing Mixed Number Fractions When dividing mixed numbers change the mixed numbers to improper fractions invert the fraction on the right of the multiply then simplify. Ex. : symbol cancel if possible Ex. : Exercise (answers on page 0) Divide the following mixed numbers. Cancel and simplify when possible Created by 00 page of

23 Fraction Word Problems (Multiplication/Division) When solving word problems make sure to UNDERSTAND THE QUESTION. Look for bits of information that will help get to the answer. Keep in mind that some sentences may not have key words or key words might even be misleading. USE COMMON SENSE when thinking about how to solve word problems. The first thing you think of might be the best way to solve the problem. Here are some KEY WORDS to look for in word problems: Product times: mean to multiply Quotient per for each average: mean to divide Ex. : If boxes of candy weigh ½ pounds find the weight per box. per means to divide pounds Ex. : If one by is actually ½ inches wide find the width of twelve by s. inches inches twelve by s here means times as wide as one by ½ inches Created by 00 page of

24 Exercise (answers on page 0) Solve the following fraction word problems. Cancel and simplify your answers.. A stack of boards is inches high. Each board is ¾ inches thick. How many boards are there?. A satellite makes revolutions of the earth in one day. How many revolutions would it make in ½ days?. A bolt has ½ turns per inch. How many turns would be in ½ inches of threads?. If a bookshelf is hold? inches long how many inch thick books will it. Deborah needs to make costumes for the school play. Each costume requires yards of material. How many yards of material will she need? Created by 00 page of

25 . The Coffee Pub has cans of coffee that weigh pounds each. The Pub has ½ cans of coffee left. What is the total weight of ½ cans?. Belinda baked pies that weigh pie weigh? 0 pounds total. How much does each. A piece of paper is 000 inches thick. How many sheets of paper will it take to make a stack inch high?. Tanya has read of a book which is 0 pages. How many pages are in the entire book? 0. DJ Gabe is going to serve ⅓ of a whole pizza to each guest at his party. If he expects guests how many pizza s will he need? Created by 00 page of

26 To the student: The fractions chapter is split into two parts. The first part introduces what fractions are and shows how to multiply and divide them. The second part shows how to add and subtract. The methods for accomplishing these operations can be confusing if studied all at once. Before proceeding with this packet please talk to your instructor about what you should do next. The Editors. Addition and Subtraction of Fractions Finding the LEAST COMMON DENOMINATOR (LCD) When adding and subtracting fractions there must be a common denominator so that the fractions can be added or subtracted. Common denominators are the same number on the bottom of fractions. There are several methods for finding the common denominator. The following is one in which we will find the least common denominator or LCD. Each set of fractions has many common denominators; we will find the smallest number that one or both fractions will change to. Ex. Suppose we are going to add these fractions: Step : Start with the largest of the denominators Ex: is the largest Step : See if the other denominator can divide into the largest without getting a remainder. If there is no remainder then you have found the LCD! Ex. divided by has a remainder of Step : If there is a remainder multiply the largest denominator by the number and repeat step above. If there is no remainder then you have found the LCD! If there is a remainder keep multiplying the denominator by successive numbers ( etc.) until there is no remainder. This process may take several steps but it will eventually get to the LCD. Ex. x = ; divides evenly into ; therefore is the LCD. Ex. : Step : is the largest denominator Step : divided by has no remainder therefore is the LCD! Created by 00 page of

27 Ex. : Step : is the largest denominator Step : divided by has a remainder. Multiply x =. divided by has a remainder x =. divided by has a remainder x = divided by has a remainder x = 0 0 divided by has NO remainder therefore 0 is the LCD! Note: You may have noticed that multiplying the denominators together also gets the LCD. This method will always get a common denominator but it may not get a lowest common denominator. Exercise (answers on page ) Using the previously shown method write just the LCD for the following sets of fractions (Do Not Solve) ) ) ) ) ) ) ) ) ) 0 0) ) ) ) ) ) Created by 00 page of

28 Getting equivalent Fractions and Reducing Fractions Once we have found the LCD for a set of fractions the next step is to change each fraction to one of its equivalents so that we may add or subtract it. An equivalent fraction has the same value as the original fraction it looks a little different! Here are some examples of equivalent fractions: etc. 0 0 etc. An equivalent fraction is obtained by multiplying both the numerator and denominator of the fraction by the same number. This is called BUILDING. Here are some examples: x x and were both multiplied by x x and were both multiplied by x x and were both multiplied by Note: the numbers used to multiply look like fraction versions of. An equivalent fraction can also obtained by dividing both the numerator and denominator of the fraction by the same number. This is called REDUCING. Here are some more examples: 0 0 and were both divided by and were both divided by and were both divided by Created by 00 page of

29 Exercise (answers on page ) Find the number that belongs in the space by building or reducing equivalent fractions. ) ) ) 0 ) ) 0 ) ) ) ) 0 0) ) 0 ) ) ) ) 0 ) ) ) 0 ) 0) ) 0 0 Created by 00 page of

30 Simplifying Improper Fractions An improper fraction is one in which the numerator is larger than the denominator. If the answer to an addition subtraction multiplication or division fraction is improper simplify it and reduce if possible. Ex. : is an improper fraction. Divide the denominator into numerator. Ex. : 0 is an improper fraction. Divide to simplify. Reduce. 0 0 Ex. : is an improper fraction. Divide to simplify. Reduce Created by 00 page 0 of

31 Exercise (answers on page ) Simplify the following fractions. Reduce if possible. ) = ) = ) = ) 0 = ) = ) = ) 0 = ) = ) = 0) = ) = ) = ) = ) = ) = ) 0 0 = ) = ) = 0 ) = 0) = ) = 0 Created by 00 page of

32 Adding and Subtracting of Fractions When adding or subtracting there must be a common denominator. If the denominators are different: (a) Write the problem vertically (top to bottom) (b) Find the LCD (c) Change to equivalent fractions (by building) (d) Add or subtract the numerators (leave the denominators the same) (e) Simplify and reduce if possible Ex. : Ex. : The denominators are the same. Add the numerators keep the denominator. This fraction cannot be simplified or reduced.? The denominators are different numbers. Therefore change to equivalent fractions. See page Ex. :? Ex. :? Ex. :? Simplifying and reducing completes addition and subtraction problems. See page & Created by 00 page of

33 Exercise (answers on page ) Add or subtract the following fractions. Simplify and reduce when possible. ) ) ) ) ) ) ) ) ) 0) ) ) ) ) ) ) ) ) 0 ) 0 0) ) ) ) ) ) ) 0 ) Created by 00 page of

34 Adding and subtracting mixed numbers A mixed number has a whole number followed by a fraction: and are examples of mixed numbers When adding or subtracting mixed numbers use the procedure from page. Note: Don t forget to add or subtract the whole numbers. Ex. :? Ex. :? Ex. :? Ex. :? When mixed numbers cannot be subtracted because the bottom fraction is larger than the top fraction BORROW so that the fractions can be subtracted from each other. The cannot be Ex. : -? The cannot be subtracted from nothing. One was borrowed from the and changed to. was changed to a.now the mixed numbers can be subtracted from each other. Ex. :? subtracted from the. One was borrowed from the changed to and then added to the to make. The whole number was changed to a. Now the mixed numbers can be subtracted. Created by 00 page of

35 Exercise (answers on page ) Add or subtract the following mixed numbers. Simplify and reduce when possible. ) = ) = ) = 0 ) = ) = ) = ) = ) = ) = 0) = ) = ) = ) = ) = ) = ) 0 = ) = ) = ) = 0) = ) 00 = Created by 00 page of

36 Fraction Word Problems (Addition/Subtraction) When solving word problems make sure to UNDERSTAND THE QUESTION. Look for bits of information that will help get to the answer. Keep in mind that some sentences may not have key words or key words might even be misleading. USE COMMON SENSE when thinking about how to solve word problems. The first thing you think of might be the best way to solve the problem. Here are some KEY WORDS to look for in word problems: Sum total more than: mean to add Difference less than how much more than: mean to subtract Ex. : If brand X can of beans weighs ounces and brand Y weighs ounces how much larger is the brand X can? Borrow from the whole number and add to the fraction Ex. : Find the total snowfall for this year if it snowed 0 means to subtract inch in November inches in December and means to add Simplify. 0 inches in January. Created by 00 page of

37 Exercise (answers on page ) Solve the following add/subtract fraction word problems. Find the total width of boards that inches wide inch wide and inches wide.. A.H tire is inches wide and a.c tire is wide. What is the difference in their widths? inches. A patient is given teaspoons of medicine in the morning and teaspoons at night. How many teaspoons total does the patient receive daily?. feet are cut off a board that is the remaining part of the board? feet long. How long is. of the corn in the U.S. is grown in Iowa. of it is grown in Nebraska. How much of the corn supply is grown in the two states? Created by 00 page of

38 . A runner jogs miles east How far has she jogged? miles south and miles west.. If ounce of cough syrup is used from a much is left? ounce bottle how. I set a goal to drink ounces of water a day. If I drink 0 ounces in the morning ounces at noon and 0 ounces at dinner how many more ounces of water do I have to drink to reach my goal for the day?. Three sides of parking lot are measured to the following lengths: 0 feet feet and feet. If the distance around the lot is feet find the fourth side. 0. Gabriel wants to make five banners for the parade. He has feet of material. The size of four of the banners are: ft. ft. ft. and ft. How much material is left for the fifth banner? Created by 00 page of

39 Answer to Intro to Fractions Exercise Exercise Exercise Exercise Exercise..... > >..... <..... = < Exercise Exercise Exercise Exercise ) )... )... ). ). ). Created by 00 page of

40 Answer to Multiplication and Division of Fractions Exercise Exercise Exercise Exercise Exercise..... boards..... revolutions..... turns..... books..... yards pounds..... pounds sheets pages pizzas Created by 00 page 0 of

41 Answers to Addition and Subtraction of Fractions Exercise Exercise Exercise Exercise Exercise Exercise ) ) ) ) ) ) inches 0 ) ) 0 ) ) ) 0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) 0 ) ) ) ) ) ) 0 ) ) ) ) ) ) 0 ) ) 0 ) ) ) 0) 0) 0) ) ) 0) ) ) ) ) ) ) ) ) ) 0 ) ) ) ) ) 0 ) ) ) ) ) ) ) ) ) ) ) ) ) 0 ) ) ) ) 0) 0) ) ) ) 0) ) ) ) ) ) ) ) ) ) 0) ) ) ) ) ) ) ) ) 0 ) 0) 0 ) ) ) ) ) ) ) ) ) 0) 0 inches teaspoons feet miles 0 ounces ounces 0 feet 0 ft. Created by 00 page of

42 Answer to Ordering Fractions Exercise A.. Exercise B.. Created by 00 page of

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