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Iulia & Teodoru GugoiuThe Book ofFractionsCopyright 2006 by La Citadellewww.la-citadelle.com

Iulia & Teodoru GugoiuThe Book of FractionsISBN 0-9781703-0-X 2006 by La Citadelle4950 Albina Way, Unit 160Mississauga, OntarioL4Z 4J6, ted by Rob CouvillonAll rights reserved. No part of this book may bereproduced, in any form or by any means,without permission in writting from thepublisher.

ContentPage 55Understanding fractionsThe graphical representation of a fractionReading or writing fractions in wordsUnderstanding the fraction notationUnderstanding the mixed numbersReading and writing mixed numbers in wordsUnderstanding mixed number notationUnderstanding improper fractionsUnderstanding improper fraction notationThe link between mixed numbers and improper fractionsConversion between mixed numbers and improper fractionsWhole numbers, proper fractions, improper fractions and mixed numbersUnderstanding the addition of like fractionsUnderstanding the addition of like fractions (II)Adding proper and improper fractions with like denominatorsAdding mixed numbers with like denominatorsAdding more than two like fractionsUnderstanding equivalent fractionsFinding equivalent fractionsSimplifying fractionsChecking fractions for equivalenceEquations with fractionsAdding fractions with unlike denominatorsAdding fractions with unlike denominators using the LCD methodUnderstanding the subtraction of fractions with like denominatorsSubtracting fractions with like denominatorsSubtracting mixed numbers with like denominatorsSubtracting fractions with unlike denominatorsSubtracting fractions with unlike denominators using the LCD methodOrder of operations (I)Multiplying fractionsMore about multiplying fractionsThe order of operations (II)Reciprocal of a fractionDividing fractionsDivision operatorsOrder of operations (III)Order of operations (IV)Raising fractions to a powerOrder of operations (V)Converting fractions to decimalsConverting decimals to fractionsOrder of operations (VI)Time and FractionsCanadian coins and fractionsFractions, ratio, percent, decimals, and proportionsFractions and Number LineComparing fractionsSolving equations by working backward methodFinal TestAnswers

Preface“The Book of Fractions" presents one of the primary concepts of middle and high school mathematics: theconcept of fractions. This book was developed as a workbook and reference useful to students, teachers, parents,or anyone else who needs to review or improve their understanding of the mathematical concept of fractions.The structure of this book is very simple: it is organized as a collection of 50 quasi-independent worksheets andan answer key. Each worksheet contains:· a short description of the concepts, notations, and conventions that constitute the topic of the worksheet;· step-by-step examples (completely solved) demonstrating the techniques and skills the student shouldgain by the end of each worksheet; and· an exhaustive test to be completed independently by the students.The concept of fractions and the relations between fractions and other types of numbers, like many abstractmathematical concepts, is not always easy to understand. Bearing this in mind, the authors of this book introduceeach topic gradually, starting with the basic concepts and operations and progressing to the more difficult ones.Geared specifically to help the beginners, the first part of the book contains graphical representations of thefractions.The techniques for solving both simple and complex equations implying fractions are explained. As well,complete worksheets are provided, starting with very simple and basic equations and progressing to extremelycomplex equations requiring the application of a full range of operations with fractions."The Book of Fractions" also presents the link between fractions and other related mathematical concepts, suchas ratios, percentages, proportions, and the application of fractions to real life concepts like time and money.The importance of the concept of fractions comes both from its link to natural numbers and its link to morecomplex mathematical concepts, like rational numbers. As such, the concept of fractions is a milestone in themathematical evolution of a student, being a concept that is simultaneously concrete (as a part of a whole) andabstract (as a set of two numbers and a hidden division operation).The concept of equivalent fractions is an essential part of understanding fractions, and a full range of techniquesis presented, starting with graphical representations (suitable for students in lower grades) and progressing toadvanced uses, like the factor tree method of finding the LCD.The order of operations is also presented, gradually, after each main operation with fractions: addition,subtraction, multiplication, and division; using multi-term expressions; expressions containing grouping symbolsof one or more levels; and more complex operations with fractions, like powers with positive and negativeexponents.Single-step questions (requiring a basic knowledge and understanding of the topic presented in the worksheets)and multi-step questions (requiring a complete understanding of all of the concepts presented in the worksheetsto that point) are presented throughout the entire book.Combining more than 15 years of academic studies and 30 years of teaching experience, the authors of this bookwrote it with the intention of sharing their knowledge, experience and teaching strategies with all the partnersinvolved in the educational process.Iulia & Teodoru Gugoiu,Toronto, 2006

The Book of FractionsIulia & Teodoru GugoiuUnderstanding fractions1. A fraction represents a part of a whole.Example 1.2. The corresponding fraction is:34The whole is divided into four equal parts.Three part are taken (considered).The numerator represents how many parts are taken.Fraction line or division barThe denominator represents the number ofequal parts into which the whole is divided.F01. Write the fraction that represents the part of the object that has been shaded:a)b)c)d)e)f)g)h)i)j)k)l)m)n)o)p)q)r)s)t)0u) La Citadellev)w)x)51y)www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuThe graphical representation of a fraction1. A fraction represents a part of a whole.Example 1.2. A corresponding graphical representation(diagram) is:34The whole is divided into four equal parts.Three part are taken (considered).The numerator represents how many parts are taken.Fraction line or division barThe denominator represents the number ofequal parts into which the whole is divided.F02. Draw a diagram to show each fraction (use the images on the bottom of this 6w)18x)749y)37100z)11180 La Citadelle16www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuReading or writing fractions in words1. You can use words to refer to a part of a whole.So one whole has:2 halves3 thirds4 quarters5 fifths6 sixths7 sevenths8 eighths9 ninths10 tenths11 elevenths12 twelfths13 thirteenths20 twentieths30 thirtieths50 fiftiethsExample 1.100 hundredths1000 thousandths1000000 millionths1000000000 billionthsThe fraction34can be written in words as:three quartersF03. Write the following fractions in F04. Find the fraction written in words:a)one thirdb)one halfc)one sixthd)two fifthse)four seventhsf)seven eighthsg)eleven fiftiethsh)seven twentiethsi)five twelfthsj)eight ninthsk)six tenthsl)nine thousandthsm)fifteen millionthsn)eight sixthso)three fiftiethsp)eleven billionthsq)twenty-three hundredthsr)seven thirteenthss)eleven twelfthst)three billionthsu)thirteen thirtiethsv)one fifthw)one eleventhx)eight ninthsy)six tenthsz)six twelfths La Citadelle7www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuUnderstanding the fraction notation1. A fraction also represents aquotient of two quantities:Example 1. The dividend (numerator) is 3.3The divisor (denominator) is 4.The fraction in words is three quarters.4dividentdivisor2. The dividend (numerator) represents how many partsare taken.The divisor (denominator) represents the number ofequal parts into which the whole is divided.A possible graphical representation of this fraction is:F05. Fill out the following (Divisor)2314c)The fraction written in wordsGraphical representationtwo thirdsthree fifthsd)e)f)3522g)53h). quartersi)five .3j)k)l) La Citadelle4. sixths5three .8www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuUnderstanding the mixed numbersfraction part1. A mixed number is an addition of wholes and a part of a whole.Example 1.whole-number part(the number ofcomplete wholes)There are one complete whole andthree quarters of the second wholeThe numerator indicates how many partsare taken from the last whole.The denominator represents the numberof equal parts into which the whole isdivided.314F06. Find the mixed number that corresponds to the shaded )w) 0120x)21F07. Find a possible graphical representation of each mixed 512x)379y)238z)634 La le.com3

The Book of FractionsIulia & Teodoru GugoiuReading and writing mixed numbers in words1. You can use words to refer to a part of a whole.So one whole has:2 halves3 thirds4 quarters5 fifths6 sixths7 sevenths8 eighths9 ninths10 tenths11 elevenths12 twelfths13 thirteenths20 twentieths30 thirtieths50 fiftiethsExample 1.The fraction 2100 hundredths1000 thousandths1000000 millionths1000000000 billionths34can be written in words as:two wholes and three quarters ortwo and three quartersF08. Write the following mixed numbers in words:a) 112b)h) 159i)o) 2750p) 2317w) 4 3v) 213c)1310j)13100q)32214x)14d)2211k)391000r) 22515y)35e) 156f)237g)3512l) 1215m) 3720n)2t)719u)73s) 11000000402160z)22589303516390F09. Find the mixed numbers written in words:a)two and two thirdsb)three and one halfc)five and five sixthsd)two and one thirde)four and five seventhsf)seven and five fiftiethsg)two and three quartersh)three and two ninthsi)six and seven hundredthsj)nine and one halfk)eight and eleven fiftiethsl)one and five billionthsm)one and two eleventhsn)eight and five sixthso)three and two twelfthsp)five and three millionthsq)twenty and three hundredthsr)six and four fifteenthss)eleven and four thirtiethst)eight and seven tenthsu)four and one thirdv)one and two fifthsw)three and two eleventhsx)eight and six ninthsy)five and nine tenthsz)one and eleven twelfths La Citadelle10www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuUnderstanding mixed number notation1. A mixed number is represented by the expression:wholesExample 1.numeratordenominator235This mixed number written in words is two wholes and three fifths.A possible graphical representation of this mixed number is:The whole-number part is 2(the number of complete wholes).The numerator is 3.The denominator is 5. 3The fraction part is:5F10. Fill out the following table:MixedNumbera)235b)213c)Numberof wholesNumeratorThe mixed numberin wordsDenominator235134Graphical representationtwo and three fifthsd)e)2f)3g)3522352three and a halfh)6i)j)3k)l) La Citadelletwo and four . and three fifths2four and . thirds2.11www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuUnderstanding improper fractions1. For an improper fraction the number of parts taken (the numerator) is equal to or greater than the number ofparts the whole is divided into (the denominator).Example 1.53This is a possible graphical representation of this improper fraction:F11. Find the improper fraction that corresponds to the shaded s)t)u)v)w)x)F12. Find a possible graphical representation of each improper )5549t)256u)179v)3012w)154x)175y)2410z)133 La Citadelle12www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuUnderstanding improper fraction notation1. An improper fraction is represented by the expression:Example 1.numeratordenominator53The numerator is 5The denominator is 3The improper fraction in words is five thirds. A possiblegraphical representation of this improper fraction is:where the numerator is equal to or greater than thedenominator.F13. Fill out the following table:Fractiona)54b)NumeratorDenominator5474c)The fraction in wordsGraphical representationfive quartersseven fifthsd)e)f)165.127g)511h). quartersi)seven .13j)k)l) La Citadelle13. sixths5thirteen .13www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuThe link between mixed numbers and improper fractions1. There is a direct link between a mixed number and an improper fraction. A mixed number is a short way to writethe sum of a whole number and a fraction.522 1 1 333Example 1:933 1 1 666Example 2:F14. Find the mixed number and the improper fraction that correspond to each )s)t)u)v)w)x) La Citadelle14www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuConversion between mixed numbers and improper fractions1. To convert a mixed number to an improper fraction,use the formula:3. To convert an improper fraction to a mixed number,divide the numerator n into the denominator d to obtainthe quotient q and the remainder r. Then write:n w d n dd3 2 5 3 10 3 13Example 1. 2 5555wnr qdd2. Fractions that have a denominator of 0 are notdefined.91 244Example 2.F15. Write each mixed number as an improper fraction:a) 112b)23c)3h) 534i)730j)o) 2350p) 445v) 029w) 20102234d)312e) 237f)5320g)51513k)289l) 356m)258n)12q)221100r)2112s) 3211t)24950u)x)320y)333z) 231071002111001510F16. Write each improper fraction as a mixed y)2015z)709 La citadelle.com

The Book of FractionsIulia & Teodoru GugoiuWhole numbers, proper fractions, improper fractions and mixed numbers1. Although written in fraction notation,some numbers are actually whole numbers.Example 1:10 52Example 2: 23 332. A whole number can be converted into a fraction.This conversion is not unique.Example:5 ndwhere n dExample:An improper fraction is: ndwhere n ³ dExample:3. A proper fraction is:A mixed number is:wA mixed numberin standard form is:wndExample:23734532nwhere n d Example: 5d34. Fractions that have a denominator of 0 are not defined.501033014 4 4 2 310103107F17. Convert fractions to whole numbers. Identify the expressions that are not defined.11b)h) 393l) 4555m) 2168n)00r)055s) 209t)06u)0100102300100000F18. Convert whole numbers to fractions (the conversion is not unique, so give at leasttwo solutions):a)1b)3c)7d)4e) 2f)5g)10h)0i)25j)100k)11l)m)13n)178F19. Identify each of the following expressions as a whole number, a proper fraction,an improper fraction, a mixed number, or a not defined expression:a)32b)1c)54d)29e) ) La 0358n)2171011150u)010www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuUnderstanding the addition of like fractionsTwo fractions with the same denominators are called like fractions. When you add two fractions, you add the partsof the whole they represent.1 2 3 4 4 4Example 1.So, by adding 1 quarter and 2 quarters you get 3 quarters.F20. Add the fractions that correspond to the shaded ) La Citadellev)w)171x)y)www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuUnderstanding the addition of like fractions (II)1. Sometimes when you add two like fractions, the number of parts you add exceeds a whole. The result is animproper fraction or a mixed number.Example 1. Or, in mathematical symbols:3 3 62 14 4 44F21. Add the fractions that correspond to the shaded regions. Express the result both as animproper fraction and as a mixed )w)x)y) La Citadelle18www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuAdding proper and improper fractions with like denominators1. To add proper or improper fractions with likedenominators (called like fractions), add the numeratorsand keep unchanged the denominator, according to therule:n1 n2 n1 n2 d dd1 2 3 5 5 5Example 1.2. If the result is an improper fraction, you canchange it to a mixed number.Example 2.3 2 51 14 4 44F22. Add the fractions:a)2 1 4 4b)1 1 3 3c)2 1 5 5d)3 5 11 11e)1 2 6 6f)2 3 7 7g)5 7 19 19h)20 10 100 100i)6 23 35 35j)2 10 41 41k)0 11 13 13l)21 11 54 54m)2 3 10 10n)5 4 12 12o)3 10 17 17p)0 0 4 4q)3 2 13 13r)73 20 20s)7 11 25 25t)5 2 9 9u)2 5 19 19v)2 3 10 10w)7 13 30 30x)3 8 13 13y)1525 100 100F23. Add the fractions. Write the result as a mixed number in standard form.a)2 2 3 3b)3 2 4 4c)3 4 5 5d)3 5 6 6e)3 5 7 7f)5 7 8 8g)7 8 9 9h)7 7 10 10i)8 10 11 11j)9 5 12 12k)10 11 15 15l)4 19 20 20m)10 20 25 25n)44 44 50 50o)99 11 100 100p)7 6 9 9q)8 9 10 10r)33 19 40 40s)4 9 3 3t)15 9 10 10u)12 5 9 9v)20 12 10 10w)22 77 50 50x)13 12 3 315 34 y) 10 10 La Citadelle19www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuAdding mixed numbers with like denominatorsExample 1. 2 3 3 4 ( 2 3) 3 4 5 7 6 21. To add mixed fractions with like denominators, addseparately the wholes and separately thenumerators, and keep the denominator unchanged:555552. In the same way you can add whole and mixed numbers.nnn nw1 1 w2 2 ( w1 w2 ) 1 2dddExample 2. 2 31010 11 2 3 (2 3) 544444F24. Add the mixed numbers. Write the result as a mixed number in standard form or asa whole number:111112133 2a) 1 2b) 1 2c) 1 2d) 1 3e) 2 1223344556 632f) 2 37723g) 2 38821h) 3 0993 7k) 2 5 546l) 4 277m) 228p) 5 299q) 22211 51010v) 21041 53030u) 221 12 119 19i) 25 3 10 10j) 356 21515112 31010n) 21211 52020o) 388 311 11r) 31122 2303079s) 3 233t) 21525 31010w) 3222 123 23x) 43322 23535y) 11595 2100100F25. Add the wholes and the mixed numbers. Write the result as a mixed number in standard formor as a whole number.12125a) 2 1b) 1 2c) 1 3d) 2 5e) 5 223456f)5 397g)57 68k)93 25l)5 1p)919 99q)20 10u)1 91919v)4 La Citadelleh)87m) 11001014 530109i)111 310n)113040233 r)11 3w)2 2201 1110j)0 20152 22220o)13 13s)9 3103t)4x)1 13735y)111 113114 410111100www.la-citadelle.com

The Book of FractionsIulia & Teodoru GugoiuAdding more than two like fractions1. To add more than two fractions or whole numbers, start to add in order, from left to right.2 5 æ2ö 52 5731 3 ç1 3 4 4 54 4 è4ø 44 444Example 1.2. Because the addition is a commutative operation, the order in which you add the fractions is not important.So, group them conveniently.23 122öæ 2 1ö æ 3 2 1 2 1 2 ç 1 ç 2 1 2 2 4 835 353355èø èøExample 2.F26. Add the fractions:a) 1 3 3 12f)21 2 31 3 3 31 2 3 4 4 4g)27317 210 10103545l)1 521 19 991979q)1v)1353 2 1444k) 1 1p) 1 219b)49u) 0 189135 210 1010e)1234 1 2 35555373 2 5050 50j)13571 2 3 466663 57 112 12 12o)1 3 5 7 9 11 11 11 11 112131 3100100t)2135 1 010 1010y)1 2 3 4 5 6 7 7 7 7 7 7c)3 511

“The Book of Fractions" presents one of the primary concepts of middle and high school mathematics: the concept of fractions. This book was developed as a workbook and reference useful to students, teachers, parents, or anyone else who needs to review or improve their understanding o