Arithmetic Review: Operations with Fractionsm21867Arithmetic Review: Operations with Fractions1.42009/03/23 02:28:30 GMT-52009/05/28 16:09:24.903 GMT-5WadeEllisWade Ellisfgafaculty@gmail.comDennyBurzynskiDenny Burzynskidenny_burzynski@westvalley.eduWadeEllisWade Ellisfgafaculty@gmail.comDennyBurzynskiDenny Burzynskidenny_burzynski@westvalley.eduLearningMateLearningMateLearningMate LearningMateabhijit.chaturvedi@learningmate.comMattGardnerMatt Gardnermgardner@wordsandnumbers.comWadeEllisWade Ellisfgafaculty@gmail.comDennyBurzynskiDenny Burzynskidenny_burzynski@westvalley.edualgebraelementaryfractionsMathematics and StatisticsThis module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.enOverview Multiplication of Fractions Division of Fractions Addition and Subtraction of Fractions Multiplication of FractionsMultiplication of FractionsTo multiply two fractions, multiply the numerators together and multiply the denominators together. Reduce to lowest terms if possible.For example, multiply 34·16.34·16=3·14·6=324Now reduce.=3·12·2·2·3=3·12·2·2·33 is the only common factor.=18Notice that we since had to reduce, we nearly started over again with the original two fractions. If we factor first, then cancel, then multiply, we will save time and energy and still obtain the correct product.Sample Set APerform the following multiplications.14·89=12·2·2·2·23·3=12·2·2·2·23·32 is a common factor.=11·23·3=1·21·3·3=2934·89·512=32·2·2·2·23·3·52·2·3=32·2·2·2·23·3·52·2·32 and 3 are common factors.=1·1·53·2·3=518Division of FractionsReciprocals Two numbers whose product is 1 are reciprocals of each other. For example, since 45·54=1,45 and 54 are reciprocals of each other. Some other pairs of reciprocals are listed below.27,7234,4361,16Reciprocals are used in division of fractions.Division of FractionsTo divide a first fraction by a second fraction, multiply the first fraction by the reciprocal of the second fraction. Reduce if possible.This method is sometimes called the “invert and multiply” method.Sample Set BPerform the following divisions.13÷34.The divisor is 34. Its reciprocal is 43.13÷34=13·43=1·43·3=4938÷54.The divisor is 54. Its reciprocal is 45.38÷54=38·45=32·2·2·2·25=32·2·2·2·252 is a common factor.=3·12·5=31056÷512.The divisor is 512. Its reciprocal is 125.56÷512=56·125=52·3·2·2·35=52·3·2·2·35=1·21=2Addition and Subtraction of FractionsFractions with Like DenominatorsTo add (or subtract) two or more fractions that have the same denominators, add (or subtract) the numerators and place the resulting sum over the common denominator. Reduce if possible.CAUTIONAdd or subtract only the numerators. Do not add or subtract the denominators!Sample Set CFind the following sums.37+27.The denominators are the same. Add the numerators and place the sum over 7.37+27=3+27=5779−49.The denominators are the same. Subtract 4 from 7 and place the difference over 9.79−49=7−49=39=13Fractions can only be added or subtracted conveniently if they have like denominators.Fractions with Unlike DenominatorsTo add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as the denominator the least common multiple of the original denominators.The least common multiple of the original denominators is commonly referred to as the least common denominator (LCD). See Section () for the technique of finding the least common multiple of several numbers.Sample Set DFind each sum or difference.16+34.The denominators are not alike. Find the LCD of 6 and 4.{6=2·34=22The LCD is 22·3=4·3=12.Convert each of the original fractions to equivalent fractions having the common denominator 12.16=1·26·2=21234=3·34·3=912Now we can proceed with the addition.16+34=212+912=2+912=111259−512.The denominators are not alike. Find the LCD of 9 and 12.{9=3212=22·3The LCD is 22·32=4·9=36.Convert each of the original fractions to equivalent fractions having the common denominator 36.59=5·49·4=2036512=5·312·3=1536Now we can proceed with the subtraction.59−512=2036−1536=20−1536=536ExercisesFor the following problems, perform each indicated operation.13·434913·2325·561356·1415916·20275123536·48552125·15149107699·663837·1418·6211415·2128·45759÷5623916÷15849÷6151092549÷49154÷2781092475÷815578÷78577710÷10738+2858311+411512+71211116−2161523−2231323311+111+5111620+120+220192038+28−181116+916−516151612+1618+125834+1358+23312467−14815−31056115+5122536−710−1180928−445815−310730116+34−3883−14+736471834−322+524