11-18-2009Adding and Subtracting FractionsNow that we have carefully defined the set of rational numbers we will describe how todo basic arithmetic in this set.Addition.Definition. For the fractionsabandcbthe sum is given byab+cb=a + cb.In words, if two fractions have the same denominator we may add them by adding thenumerators.It is helpful to illustrate this definition with some of the models of the previous section.See page 364.Since we now know how to add fractions with a common denominator we can addfractions with unlike denominators by doing the following procedure:Procedure (Adding Fractions Whose Denominators are Different).1. Rewrite both fractions over a common denominator2. Add the numerators of the converted fractions and put the result over the commondenominator3. Reduce to lowest termsIn general we have:ab+cd=a · db · d+b · cb · d=ad + bcbd.Fraction strips are an excellent way to demonstrate that this procedure works. See page364.Subtraction.If two fractions have the same denominator you compute their difference by subtract-ing the numerators. If the fractions have unlike denominators first rewrite the fractionsover a common denominator. Then compute the difference by subtracting the numera-tors, putting the answer over the common denominator, and reducing to lowest terms.For example,23−15=1015−315=715Models can again help to understand the subtraction of fractions. See page 369.As before, we can recast a fraction subtraction problem as a fraction addition problemwith a missing addend model:Theorem.ab−cd=efif and only ifab=ef+cd.This means that to solve the problem a/b − c/d you could find what fraction you addto c/d to get back to a/b. To see how this model works, try some examples in one ormore of the fraction models.Proper Fractions and Mixed Numbers When doing arithmetic with fractions it isoften helpful to find the simplest possible express fractions.Definition. Consider the fractionab. If a < b thenabis called a proper fraction. Ifa ≥ b thenabis called an improper fraction.The fraction23is proper while the fractions32and33are improper. If the fractionabis improper b divides into a one or more times. So we can think ofabas a whole numberand a fraction:Definition. The sum of a natural number and a positive fraction is called a mixednumber.For example 2 +13is a mixed number. When writing a mixed number we will usuallydrop the plus sign and just write the whole part and the fractional part side by side:2 +13= 213.To convert the mixed Aabto an improper fraction, multiply the whole number part Aby the denominator b and add the result to the numerator:Aab=A · b + ab.To convert the improper fractionabto a mixed number, divide b into a. If a÷ b = qR r,then the mixed number expression forabis qrb.The easy way to add or subtract mixed numbers is to add the whole number partsand the fraction parts separately and then combine the answer as a new mixed number:Aab+ Bcd= (A + B)ab+cd.You could also convert everything to improper fractions first, do the addition orsubtraction, and then convert back to a mixed
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