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335 àï2.2è Reducing Fractions. äè Please Reduce the following fractions. âSêë6êè2∙3êè 3ê 3 #êêï1)ï─è =è ─────è =è ───è=è─ êêë 8êï2∙2∙2êï2∙2ê4 êêë 12êï2∙2∙3êï2∙3êè6 #êêï2)ï──è =è ─────è =è ───è =è ── êêë 38êè2∙19êè19êï19 éS To Reduce the fraction,ï12/38 , you should first express the top and bottom in prime factored form. êêêê12êï2∙2∙3 #êêêê──è =è ───── êêêê38êè2∙19 Next, cancel out any like factors that occur on both the top and bottom and multiply the remaining factors. #êê12êï╖∙2∙3êï2∙3êè6 #êê──è =è ─────è =è ───è =è ── #êê38êè╖∙19êè19êï19 Thus, the fraction,ï12/38 , reduces toï6/19.ïThese two fractions are actually equal to each other, and they occupy the same location on the number line.ïConsider reducing the fraction,ï12/10. #êè12êï╖∙2∙3êï2∙3êï6êè1 #êè──è =è ─────è =è ───è =è ─èorè1 ─ #êè10êè╖∙5êè 5êè5êè5 The answer,ï6/5 , can be left in this form or changed to the mixed number,ï1 1/5 .ïIt is not more correct to write the answer as a mixed number, however some books ask you to write the answer as a mixed number. 1êêêêê 14 #êêêïReduce the fraction,ï──. êêêêêêè28 êë2êêè1êêè 7 #ê A)ï─êë B)ï─êë C)ï──êëD)ïå êë4êêè2êêè14 ü #êêêè14êè╖∙╝êè1 #êêêè──è =è ─────è =è ─ #êêêè28êï╖∙2∙╝êï2 Ç B 2êêêêê 24 #êêêïReduce the fraction,ï──. êêêêêêè36 êë2êêè6êêè12 #ê A)ï─êë B)ï─êë C)ï──êëD)ïå êë3êêè9êêè18 ü #êêê24êï╖∙╖∙2∙╕êï2 #êêê──è =è ───────è =è ─ #êêê36êï╖∙╖∙╕∙3êï3 Ç A 3êêêêêï0 #êêêïReduce the fraction,ï──. êêêêêêè10 êë 1 #ê A)ï──êëB)ï10êëC)ï0êë D)ïå êë10 ü êêêêè0 #êêêêï──è =è 0 êêêêï10 Ç C 4êêêêê 45 #êêêïReduce the fraction,ï──. êêêêêêè15 êë1êêè 1 #ê A)ï─êë B)ï──êëC)ï3êë D)ïå êë3êêè15 ü #êêë 45êï╕∙3∙║êï3 #êêë ──è =è ─────è =è ─è =è 3 #êêë 15êè ╕∙║êï1 Ç C 5êêêêê 20 #êêêïReduce the fraction,ï──. êêêêêêè28 êë 5êêï5êêè10 #ê A)ï──êëB)ï─êë C)ï──êëD)ïå êë14êêï7êêè14 ü #êêê20êï╖∙╖∙5êï5 #êêê──è =è ─────è =è ─ #êêê28êï╖∙╖∙7êï7 Ç B 6êêêêê 18 #êêêïReduce the fraction,ï──. êêêêêêè12 êè3ê1êè9ê3êè 2 #ë A)ï─ïor 1 ─ë B)ï─ïor 1 ─êC)ï─êèD)ïå êè2ê2êè6ê6êè 3 ü #êêè 18êï╖∙╕∙3êï3êè1 #êêè ──è =è ─────è =è ─èorè1 ─ #êêè 12êï╖∙2∙╕êï2êè2 Ç A 7êêêêê 60 #êêêïReduce the fraction,ï──. êêêêêêè80 êë15êêï2êêè3 #ê A)ï──êëB)ï─êë C)ï─êë D)ïå êë20êêï3êêè4 ü #êêë 60êè╖∙╖∙3∙║êè3 #êêë ──è =è ─────────è =è ─ #êêë 80êï╖∙╖∙2∙2∙║êï4 Ç C 8êêêêêï48 #êêêïReduce the fraction,ï───. êêêêêêè144 êë1êêè5êêè 3 #ê A)ï─êë B)ï─êë C)ï──êëD)ïå êë3êêè9êêè17 ü #êêë48êï╖∙╖∙╖∙╖∙╕êè 1 #êêè ───è =è ───────────è =è ─ #êêè 144êï╖∙╖∙╖∙╖∙╕∙3êï3 Ç A 9êêêêê 36 #êêêïReduce the fraction,ï──. êêêêêêè16 êè18ê 2êè 9ê1êë3 #ë A)ï──ïorï2 ─êB)ï─ orï2 ─ê C)ï─êïD)ïå êè 8ê 8êè 4ê4êë4 ü #êêï36êï╖∙╖∙3∙3êï9êë1 #êêï──è =è ───────è =è ─è orè 2 ─ #êêï16êï╖∙╖∙2∙2êï4êë4 Ç B 10êêêêë19 #êêêReduce the fraction,ï──. êêêêêê 47 êè3êêë5êêë 8 #ë A)ï─êê B)ï─êê C)ï──êïD)ïå êè8êêë7êêë13 ü êêêè19 #êêêè──èis already Reduced. êêêè47 Ç D äèPlease find the missing numerator (top number) in order êê to construct an equivalent fraction. âSë1)ï2ê?êêè2)ï5ê ? #êêè─è=è─êêê─è=è── êêè3ê6êêê7ê49 êêè2ê4êêê5ê35 #êêè─è=è─êêê─è=è── êêè3ê6êêê7ê49 éS In order to find the missing numerator in the equation, êêêê 5êè? #êêêê ─è =è ── êêêê 7êï49 it is necessary to divide the "7" into "49".ïThe answer, 7, is then multiplied times "5" to get "35".ïThis is the missing numerator. êêêê 5êï35 #êêêê ─è =è ── êêêê 7êï49 The fraction,ï5/7 , is actually equal to the fraction,ï35/49.ïIt will be necessary in the next section, which is about adding fractions, to find missing numerators. 11êêêêêêè3ê ? #êèFind the missing numerator in the equation,ï─è=è──. êêêêêêêë 4ê16 ë3ë12êê3ë10êê3ë15 # A)ï─ï=ï──êèB)ï─ï=ï──êèC)ï─ï=ï──êD)ïå ë4ë16êê4ë16êê4ë16 ü êêêê 3êï12 #êêêê ─è =è ── êêêê 4êï16 Ç A 12êêêêêêè5ê? #êïFind the missing numerator in the equation,ï──è=è──. êêêêêêêë17ê51 ê5ë20êë 5ë15êë 5ë10 #ïA)ï──ï=ï──ê B)ï──ï=ï──ê C)ï──ï=ï──ëD)ïå ë 17ë51êë17ë51êë17ë51 ü êêêê 5êï15 #êêêê──è =è ── êêêê17êï51 Ç B 13êêêêêêï3ê ? #êïFind the missing numerator in the equation,ï─è=è──. êêêêêêêë8ê32 ë3ë 9êê3ë14êê3ë12 # A)ï─ï=ï──êèB)ï─ï=ï──êèC)ï─ï=ï──êD)ïå ë8ë32êê8ë32êê8ë32 ü êêêê 3êï12 #êêêê ─è =è ── êêêê 8êï32 Ç C 14êêêêêêè5ê? #êïFind the missing numerator in the equation,è─è=è─. êêêêêêêë 1ê4 ë 5ë25êë 5ë20êë 5ë15 # A)è─ï=ï──ê B)è─ï=ï──ê C)è─ï=ï──ëD)ïå ë 1ë 4êë 1ë 4êë 1ë 4 ü êêêêï5êï20 #êêêêï─è =è ── êêêêï1êè4 Ç B 15êêêêêêï1ê ? #êïFind the missing numerator in the equation,ï─è=è──. êêêêêêêë4ê60 ë1ë15êê1ë12êê1ë20 # A)ï─ï=ï──êèB)ï─ï=ï──êèC)ï─ï=ï──êD)ïå ë4ë60êê4ë60êê4ë60 ü êêêêï1êï15 #êêêêï─è =è ── êêêêï4êï60 Ç A 16êêêêêêï11ê? #êïFind the missing numerator in the equation,ï──è=è──. êêêêêêêë16ê64 ë11ë22êë11ë33êë11ë44 # A)ï──ï=ï──ê B)ï──ï=ï──ê C)ï──ï=ï──ëD)ïå ë16ë64êë16ë64êë16ë64 ü êêêê 11êï44 #êêêê ──è =è ── êêêê 16êï64 Ç C äïPlease find the Least Common Multiple of the following êênumbers. âS êêThe Least Common Multiple of 8 and 14 is 56. êêThe Least Common Multiple of 3 and 4 is 12. éS To find the Least Common Multiple of 8 and 14, you should express each number in prime factored form. êêêêè8è =è 2∙2∙2 êêêêï14è =è 2∙7 Now construct a third number that has all the factors that occur in the factorizations of 8 and 14. #êêï8 ──── 2∙2∙2 ───┐ #êêêêè ┼─── 2∙2∙2∙7 ──── 56 #êê 14 ──── 2∙7──────┘ Notice that there must be three factors of "2" in the Least Common Multiple.ïEvery factor of 8 and 14 has to be present, in the Least Common Multiple, the largest number of times it occurs in any individual factorization.ïThe Least Common Multiple is the number 56.ïIt is the smallest number that both 8 and 14 divide into evenly.ïWe will need to find Least Common Multiples (denominators) in the next section on adding fractions. 17 êê Find the Least Common Multiple of 7 and 21. êA)ï42êëB)ï21êëC)ï35êëD)ïå ü #êêè 7 ───────────────┐ #êêêêê┼───── 3∙7 ───── 21 #êêè21 ───── 3∙7 ─────┘ Ç B 18 êê Find the Least Common Multiple of 14 and 42. êA)ï42êëB)ï84êëC)ï56êëD)ïå ü #êêï14 ───── 2∙7──────┐ #êêêêë ┼───── 2∙3∙7──── 42 #êêï42 ─── 2∙3∙7 ─────┘ Ç A 19 êêFind the Least Common Multiple of 44 and 12. êA)ï144êè B)ï660êè C)ï132êè D)ïå ü #êêï44 ─── 2∙2∙11 ────┐ #êêêêë ┼─── 2∙2∙3∙11 ── 132 #êêï12 ─── 2∙2∙3 ─────┘ Ç C 20 êêFind the Least Common Multiple of 3, 9, and 21. êA)ï72êëB)ï63êëC)ï36êëD)ïå ü #êêè 3 ────────────────┐ #êêêêê │ #êêè 9 ──── 3∙3 ───────┼─── 3∙3∙7 ───── 63 #êêêêê │ #êêè 21 ─── 3∙7 ───────┘ Ç B