Comparing FractionsDate: 03/11/99 at 21:12:40 From: ERIC Subject: Comparison of Fractions What is the easiest way of telling that one fraction is bigger than another (e.g. 5/9 and 5/6)? Date: 03/12/99 at 12:22:32 From: Doctor Peterson Subject: Re: Comparison of Fractions For the particular problem you gave, there is a very easy way! But let us look at three different kinds of problems, each of which can be done a different way. First, suppose the denominators are the same, like 5/9 and 7/9. Then it is pretty easy: just compare the numerators, which tell how many pieces (ninths, in this case) there are. Since the "pieces" are the same size in each fraction, the more pieces, the bigger the fraction. So the rule is: a b --- > --- if a > b n n Since 5 < 7, 5/9 < 7/9. They are in the same order as their numerators. Second, suppose the numerators are the same, as in your problem, 5/9 and 5/6. Then it works the opposite way: since you have the same number of pieces, you compare the denominators, which tell how big the pieces are. The bigger the denominator, the smaller the fraction. So the rule is: n n --- > --- if a < b a b Since 9 > 6, 5/9 < 5/6. They are in the reverse order of their denominators. Finally we have the general case, which is harder than the other two ways, but works for any fractions at all. You simply convert the fractions to the first case, by giving them a common denominator. You do not really have to worry about finding the least common denominator, though sometimes that will save a lot of work. Let us compare 5/9 and 4/7. Since we do not see any common factors immediately (and in fact there are not any), we can just multiply the denominators to get a common denominator, 63. To convert 5/9 to 63rds, we multiply by 7; to convert 4/7 to 63rds, we multiply by 9: 5 4 --- ? --- 9 7 5*7 4*9 --- ? --- 9*7 7*9 35 < 36, so 5 4 --- < --- 9 7 You may notice that I did not have to bother calculating the 63. Since I know the denominators are the same, I only need the numerators. What I have really done is just to multiply the 5 by 7 and the 4 by 9 and compare the results. Some people call this "cross multiplication": 5*7=35 4*9=36; since this is bigger, 4/7 is bigger \ / 5 4 --- ? --- 9 7 This is what I DO, but what I THINK is what I described above - otherwise I would never remember which product has to be bigger to make which fraction bigger! And this also lets me think in terms of an LCD if that helps. For instance, which is bigger, 5/9 or 44/81? I see that 81 is a multiple of 9, so I do not have to go to the trouble of multiplying 5 by 81 and 44 by 9; I just multiply 5 by 9 and compare to 44: 5 44 --- ? -- 9 81 5*9 44 --- ? -- 9*9 81 45 > 44 so 5 44 --- > -- 9 81 A little thinking saved a lot of work. And that is what math is all about! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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