Putting Rational Numbers in Order
Date: 09/22/2002 at 18:08:16 From: Eric Subject: Ordering rational numbers Hello, I am stuck on a problem and was wondering if you could walk me through it. I have to order 4/15, 6/17, and 3/16 from least to greatest. I can't use a calculator so it's hard to find a common denominator for all three of those fractions. How would I solve this and other similar problems?
Date: 09/22/2002 at 19:26:58 From: Doctor Sarah Subject: Re: Ordering rational numbers Hi Eric - thanks for writing to Dr. Math. If you can't use a calculator, you could multiply together the denominators (15 * 16 * 17) and use that denominator: 15 * 16 * 17 = 4080 4080/15 = 272 1/15 = 408/4080 4/15 =1088/4080 4080/16 = 255 1/16 = 255/4080 3/16 = 765/4080 4080/17 = 240 1/17 = 240/4080 6/17 = 1440/4080 If that looks like too much calculating, try division: __.187_ 16 )3.000 3/16 = .187 __.266_ 15 )4.000 4/15 = .266 __.352_ 17 )6.000 6/17 = .352 - Doctor Sarah, The Math Forum http://mathforum.org/dr.math/
Date: 09/22/2002 at 23:51:42 From: Doctor Greenie Subject: Re: Ordering rational numbers Hi, Eric - Here is another method which can sometimes be used (it works very nicely on this problem). Let's think of each fraction as representing the record of some sports team. For example, the fraction 7/10 would represent the record of a team which has won 7 out the 10 games it has played. Now let's compare the fractions 3/16 and 4/15. A team whose record is represented by the fraction 3/16 has played one more game than the team whose record is represented by the fraction 4/15, but they have won one game less. If they have played more total games but have won fewer, then clearly their record is worse - that is, their winning percentage is lower. So we know that 3/16 is less than 4/15 without doing any difficult arithmetic. Now let's compare the fractions 4/15 and 6/17 in a slightly different but still very similar manner. This time we can think of these two fractions as representing the record of the same team, but a couple of games apart. So the fraction 4/15 represents the fact that the team at one point had played 15 games and won 4; the fraction 6/17 represents the fact that, after the team had played 17 games, it had won 6 of them. In going from its earlier record of winning 4 out of 15 games to its later record of winning 6 out of 17 games, it must have won 2 out of 2 games. The winning percentage for winning 2 out of 2 games is much higher than the winning percentage for winning 4 out of 15 games; so winning those 2 out of 2 games must have raised the team's winning percentage. That is, the fraction 6/17 must be bigger than the fraction 4/15. I hope you find this method of comparing fractions useful in at least some problems of this type. Please write back if you have any questions about this method. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/
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