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December 23rd, 2015, 07:48 PM  #1 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18  A fraction problem
The book I'm reading provides the following definition of "less than" and "equal to" for rational numbers x/y: 1) a/b < c/d IFF ad < bc 2) a/b = c/d IFF ad = bc Then it asks a question as follows: True or false? 15/(9) > (13)/4 To answer my question I use one of the definitions above i.e. I cross multiply: (15)(4) = 60 (9)(13) = 117 So according to the definition 15/(9) < (13)/4 since 15*4 < (9)*(13) But if you do division you get 15/(9) = 1.6666... and (13)/4 = 3.25 But 1.666...> 3.25 So 15/(9) > (13)/4 How come then that doing it one way results in 15/(9) < (13)/4 and doing it in another way results in 15/(9) > (13)/4? Thanks 
December 23rd, 2015, 08:01 PM  #2 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
You're cross multiplying be a negative number, so the inequality sign should flip. I think the books definition of less than is only for nonnegative rational numbers.

December 24th, 2015, 01:21 AM  #3 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18  
December 24th, 2015, 02:05 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,629 Thanks: 2077 
The a/b < c/d IFF ad < bc rule applies whenever bd > 0, so it applies if b and d are both negative. To apply this to 15/(9) > (13)/4, rewrite the righthand side as 13/(4). 15/(9) > 13/(4) if and only if 15(4) > 13(9), i.e., if and only if 60 > 117, so the original inequality is true. 
December 29th, 2015, 09:36 AM  #5 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 390 Thanks: 70 
15/9 = (15/9) and 13/4 = (13/4) Cross multiply without the negative signs: 60<117 implies 15/9 < 13/4 But with negative signs, the greater number becomes lesser. Thus: (15/9)> (13/4) This implies: 15/9 > 13/4 
December 29th, 2015, 09:43 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  

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