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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 non-negative rational numbers. Thanks from shunya
December 24th, 2015, 01:21 AM   #3
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Quote:
 Originally Posted by Azzajazz 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 non-negative rational numbers.
As you said this a/b < c/d IFF ad < bc is true only if b > 0 and d > 0.

 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 right-hand 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. Thanks from shunya
 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
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Quote:
 Originally Posted by shunya As you said this a/b < c/d IFF ad < bc is true only if b > 0 and d > 0.
No. It is also true if both b<0 and d< 0, then a/b< c/d if ad< bc.

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