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December 23rd, 2015, 07:48 PM   #1
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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?

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December 23rd, 2015, 08:01 PM   #2
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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.
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December 24th, 2015, 01:21 AM   #3
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Quote:
Originally Posted by Azzajazz View Post
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.
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December 24th, 2015, 02:05 AM   #4
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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.
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December 29th, 2015, 09:36 AM   #5
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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
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December 29th, 2015, 09:43 AM   #6
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Quote:
Originally Posted by shunya View Post
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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