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Is a/b < c/d?

This topic has 3 expert replies and 2 member replies
catty2004 Newbie | Next Rank: 10 Posts
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Is a/b < c/d?

Sun Jul 08, 2012 5:19 pm
Elapsed Time: 00:00
• Lap #[LAPCOUNT] ([LAPTIME])
If a, b, c, and d, are positive numbers, is a/b < c/d?

1) 0 < (c-a) / (d-b)

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eagleeye Legendary Member
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Sun Jul 08, 2012 5:39 pm
catty2004 wrote:
If a, b, c, and d, are positive numbers, is a/b < c/d?

1) 0 < (c-a) / (d-b)

We are told that a,b,c,d are positive integers, we need to find if a/b < c/d
Now:
a/b < c/d
=> a/b - c/d < 0
=> ad-bc<0 (b and d are positive, Hence multiplying both sides by bd doesn't change the sign)
So we need to find it ad < bc

With that in mind let's look at the options:
1) 0 < (c-a) / (d-b)
This means that both c-a and d-b have the same sign:
If both are positive c-a>0, and d-b>0 => c>a and d>b. So we can't say anything about ad>bc.
We need not check the both less than 0 case now, because the both are positive case already led us to Insufficient data condition. Hence Insufficient.

Hence B is the correct answer.

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KapTeacherEli GMAT Instructor
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Sun Jul 08, 2012 5:46 pm
catty2004 wrote:
If a, b, c, and d, are positive numbers, is a/b < c/d?

1) 0 < (c-a) / (d-b)

Hi catty,

We're looking for whether a/b < c/d. Fortunately, we're told a useful bit of info in the question stem. All four terms are positive. That's very important with inequalities, because it means that we can multiply and divide without having to worry about the direction of the inequality signs. In this case, we could rephrase the question to whether ad < bc by cross-multiplying. This will be useful laters.

Statement 1) is not useful, however. (c-a) and (d-b) could both be positive or negative; that means that when me multiply to get rid of a term, we might or might not have to flip the terms. Since any of the variables could be greater or less than any of the other variables, this statement is insufficient.

Statement 2) gives us exactly what we want. Here, with no subtraction, everything stays positive. That means we can divide out (ad/bc) from both sides without flipping the inequality. We get ad/bc < 1, and can cross-multiply to get ad < bc. That answers our question with a definite yes, so it's sufficient and the answer is (B)

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Nina1987 Senior | Next Rank: 100 Posts
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Sun Dec 27, 2015 7:05 am
Is it possible to prove St1 insufficient w/o resorting to number picking? thanks

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Matt@VeritasPrep GMAT Instructor
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Sun Dec 27, 2015 5:16 pm
There is, but it's a little time consuming. I think S1 is mostly there to test your understanding of fractions; i.e. whether you recognize that (c/d) - (a/b) is NOT the same thing as (c - a)/(d - b).

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ceilidh.erickson GMAT Instructor
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Mon Oct 09, 2017 7:45 am
Nina1987 wrote:
Is it possible to prove St1 insufficient w/o resorting to number picking? thanks
This problem can be solved without choosing numbers, if you rephrase the question:
a/b < c/d --> Because all of these variables are positive, we're allowed to cross-multiply here:

Target question: is ad < bc?
Often it's easier to deal with products than with ratios on DS questions.

Statement 1: 0 < (c-a)/(d-b)
Here, we can't cross-multiply, because we don't know if (d - b) is positive or negative. Our question was asking us about ratios/products... in other words, PROPORTIONAL relationships. This statement is giving us information about DIFFERENCES, which cannot answer a proportion question. (You can certainly test numbers to prove the point, though, as Mitch pointed out).
Insufficient

Here, we're given a proportion, which is already more helpful. Let's rephrase:
(ad/bc)² < (ad)/(bc) Because everything is positive, we know that all products and ratios will be positive. If the square of the ratio (ad/bc) is less than the ratio itself, what does that mean? It means the ratio must be a POSITIVE FRACTION.
Multiply both sides by bc:
This is our target question. Sufficient.

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