Is a/b

Question

If a, b, c, and d, are positive numbers, is a/b < c/d? 1) 0 < (c-a) / (d-b) 2) (ad/bc)^2 < (ad)/(bc)

eagleeye · Answer

catty2004 wrote:If a, b, c, and d, are positive numbers, is a/b < c/d?

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

2) (ad/bc)^2 < (ad)/(bc)

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)/(bd)<0
=> ad-bc<0 (b and d are positive, Hence multiplying both sides by bd doesn't change the sign)
=> ad<bc
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.

2) (ad/bc)^2 <(ad/bc)
Since ad/bc is positive, dividing both sides by ad/bc
ad/bc <1 => ad < bc. So we get that ad < bc. Hence ad is definitely not greater than bc. Sufficient.

Hence B is the correct answer.

KapTeacherEli · Answer

catty2004 wrote:If a, b, c, and d, are positive numbers, is a/b < c/d?

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

2) (ad/bc)^2 < (ad)/(bc)

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)/(bd)<0
=> ad-bc<0 (b and d are positive, Hence multiplying both sides by bd doesn't change the sign)
=> ad<bc
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.

2) (ad/bc)^2 <(ad/bc)
Since ad/bc is positive, dividing both sides by ad/bc
ad/bc <1 => ad < bc. So we get that ad < bc. Hence ad is definitely not greater than bc. Sufficient.

Hence B is the correct answer.

Nina1987 · Answer

Is it possible to prove St1 insufficient w/o resorting to number picking? thanks

Matt@VeritasPrep · Answer

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).

ceilidh.erickson · Answer

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:
ad < bc

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

[color=]Statement 2: (ad/bc)² < (ad)/(bc) [/color]
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.
ad/bc < 1
Multiply both sides by bc:
[color=]ad < bc [/color]
This is our target question. Sufficient.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

Scott@TargetTestPrep · Answer

catty2004 wrote: ↑Sun Jul 08, 2012 5:19 pm If a, b, c, and d, are positive numbers, is a/b < c/d? 1) 0 < (c-a) / (d-b) 2) (ad/bc)^2 < (ad)/(bc) Solution: Question Stem Analysis: We need to determine whether a/b < c/d given that a, b, c and d are positive. Since all quantities are positive, we can cross-multiply, obtaining ad < bc. In other words, we can rewrite the question as: Is ad < bc ? Statement One Alone: This is not sufficient. For example, if a = b = 1, c = 3 and d = 2, then ad = 2 and bc = 3, and the answer to the question is yes.. However, if a = 3, b = 2, and c = d = 1, we see that ad = 3 and bc = 2, and the answer to the question is no.. Statement Two Alone: The square of a quantity is less than itself if the quantity has a value between 0 and 1. Therefore, ad/bc < 1. Since bc is positive, multiplying both sides by bc, we have ad < bc. Statement two alone is sufficient. Answer: B

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