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
What is the significance of the statement#1 starting with 0 which is an unusual notation.
Secondly, is there an algebraic way (not plugging in numbers as the OG explains) to prove statement#1 insufficiency.
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OG Q#92
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Statement 1: (c-a) / (d-b) > 0If 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)
Make c and d both greater than a and b.
Try two cases:
c > d, so that c/d > 1.
c < d, so that c/d < 1.
Case 1: a=1, b=1, c=3, and d=2.
In this case, a/b = 1 and c/d = 3/2, so a/b < c/d.
Case 2: a=1, b=1, c=2, and d=3.
In this case, a/b = 1 and c/d = 2/3, so a/b > c/d.
INSUFFICIENT.
Statement 2: (ad/bc)² < (ad)/(bc)
Since all of the values are positive, we can rephrase the question stem by cross-multiplying:
a/b < c/d
ad < bc.
Question stem rephrased: Is ad < bc?
Since all of the values are positive, we can divide each side of statement 2 -- (ad/bc)² < (ad)/(bc) -- by ad/bc, yielding the following:
ad/bc < 1
ad < bc.
SUFFICIENT.
The correct answer is B.
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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
Explanation :
Statement 1:
c-a/d-b > 0 : P/Q > 0 means either both are positive or negative for the ratio to be positive.
Consider the case when the numerator and denominator are positive : so, c-a > 0 and d-b > 0 which means c>a and d>b. Does this mean you can divide the two inequalities? No.
Does that tell us anything about the relationship between the ratio a/b and c/d ? Not really.
So statement 1 is insufficient.
A food for thought is if there are two inequalities of the same sign, which all operations can be performed with the two inequalities: add, multiply, subtract, divide?And under what conditions?
Statement 2 :
Since both the sides of the inequality are positive(since a,b,c and d are positive), we can multiply both the sides by bc/ad, and we get:
ad/bc < 1
ad < bc
similarly, dividing both the sides by bd, we get :
a/b < c/d
Hence, Statement 2 is sufficient.
Answer is B
1) 0 < c-a/d-b
2) (ad/bc)^2 < ad/bc
Explanation :
Statement 1:
c-a/d-b > 0 : P/Q > 0 means either both are positive or negative for the ratio to be positive.
Consider the case when the numerator and denominator are positive : so, c-a > 0 and d-b > 0 which means c>a and d>b. Does this mean you can divide the two inequalities? No.
Does that tell us anything about the relationship between the ratio a/b and c/d ? Not really.
So statement 1 is insufficient.
A food for thought is if there are two inequalities of the same sign, which all operations can be performed with the two inequalities: add, multiply, subtract, divide?And under what conditions?
Statement 2 :
Since both the sides of the inequality are positive(since a,b,c and d are positive), we can multiply both the sides by bc/ad, and we get:
ad/bc < 1
ad < bc
similarly, dividing both the sides by bd, we get :
a/b < c/d
Hence, Statement 2 is sufficient.
Answer is B
