samichange wrote:
If abcd is not equal to zero, is abcd <0 ?
(1) a/b > c/d
(2) b/a > d/c
APPROACH #1Target question: Is abcd <0 Statement 1: a/b > c/d There are several values of a, b, c and d that satisfy statement 1. Here are two:
Case a: a = 1, b = 1, c = 1, d = 2. Plugging these values into the statement 1 inequality, we get 1/1 > 1/2, which works. In this case, abcd = (1)(1)(1)(2) = 2. So,
abcd > 0Case b: a = 1, b = 1, c = -1, d = 2. Plugging these values into the statement 1 inequality, we get 1/1 > -1/2, which works. In this case, abcd = (1)(1)(-1)(2) = -2. So,
abcd < 0Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: b/a > d/cThere are several values of a, b, c and d that satisfy statement 2. Here are two:
Case a: a = 1, b = 1, c = 2, d = 1. Plugging these values into the statement 2 inequality, we get 1/1 > 1/2, which works. In this case, abcd = (1)(1)(2)(1) = 2. So,
abcd > 0Case b: a = 1, b = 1, c = 2, d = -1. Plugging these values into the statement 2 inequality, we get 1/1 > -1/2, which works. In this case, abcd = (1)(1)(2)(-1) = -2. So,
abcd < 0Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 1 tells us that a/b > c/d
Statement 2 tells us that b/a > d/c
So, the fraction a/b is greater than the fraction c/d
When we invert the two fractions, b/a is also greater than d/c
This should strike us as odd.
In MOST cases, when we invert two fractions in an inequality, the inequality symbol should reverse.
For example, 2/3 < 7/6 and 3/2 > 6/7
Likewise, 1/30 > 1/50 and 30/1 < 50/1
Notice that in my above examples, both fractions are POSITIVE
The same holds true when both fractions are NEGATIVE
For example, -2/3 < -1/6 and -3/2 > -6/1
Likewise, -5/7 > -10/3 and -7/5 < -3/10
The COMBINED statements tell a different story. Here, when we invert the fractions, the inequality symbol does NOT reverse.
This means that it is not the case that both fractions are POSITIVE, and it is not the case that both fractions are NEGATIVE
So,
one fraction must be positive and one fraction must be negative.
In both inequalities, the fractions with the c and d are LESS THAN the fractions with a and b
So, it must be the case that the fractions c/d and d/c are both NEGATIVE
And the fractions a/b and b/a are both POSITIVE
If fractions c/d and d/c are both NEGATIVE, then the product
cd is also NEGATIVEIf fractions a/b and b/a are both POSITIVE, then the product
ab is POSITIVESo,
abcd = (
POSITIVE)(
NEGATIVE) = SOME NEGATIVE VALUE
In other words,
abcd < 0Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
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APPROACH #2Target question: Is abcd <0 We can test values (as I did above) to show that statement 1 alone and statement 2 alone are insufficient.
Statements 1 and 2 combined Statement 1 says: \(\frac{a}{b} > \frac{c}{d}\)
Rewrite as: \(\frac{a}{b} - \frac{c}{d} > 0\)
Find common denominators and combine to get: \(\frac{ad - bc}{bd} > 0\)
Statement 2 says: \(\frac{b}{a} > \frac{d}{c}\)
Rewrite as: \(\frac{d}{c} - \frac{b}{a} < 0\)
Find common denominators and combine to get: \(\frac{ad - bc}{ac} < 0\)
This tells us that \(\frac{ad - bc}{bd}\) is POSITIVE, and \(\frac{ad - bc}{ac} \) is NEGATIVE
So, \((\frac{ad - bc}{bd})(\frac{ad - bc}{ac}) = \) (POSITIVE)(NEGATIVE) \(=\) NEGATIVE
In other words: \(\frac{(ad - bc)^2}{abcd}\) is NEGATIVE
Since \((ad - bc)^2\) is POSITIVE, it must be the case that
abcd is NEGATIVEIn other words,
abcd < 0Since we can answer the
target question with certainty, the combined statements are SUFFICIENT