Here's the original question
If ab > cd and a,b,c and d are all greater than zero, which of the following cannot be true?
a) c > b
b) d > a
c) b/c < d/a
d) a/c > d/b
e)(cd)² < (ab)²
For this question, we can either ELIMINATE any answer choice that CAN be true, or we can look for a statement that can NEVER be true.
a) c > b
This can be true (if a=8, b=1, c=3, d=2), so we'll eliminate A
b) d > a
This can be true (if a=1, b=8, c=3, d=2), so we'll eliminate B
c) b/c < d/a
If we multiply both sides by c and by a, we get ab < cd,
BUT we're told that ab > cd
So, it can never be true that b/c < d/a, which means C must be the correct answer.
Nevertheless, let's check the remaining answer choices for "fun".
Aside: since a and c are both positive, it's perfectly acceptable to multiply both sides of the inequality by a and c. Had we not been certain that a and c are positive, it would not be acceptable to multiply both sides by a and c.
d) a/c > d/b
If we multiply both sides by c and by b, we get ab > cd.
This confirms the given information (that ab > cd)
Since the statement that a/c > d/b must be true, we'll eliminate D
e)(cd)² < (ab)²
This can be true (if a=1, b=8, c=3, d=2), so we'll eliminate E
Aside: We can also use a helpful rule to show why E it always true. The rule that says,
If 0 < x < y, then 0 < x² < y²
Since ab > 0 and cd > 0, and since 0 < cd < ab, then it must be true that 0 < (cd)² < (ab)²
Answer =
C
Cheers,
Brent
