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If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?
I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
06 Apr 2023, 03:54
Kudos
Bookmarks
Official Solution:
If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?
I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Given that the product of five numbers, \(a\), \(b\), \(c\), \(d\), and \(e\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(a > b > c > d > e\), we can have the following three cases:
\(a\; |\; b\; |\; c\; |\; d\; |\; e\)
\(+ | + | + | + | +\) (no negative numbers)
\(+ | + | + | - | -\) (two negative numbers)
\(+ | - | - | - | -\) (four negative numbers)
Let's analyze each option, taking into consideration that the question asks which of them MUST be true, not COULD be true.
I. \(ab > 0\)
If we have the third case, then this option is not true. Eliminate.
II. \(bc > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
III. \(de > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
Consequently, only options II and III are always true, and thus the answer is D.
Answer: D
If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?
I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Given that the product of five numbers, \(a\), \(b\), \(c\), \(d\), and \(e\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(a > b > c > d > e\), we can have the following three cases:
\(a\; |\; b\; |\; c\; |\; d\; |\; e\)
\(+ | + | + | + | +\) (no negative numbers)
\(+ | + | + | - | -\) (two negative numbers)
\(+ | - | - | - | -\) (four negative numbers)
Let's analyze each option, taking into consideration that the question asks which of them MUST be true, not COULD be true.
I. \(ab > 0\)
If we have the third case, then this option is not true. Eliminate.
II. \(bc > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
III. \(de > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
Consequently, only options II and III are always true, and thus the answer is D.
Answer: D
