If d > 0 and 0 < 1 - c/d < 1, which of the following must be true?
I. c > 0
II. c/d < 1
III. c^2 + d^2 > 1
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II and III
[spoiler]OA: C[/spoiler]
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Fractions problem
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tutorphd
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I would classify this as inequality problem with number properties (positive/negative), it has no fraction manipulation in it.
The double inequality 0 < 1 - c/d < 1 simplifies to 1 > c/d > 0 which implies that II is true.
c/d>0 means c and d have the same sign, so if d>0, then c>0, which is I itself.
You can show III is not always true by plugging numbers: keep the ratio c/d<1 but choose very small or very big c and d.
Ans: C
The double inequality 0 < 1 - c/d < 1 simplifies to 1 > c/d > 0 which implies that II is true.
c/d>0 means c and d have the same sign, so if d>0, then c>0, which is I itself.
You can show III is not always true by plugging numbers: keep the ratio c/d<1 but choose very small or very big c and d.
Ans: C
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GMATGuruNY
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0 < 1 - c/d < 1 [Subtract 1 from each part]Troika wrote:If d > 0 and 0 < 1 - c/d < 1, which of the following must be true?
I. c > 0
II. c/d < 1
III. c^2 + d^2 > 1
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II and III
[spoiler]OA: C[/spoiler]
-1 < -c/d < 0 [Multiply through by -1, remembering to switch the direction of the inequality signs]
1 > c/d > 0 [Multiply by d -- since d>0, no need to worry about switching the direction of the inequality signs]
d > c > 0.
The portions in red show that both II and I must be true.
Eliminate A, B and D.
If c and d are fractions between 0 and 1, then III does not have to be true.
To illustrate:
If d = 1/2 and c = 1/4, then c² + d² = (1/4)² + (1/2)² = 1/16 + 1/4 = 5/16, which is not greater than 1.
Eliminate E.
The correct answer is C.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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