If \(d > 0\) and \(0 < 1 - \dfrac{c}{d} < 1,\) which of the following must be true?
I. \(c > 0\)
II. \(\dfrac{c}{d} < 1\)
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
Answer: C
Source: Official Guide
If \(d > 0\) and \(0 < 1 - \dfrac{c}{d} < 1,\) which of the following must be true?
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Solution:Gmat_mission wrote: ↑Thu Apr 29, 2021 7:34 amIf \(d > 0\) and \(0 < 1 - \dfrac{c}{d} < 1,\) which of the following must be true?
I. \(c > 0\)
II. \(\dfrac{c}{d} < 1\)
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
Answer: C
Source: Official Guide
Let’s simplify the double inequality:
0 < 1 - c/d < 1
-1 < -c/d < 0
Multiplying both sides by -d (which is a negative quantity since d > 0), we have:
d > c > 0
We see that c > 0, so statement I is true. Now, dividing both sides by d, we have:
1 > c/d > 0
We see that c/d < 1, so statement II is true.
However, statement III might not be true. For example, if d = 0.4 and c = 0.3, then c^2 + d^2 = 0.09 + 0.16 = 0.25, which is not greater than 1.
Answer: C
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