Although, I got this question right while I was solving the problem, it took me about 4 mins to reach the answer. Any faster way of reaching the solution to problems such as listed below will be REALLY helpful:
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
Approach:
d>0: Let d=1/2, 1, 2
c>0 (?) True
d=1/2: 0 < 1-2c < 1
=> 0 < 1-2c => c < 1/2
1-2c < 1 => -2c < 0 => 2c > 0 => c > 0 . True
d=1: 0 < 1-c < 1
0 < 1-c => c < 1
1-c < 1 => -c < 0 => c > 0. True
[II] (c/d) > 1 (?) True
d=1/2: 2c > 1 => c > 1/2
c/d = 1/1/2 = 2 > 1. True
d=1: c > 1
c/d = 2/1 = 2 > 1. True
[III] c^2 + d^2 > 1 (?) False
d=1/2 and c > 0: 1 + (1/4) = 5/4 > 1. True
d=1 and c > 0: (1/4) + (1/4) = 1/2 < 1. False
Hence [spoiler] and [II] --> [C][/spoiler]
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
Approach:
d>0: Let d=1/2, 1, 2
c>0 (?) True
d=1/2: 0 < 1-2c < 1
=> 0 < 1-2c => c < 1/2
1-2c < 1 => -2c < 0 => 2c > 0 => c > 0 . True
d=1: 0 < 1-c < 1
0 < 1-c => c < 1
1-c < 1 => -c < 0 => c > 0. True
[II] (c/d) > 1 (?) True
d=1/2: 2c > 1 => c > 1/2
c/d = 1/1/2 = 2 > 1. True
d=1: c > 1
c/d = 2/1 = 2 > 1. True
[III] c^2 + d^2 > 1 (?) False
d=1/2 and c > 0: 1 + (1/4) = 5/4 > 1. True
d=1 and c > 0: (1/4) + (1/4) = 1/2 < 1. False
Hence [spoiler] and [II] --> [C][/spoiler]
Want to Beat GMAT.
Always do what you're afraid to do. Whoooop GMAT
Always do what you're afraid to do. Whoooop GMAT
An excellent technique.
