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If d > 0 and 0 < 1 - c/d < 1, which of the following must be

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If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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New post 06 Mar 2014, 03:05
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 134
Category: Algebra Inequalities
Page: 79
Difficulty: 600


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Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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SOLUTION

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

\(0<1-\frac{c}{d}<1\) --> add \(-1\) to all three parts of this inequality --> \(-1<-\frac{c}{d}<0\) --> multiply by \(-1\) and as multiplying by negative, flip signs --> \(1>\frac{c}{d}>0\).

So we have that: \(1>\frac{c}{d}>0\)

I. \(c>0\) --> as \(\frac{c}{d}>0\) and \(d>0\), then \(c>0\). Always true.

II. \(\frac{c}{d}<1\) --> directly given as true.

III. \(c^2 + d^2 > 1\) --> if \(c=1\) and \(d=2\), then YES, but if \(c=0.1\) and \(d=0.2\), then No, hence this one is not always true.

Answer: C (I and II only).
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Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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New post 06 Mar 2014, 03:58
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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


Sol: Given d>0 and 0 < 1 - c/d < 1---------> Eq a

Subtract from Eq a we get -1<-c/d<0

Mulitply by each term by -1 we get 1>c/d>0

Since d>0 and c/d>0 this implies c>0

also we can clearly see c/d> 1. At this stage options A,B and D can be rejected

Consider c^2+d^2> 1

If c=3 and d=5 then c^2+d^2>1
But if c=1/2 and d=3/4 then we have 1/4+9/25 or (25+36)/100 or 61 /100 which is less than 1

Ans is C
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Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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New post 08 Mar 2014, 12:55
SOLUTION

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

\(0<1-\frac{c}{d}<1\) --> add \(-1\) to all three parts of this inequality --> \(-1<-\frac{c}{d}<0\) --> multiply by \(-1\) and as multiplying by negative, flip signs --> \(1>\frac{c}{d}>0\).

So we have that: \(1>\frac{c}{d}>0\)

I. \(c>0\) --> as \(\frac{c}{d}>0\) and \(d>0\), then \(c>0\). Always true.

II. \(\frac{c}{d}<1\) --> directly given as true.

III. \(c^2 + d^2 > 1\) --> if \(c=1\) and \(d=2\), then YES, but if \(c=0.1\) and \(d=0.2\), then No, hence this one is not always true.

Answer: C (I and II only).
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

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Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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If d > 0 and 0 < 1 - c/d < 1, which of the following must be [#permalink]

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New post 30 Aug 2017, 08:59
Answer is C:

0<1-c/d<1
1-c/d>0
-c/d>-1
c/d<1 ________________________- ii point is proved

1-c/d<1
-c/d<0
c/d>0 , d>0 given so c/d to be >0 , implies c>0 ______i point proved

0<c/d<1
c=1 d=2______________c^2+d^2 =5 wow
c=0.1 d=0.2__________c^2 + d^2 = 0.05 ohh no.

hence only 1 and 2 are correct....
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If d > 0 and 0 < 1 - c/d < 1, which of the following must be   [#permalink] 30 Aug 2017, 08:59
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