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Math Expert V
Joined: 02 Sep 2009
Posts: 53796
If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 52% (02:14) correct 48% (02:15) wrong based on 766 sessions

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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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Math Expert V
Joined: 02 Sep 2009
Posts: 53796
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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1
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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Senior Manager  P
Joined: 29 Jun 2017
Posts: 454
GPA: 4
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If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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1

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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Joined: 04 May 2014
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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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0<1-c/d<1

0<d-c/d<1

0<d-c<d

Ie d is >d-c and both d and c are positive (if c is -ve then d-c cannot be <d) and d is>c.

c>0 proved above

c/d<1 a smaller postive number divided by a larger +ve no will result in a decimal-Possible

c^2+d^2>1 if c and d are decimals then this inequality is not true.

Only 1 and II are a 'must' case.
Intern  B
Joined: 16 Oct 2017
Posts: 37
If d > 0 and 0 < 1 - c/d < 1, which of the following must be tru  [#permalink]

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Bunuel, can we plug in smart numbers for statement #1? When I plug in a "2" for "d", it turns out to be false.

0 < 1 - c/2 < 1

According to I, C is supposed to be a positive number. This statement works when C = 1, but it doesn't work when C = 2.

Plugging in numbers works with #2, making it true.
Math Expert V
Joined: 02 Sep 2009
Posts: 53796
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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OCDianaOC wrote:
Bunuel, can we plug in smart numbers for statement #1? When I plug in a "2" for "d", it turns out to be false.

0 < 1 - c/2 < 1

According to I, C is supposed to be a positive number. This statement works when C = 1, but it doesn't work when C = 2.

Plugging in numbers works with #2, making it true.

It's the other way around. We are given that $$d >0$$ and $$0 < 1 - \frac{c}{d} < 1$$ are true. If these two inequalities are true, then c will be positive. So you should consider only those numbers which satisfy $$d >0$$ and $$0 < 1 - \frac{c}{d} < 1$$. For c and d, which satisfy $$d >0$$ and $$0 < 1 - \frac{c}{d} < 1$$, c will turn out to be positive.
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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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rahulraao 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

Simplifying the given inequality we have:

0 < 1 - c/d < 1

-1 < -c/d < 0

-d < -c < 0

d > c > 0

So we see that c must be positive but less than d. Thus, I and II must be true. III does not have to be true since c and d could be some proper fractions, say both less than 1/2.

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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] 28 Feb 2019, 10:09
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