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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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Updated on: 17 Jul 2018, 04:26
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Difficulty:

65% (hard)

Question Stats:

58% (01:21) correct 42% (01:23) wrong based on 741 sessions

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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-d-0-and-0 ... 68429.html

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Cheers, Rahul.

Originally posted by rahulraao on 03 Sep 2005, 06:16.
Last edited by Bunuel on 17 Jul 2018, 04:26, edited 2 times in total.
Edited the question and added the OA
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Posts: 273
Location: CA, USA
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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03 Sep 2005, 13:13
0 < 1 - c/d < 1 => 0<c/d<1
with d>0

we get: c>0 and c/d<1

not true for c^2 + d^2 > 1
example: c=0.1 d=0.2

C
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Posts: 47967
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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01 Apr 2012, 13:09
14
10
If $$d >0$$ and $$0 < 1 - \frac{c}{d} < 1$$, which of the following must be true?

I. $$c > 0$$
II. $$\frac{c}{d} < 1$$
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[/quote]

$$0<1-\frac{c}{d}<1$$ --> add $$-1$$ to all three parts of this inequality --> $$-1<-\frac{c}{d}<0$$ --> mutliply 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, hences this one is not always true.

Answer: C (I and II only).

Hope it's clear.
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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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01 Apr 2012, 14:27
Inequalities !!
is it possible to get some tips and tools to solve these "sign" (< = > ) questions?
Math Expert
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Posts: 47967
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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01 Apr 2012, 15:12
2
4
kashishh wrote:
Inequalities !!
is it possible to get some tips and tools to solve these "sign" (< = > ) questions?

DS questions on inequalities: search.php?search_id=tag&tag_id=184
PS questions on inequalities: search.php?search_id=tag&tag_id=189

Hard inequality and absolute value questions with detailed solutions check this: inequality-and-absolute-value-questions-from-my-collection-86939-40.html

x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.
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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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02 Apr 2012, 18:18
1
kashishh wrote:
Inequalities !!
is it possible to get some tips and tools to solve these "sign" (< = > ) questions?

i dunno if my method can help you but i solved it in two ways.

0 < 1 - (c/d) < 1

the way i looked at this problem was to first simply the equation by subtracting 1 in the middle to both sides:
-1 < -(c/d) < 0

and the given statement is that d is greater than zero, so d must be positive. if we know a variable is positive, you can safely multiply the variable without changing the signs. so i multiplied d to isolate c.
-1(d) < -c < 0(d)

so we get
-d < -c < 0

and then i divided the negative out and switched the signs.
d > c > 0

the above answers statement I. c > 0 is true.
for statement II, i manipulated the above equation by dividing d and i get
1 > c/d > 0
this answers statement II. c/d < 1

the last statement i just tested numbers using the d > c > 0 by using whole numbers and decimals.

the second way i solved this was using bunuel's way.
Math Expert
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Posts: 47967
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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19 Jun 2013, 04:51
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

All Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

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Concentration: Strategy, Finance
GMAT 1: 710 Q49 V39
WE: Engineering (Energy and Utilities)
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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19 Jun 2013, 11:29
1
0 < 1 - c/d < 1

Multiply by d,
0 < d - c < d

Implies that c>0

Now, d - c > 0 implies that d>c and hence c/d < 1

information we now have is c>0, d>0 and d>c

c^2 + d^2 > 0 but it may or may not be greater than 1.

Hence I & 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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21 Sep 2013, 16:58
1
$$0<1-\frac{c}{d}<1$$

First Part:
$$0<1-\frac{c}{d}$$
$$\frac{c}{d}<1$$ This satisfies Choice II

Second Part:
$$1-\frac{c}{d}<1$$
$$0<\frac{c}{d}$$ This satisfies Choice I as $$d>0$$

As for Choice III, if c and d are decimals then this gives us "False" but if they are integers, then "True"
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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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08 Nov 2014, 17:16
1
Hey Guys

May I raise a small question regarding the third statement? If it would be INSTEAD of c^2 + d^2 > 1 be c^2 + d^2 > 0 , would the third statement be considered sufficient? My thinking is as follows:

1. as we know (out of proven first statement) that c>0: multiply the IE with itself leads to: c^2 >0
2. as we know (out of the question stem) that d>0: multiply the IE with itself leads to: d^2 >0
3. Combining the two IE from above by adding them up: c^2 + d^2 >0
4. The MODIFIED Statement 3 is considered sufficient

Would highly appreciate your inputs on my thought...

Cheers, Rene
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Posts: 47967
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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09 Nov 2014, 05:19
DarkMight wrote:
Hey Guys

May I raise a small question regarding the third statement? If it would be INSTEAD of c^2 + d^2 > 1 be c^2 + d^2 > 0 , would the third statement be considered sufficient? My thinking is as follows:

1. as we know (out of proven first statement) that c>0: multiply the IE with itself leads to: c^2 >0
2. as we know (out of the question stem) that d>0: multiply the IE with itself leads to: d^2 >0
3. Combining the two IE from above by adding them up: c^2 + d^2 >0
4. The MODIFIED Statement 3 is considered sufficient

Would highly appreciate your inputs on my thought...

Cheers, Rene

______________
Yes, that's correct.
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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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09 Nov 2014, 06:18
Bunuel, thanks very much for the feedback
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GMAT 1: 630 Q42 V35
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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21 Nov 2014, 08:42
Bunuel I have just one confusion here, 1> c/d > 0, however st 2 says c/d <1, doesnt the statement also include the cases where c/d<0, where the inequality wont hold so it isnt a MUST BE TRUE statement.
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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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21 Nov 2014, 09:50
1
Bunuel I have just one confusion here, 1> c/d > 0, however st 2 says c/d <1, doesnt the statement also include the cases where c/d<0, where the inequality wont hold so it isnt a MUST BE TRUE statement.

We got that $$0 < \frac{c}{d} < 1$$. So, the answer to the question whether $$\frac{c}{d} < 1$$ is true, is YES. How else?
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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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07 Sep 2015, 05:13
1
Bunuel wrote:
I have just one confusion here, 1> c/d > 0, however st 2 says c/d <1, doesnt the statement also include the cases where c/d<0, where the inequality wont hold so it isnt a MUST BE TRUE statement.

We got that $$0 < \frac{c}{d} < 1$$. So, the answer to the question whether $$\frac{c}{d} < 1$$ is true, is YES. How else?

I think what he meant was a different thing.

He assumes that c/d can also be negative. This is not possible, because c has to be bigger than 0 and d is positive (given).

Therefore c/d has to be smaller than 1 (and is NEVER negative).

Hope this helps people who struggeld with this part (like I did )
Math Expert
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Posts: 47967
Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be  [#permalink]

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17 Jul 2018, 04:27
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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-d-0-and-0 ... 68429.html
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Re: If d > 0 and 0 < 1 - c/d < 1, which of the following must be &nbs [#permalink] 17 Jul 2018, 04:27
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