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655-705 Level|   Inequalities|   Must or Could be True Questions|                     
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Bunuel
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If d > 0 and 0 < 1 - c/d < 1, which of the following must be true? 15 Feb 2023, 08:43
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tabish314
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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).


Hi Chetan,
I have the same confusion as Sahil. Isnt what you just said makes II must be true if II would have said 0<c/d<1.
Statement II just asks if c/d >1 must be true for 0 < 1-(c/d) < 1
I didnt understand how statement II simply assumed that <1 is >0.
Had statement II stated that c/d>0, would we assume that >0 means <1 ?
Show more

So, you ask if we got \(0<\frac{c}{d}<1\), why is II. \(\frac{c}{d}<1\) true?

For any number from the range we got (\(0<\frac{c}{d}<1\)) will be true to say that it's less than 1, so II must be true.

Hope it's clear.
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If d > 0 and 0 < 1 - c/d < 1, which of the following must be true? 15 Feb 2023, 13:30
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If d > 0 and 0 < 1 - c/d < 1, which of the following must be true?

III. c^2 + d^2 > 1
Show more

An alternate line of reasoning for Statement III:

\(c^2+d^2=1 \) is the equation for a circle centered at the origin with a radius of 1.
Statement III implies that (c, d) must lie OUTSIDE this circle.
If c=0.1 and d=0.2, then (c, d) will clearly lie INSIDE this circle, indicating that Statement III does not have to true.
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If d > 0 and 0 < 1 - c/d < 1, which of the following must be true? 17 Mar 2025, 13:22
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Hi Brent, c/d is less than one but greater than 0. In option it says c/d<1, and less than 1 which also includes negatives which wont be true. Then why are we also choosing this option?
BrentGMATPrepNow
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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. c2 + d2 > 1

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II and III

Take the inequality: 0 < 1 - c/d < 1
Subtract 1 from all sides: -1 < -c/d < 0
Multiply all sides by -1 to get: 1 > c/d > 0 [when we multiply an inequality by a NEGATIVE value, we must REVERSE the direction of the inequality symbols]
Rearrange to get: 0 < c/d < 1

We can immediately see that statement II (c/d < 1) is true

Also, since 0 < c/d, we know that c/d is positive.
Since we're told d is positive, we can conclude that c must also be positive
In other words, c > 0, which means statement I is true.

Now let's analyze statement III
Notice that the values c = 0.1 and d = 0.2 satisfy all of the given information.
Now recognize that c2 + d2 = 0.12 + 0.22 = 0.01 + 0.04 = 0.05
So, it's not necessarily the case that c2 + d2 > 1
In other words, statement III need not be true.

Answer: C
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If d > 0 and 0 < 1 - c/d < 1, which of the following must be true? 17 Mar 2025, 14:14
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Hi Brent, c/d is less than one but greater than 0. In option it says c/d<1, and less than 1 which also includes negatives which wont be true. Then why are we also choosing this option?
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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. c2 + d2 > 1

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II and III

Take the inequality: 0 < 1 - c/d < 1
Subtract 1 from all sides: -1 < -c/d < 0
Multiply all sides by -1 to get: 1 > c/d > 0 [when we multiply an inequality by a NEGATIVE value, we must REVERSE the direction of the inequality symbols]
Rearrange to get: 0 < c/d < 1

We can immediately see that statement II (c/d < 1) is true

Also, since 0 < c/d, we know that c/d is positive.
Since we're told d is positive, we can conclude that c must also be positive
In other words, c > 0, which means statement I is true.

Now let's analyze statement III
Notice that the values c = 0.1 and d = 0.2 satisfy all of the given information.
Now recognize that c2 + d2 = 0.12 + 0.22 = 0.01 + 0.04 = 0.05
So, it's not necessarily the case that c2 + d2 > 1
In other words, statement III need not be true.

Answer: C
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The question asks us to select statements that must be true. The second statement does not explicitly say that c/d < 0, it's an assumption you are making, if something is less than 1, it isn't necessarily less than 0

Think of it this way, if someone asks you, c/d < 1, must be true or not? Answer in yes or no; what will your answer be?
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