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# If 0 < a < b < c, which of the following statements must be true?

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If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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14 Jun 2016, 14:08
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If 0 < a < b < c, which of the following statements must be true?

I. 2a > b + c
II. c – a > b - a
III. $$\frac{c}{a}$$ < $$\frac{b}{a}$$

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

OG 2017 New Question
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If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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Updated on: 03 Mar 2018, 14:51
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Top Contributor
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AbdurRakib wrote:
If 0 < a < b < c, which of the following statements must be true?

I. 2a > b + c
II. c – a > b - a
III. $$\frac{c}{a}$$ < $$\frac{b}{a}$$

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

OG 2017 New Question

I. 2a > b + c
Consider this scenario: a = 1, b = 2 and c = 3. This meets the given condition that 0 < a < b < c.
HOWEVER, if we plug these values into statement I, we see that it is NOT the case that 2a > b + c
So, statement I NEED NOT BE TRUE

II. c – a > b - a
It's already given that c > b
If we subtract ANY VALUE (such as a) from both sides, the inequality remains valid.
So, statement II MUST BE TRUE

III. c/a < b/a
Consider this scenario: a = 1, b = 2 and c = 3. This meets the given condition that 0 < a < b < c.
HOWEVER, if we plug these values into statement III, we see that it is NOT the case that c/a < b/a
So, statement III NEED NOT BE TRUE

Cheers,
Brent
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Originally posted by BrentGMATPrepNow on 14 Jun 2016, 14:39.
Last edited by BrentGMATPrepNow on 03 Mar 2018, 14:51, edited 1 time in total.
##### General Discussion
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Joined: 01 May 2015
Posts: 35
Re: If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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16 Jun 2016, 04:39
4
I. obvously we cannot say that it "must" be true.

II. c – a > b - a
Cancelling a on both sides
c > b
Definitely true as per question.

III. c/a < b/a
Multiply both sides by "a" (positive)
c < b
Definitely "not" true as per question.

So, only II is correct.
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Re: If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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24 Jun 2016, 13:21
Hey,

For (1),
0<a<b<c, can't we write it as
a<b....i
a<c...ii

2a<b+c ???? (addition property of inequalities)

Or
Is it like this property holds true only when abcd are different numbers??

Appreciate a clarification!

TIA.
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Re: If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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26 Jun 2016, 01:31
Nothing indicates what a may be greater than, so I is wrong.
III is always wrong because the order should be reversed.

II is correct.

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Re: If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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09 Mar 2019, 07:40
[1] Assign Test Values

Must follow rule: 0 < a < b < c,

a = 2
b = 3
c = 4

[2] Plug In & Calculate

$$I. 2a > b + c \rightarrow 2(2) > 3 + 4 \rightarrow 4 > 7 \space (FALSE)$$

$$II. c – a > b - a \rightarrow 4 - 2 > 3 - 2 \rightarrow 2 > 1 \space (TRUE)$$

$$III. \frac{c}{a} < \frac{b}{a} \rightarrow \frac{4}{2} < \frac{3}{2} \rightarrow 2 < 1.5 \space (FALSE)$$

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Re: If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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16 Apr 2019, 06:33
Shouldn't the question say a,b,c are integers?
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Re: If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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11 May 2019, 22:57
AbdurRakib wrote:
If 0 < a < b < c, which of the following statements must be true?

I. 2a > b + c
II. c – a > b - a
III. $$\frac{c}{a}$$ < $$\frac{b}{a}$$

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

OG 2017 New Question

We can let smart numbers
a=2
so,b=3
c=4
Now, we'll apply on each options given☺
And we get option B(||) only matches

Posted from my mobile device
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Re: If 0 < a < b < c, which of the following statements must be true?  [#permalink]

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04 Mar 2020, 07:12
AbdurRakib wrote:
If 0 < a < b < c, which of the following statements must be true?

I. 2a > b + c
II. c – a > b - a
III. $$\frac{c}{a}$$ < $$\frac{b}{a}$$

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

OG 2017 New Question

I. 2a > b + c
Taking numbers a = 1, b = 2 and c = 3 this is not true

II. c – a > b - a
In 0 < a < b < c, subtracting a from each
0 - a < a - a < b - a < c - a
Hence c - a > b - a True

III. $$\frac{c}{a}$$ < $$\frac{b}{a}$$
As b < c
dividing by 'a' which is positive doesn't impact inequality
Hence $$\frac{b}{a} < \frac{c}{a}$$
not true

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Re: If 0 < a < b < c, which of the following statements must be true?   [#permalink] 04 Mar 2020, 07:12