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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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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

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[Reveal] Spoiler: OA

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

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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

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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

Answer:
[Reveal] Spoiler:
B


Here are two videos that cover some important rules about inequalities on the GMAT:


and

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

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New post 16 Jun 2016, 04:39
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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New post 24 Jun 2016, 13:21
Hey,

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

adding i and 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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New post 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.

Answer choice B 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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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


We are given that 0 < a < b < c and need to determine which statements are true. Let’s analyze each Roman numeral.

I. 2a > b + c

2a cannot be greater than the sum of b and c. Since a + a = 2a, a < b, and a < c, a + a < b + c, or 2a < b + c. Statement I is FALSE.

II. c – a > b - a

We can simplify the inequality to c > b. Since we are given that c is greater than b in the stem, c – a is greater than b - a. Statement II is TRUE.

III. c/a < b/a

We can multiply both sides by a and we have c < b (note: we don’t need to switch the inequality sign because a is positive). However, we are given that c is greater than b, so c/a can’t be less than b/a. Statement III is FALSE.

Thus, only Roman numeral II is true.

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

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Re: If 0 < a < b < c, which of the following statements must be true?   [#permalink] 08 Dec 2017, 20:30
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