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## Number Properties question?

tagged by: ceilidh.erickson

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AbeNeedsAnswers Master | Next Rank: 500 Posts
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#### Number Properties question?

Mon Jul 03, 2017 8:23 pm
Elapsed Time: 00:00
• Lap #[LAPCOUNT] ([LAPTIME])
Is there a secret to these question types?

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

I. 2a > b + c
II. c - a > b - a
III. c/a < b/a

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

B

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### GMAT/MBA Expert

Ian Stewart GMAT Instructor
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Fri Jul 07, 2017 4:01 pm
I does not need to be true : just let a = 1, b = 2, c = 3. In fact, I can never be true, because if a < b and a < c, then by adding those two inequalities together, you find that 2a < b + c.

For II, we know c > b from the question stem. Subtract a on both sides of that inequality, and you get c - a > b - a, so that must be true.

For III, we know c > b from the question stem. We can divide by a on both sides, because we know a is positive. We then get c/a > b/a, so III must be false.

In general for this kind of question, if you see something algebraic you can do to make the information in the stem more closely resemble the things you're trying to prove, try that and see where it leads. If you can't see anything like that to try, and can't see a conceptual reason why something ought to be true or false, then try plugging in two extremely different sets of numbers, and see what happens. Here I'd try simple values very close together and very far apart. If you find one of the roman numeral items is true for both of your very different sets of numbers, then there's a good chance it's always true. If you find it's false for one or both of your sets of numbers, then you've proved that it does not need to be true. That's not a foolproof strategy, because sometimes you'll miss some exceptional case, but if you don't see anything else to try, it gives you a good chance to get a correct answer.

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Jay@ManhattanReview GMAT Instructor
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Sun Jul 09, 2017 6:04 am
Is there a secret to these question types?

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

I. 2a > b + c
II. c - a > b - a
III. c/a < b/a

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

B
There is no secret. BTG is a forum in which experts will introduce you to all the possible approaches to any question.

This one is a Must be True kind of question. In Must be True kind of questions, under all the circumstances, a statement under consideration must be true.

Let's take this one.

We have 0 < a < b < c. What do you make out of this information?

1. a, b and c all are positive numbers.
2. a, b and c are all not necessarily positive integers. One or none of them can be integers.
3. All can be less than 1. For example, a = 1/4; b = 1/3; c = 1/2.

Keeping these in mind, let's take the three statements.

I. 2a > b + c:

=> 2*SMALLEST number > A SMALLER number + A SMALL number
This is not possible as the sum of a SMALLER number & a SMALL number would always be greater than twice the SMALLEST number.

For example, say a = 1, b = 2, and c = 3.
2*1 < 2 + 3 => 2 < 5. This statement is false.

II. c - a > b - a:
=> c > b; cancelling out '-a' from both the sides. This is a must be a true statement. it is given that 0 < a < b < c

III. c/a < b/a:

=> c < b; cancelling 'a' from both the sides. Since 'a' is positive, we can cancel it without reversing the sign of inequality. Had it 'a' been negative, we would have to reverse the sign on the inequality. Had it been not known whether 'a' is positive or negative, we cannot cancel 'a.'

As seen in Statement II, this is false. We know that b < c.

Hope this helps!

-Jay

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ceilidh.erickson GMAT Instructor
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Tue Jul 11, 2017 9:55 am
Is there a secret to these question types?
If by that you mean Roman Numeral questions, then... no. RN's make it impossible to backsolve, estimate, or use a lot of the other tactics we can use on other PS questions. Don't try to game it based on whether I, II, or III shows up most often in answer choices.

They come in 2 flavors:

Which of the following MUST be true?
For these, try to DISPROVE each RN option. Try to think of a value / circumstance in which it would be untrue. Make sure to test extremes (very big & small), zero, negatives, fractions, etc, where applicable.

Which of the following COULD be true?
These are less common. For these, you just need to come up with an example that could be true in 1 circumstance.

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### GMAT/MBA Expert

ceilidh.erickson GMAT Instructor
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Tue Jul 11, 2017 9:59 am

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Ceilidh Erickson
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