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Median data sufficiency

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Median data sufficiency

by SpencerP » Thu May 07, 2015 11:52 am
If 2 < x < 4, what is the median of the numbers 0, 5, x, 1, 7, and 3?

(1) 2x - 5 = 0
(2) 2x^2 - 7x + 5 = 0

The correct answer is D. Can someone please explain how B is sufficient? Thanks.
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by GMATGuruNY » Thu May 07, 2015 12:58 pm
SpencerP wrote:If 2 < x < 4, what is the median of the numbers 0, 5, x, 1, 7, and 3?

(1) 2x - 5 = 0
(2) 2x^2 - 7x + 5 = 0
Statement 1: 2x-5 = 0
Here, we can solve for x.
Since all 6 numbers are known, the median can be determined.
SUFFICIENT.

Statement 2: 2x² - 7x + 5 = 0
Since the two statements cannot contradict each other, one of the factors of 2x² - 7x + 5 is almost certainly (2x-5).
To yield 2x² - 7x + 5, the other factor must be (x-1).
Thus, if we factor 2x² - 7x + 5 = 0, we get:
(2x-5)(x-1) = 0
x = 5/2 or x=1.

Since the prompt requires that 2 < x < 4, only x = 5/2 is viable.
Since all 6 numbers are known, the median can be determined.
SUFFICIENT.

The correct answer is D.
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by toby001 » Sun Apr 30, 2017 3:09 pm
GMATGuruNY wrote:
SpencerP wrote:If 2 < x < 4, what is the median of the numbers 0, 5, x, 1, 7, and 3?

(1) 2x - 5 = 0
(2) 2x^2 - 7x + 5 = 0
Statement 1: 2x-5 = 0
Here, we can solve for x.
Since all 6 numbers are known, the median can be determined.
SUFFICIENT.

Statement 2: 2x² - 7x + 5 = 0
Since the two statements cannot contradict each other, one of the factors of 2x² - 7x + 5 is almost certainly (2x-5).
To yield 2x² - 7x + 5, the other factor must be (x-1).
Thus, if we factor 2x² - 7x + 5 = 0, we get:
(2x-5)(x-1) = 0
x = 5/2 or x=1.

Since the prompt requires that 2 < x < 4, only x = 5/2 is viable.
Since all 6 numbers are known, the median can be determined.
SUFFICIENT.

The correct answer is D.
Thanks, Mitch, other than spotting one of the factors from statement 1, what's a quick way to approach such a quadratic? I can solve these that are not so straightforward, but not in 2 minutes. I got to this point in the question and guessed.
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by [email protected] » Sun Apr 30, 2017 6:46 pm
Hi toby001,

The GMAT will sometimes test you on concepts that you know, but in a way that you're not used to thinking about.

For example, you can probably FOIL this calculation rather easily: (X+1)(X-2). You know the 'steps' and should be able to write everything out. In that same way, you should be able to FOIL something that looks more complex, such as (2X - 3)(3X + 15). There would be a bit more 'work', but the steps are the same.

The same can be said for factoring a Quadratic. Factoring down X^2 + 4X + 3 wouldn't take you too long because you know how to spot the 'clues.' With something a bit more complex, such as 2X^2 - 7X + 5 = 0, the same types of clues are still there - but you might have to 'play around' with the pieces to find the exact solution.

Since the first term is 2X^2, you know that the two parentheses will coin an X and a 2X. Since the third term is +5, the two parentheses will either contain +1 and +5 OR - 1 and -5. Given those limited possibilities, and the fact that the 2nd term is -7X (hint: it's NEGATIVE...), how long would it really take you to try out all of the possibilities and find the solution?

As an aside, attempting to answer each Quant question in under 2 minutes is a bad goal. Your actual goal should be to be 'efficient'; that doesn't mean "less than 2 minutes."

GMAT assassins aren't born, they're made,
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Contact Rich at [email protected]
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by Jay@ManhattanReview » Sun Apr 30, 2017 9:31 pm
SpencerP wrote:If 2 < x < 4, what is the median of the numbers 0, 5, x, 1, 7, and 3?

(1) 2x - 5 = 0
(2) 2x^2 - 7x + 5 = 0

The correct answer is D. Can someone please explain how B is sufficient? Thanks.
Median of a set is the value of the middle-most term when the terms are arranged in an ascending or in descending order.

Since we know that 2 < x < 4, x would lie between 1 and 5 of the given six terms.

So, the arranreged terms would be {0, 1, x, 3, 5, 7} or {0, 1, 3, x, 5, 7}

Since the number of terms is even, we cannot have the middle-most term; however, we can still get the median.

Median would be the arithmetic mean of the two middle-most terms, here, x and 3.

If we get the unique value of x, we get the answer.

With that in mind, let's see each statement one by one.

Statement 1: 2x - 5 = 0

This is a linear equation and we are bound to get a unique value of x. Sufficient. There is no need to calculate the median.

However, for the sake of understanding, let's calculate it.

2x - 5 = 0 => x = 5/2 = 2.5.

Median = (2.5+3)/2 = 2.75.

Statement 2: 2x^2 - 7x + 5 = 0

Above is a quadratic equation and we may get two values of x, leading to two values of the median, implying INSUFFICIENCY!

However, a quadratic equation does not necessarily render TWO values. So, we cannot tag statement 2 as insufficient and move ahead.

We have 2x^2 - 7x + 5 = 0

=> 2x^2 - 5x -2x + 5 = 0

Looking at above form of the equation, we are pretty sure that there would be two different values of x, leading to two values of the median, implying INSUFFICIENCY!

However, there is more to it than what meets the eye.

2x^2 - 5x -2x + 5 = 0 => x(2x - 5) - 1(2x - 5) = 0 => (2x - 5) (x - 1) = 0 => x = 5/2 and 1

Since it is given that 2 < x < 4, x = 1 does not qualify.

Thus, x = 5/2. It is the same value that we get in statement 1. Sufficient.

The correct answer: D

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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by toby001 » Mon May 01, 2017 8:00 am
[email protected] wrote:Hi toby001,

The GMAT will sometimes test you on concepts that you know, but in a way that you're not used to thinking about.

For example, you can probably FOIL this calculation rather easily: (X+1)(X-2). You know the 'steps' and should be able to write everything out. In that same way, you should be able to FOIL something that looks more complex, such as (2X - 3)(3X + 15). There would be a bit more 'work', but the steps are the same.

The same can be said for factoring a Quadratic. Factoring down X^2 + 4X + 3 wouldn't take you too long because you know how to spot the 'clues.' With something a bit more complex, such as 2X^2 - 7X + 5 = 0, the same types of clues are still there - but you might have to 'play around' with the pieces to find the exact solution.

Since the first term is 2X^2, you know that the two parentheses will coin an X and a 2X. Since the third term is +5, the two parentheses will either contain +1 and +5 OR - 1 and -5. Given those limited possibilities, and the fact that the 2nd term is -7X (hint: it's NEGATIVE...), how long would it really take you to try out all of the possibilities and find the solution?

As an aside, attempting to answer each Quant question in under 2 minutes is a bad goal. Your actual goal should be to be 'efficient'; that doesn't mean "less than 2 minutes."

GMAT assassins aren't born, they're made,
Rich
Wow! That's brilliant Rich! Thanks! I knew it'd be 2x and x. I just tried -5 and -1 and it worked. If not, then I'd have to try 1 and 5. That's all I'd have to do here. Thank you!
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