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Source: — Data Sufficiency |
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by Matt@VeritasPrep » Tue Nov 05, 2013 4:13 pm
Let's make sense of the stem first:

3 ≤ median ≤ 8

Mean = (x+1+3+8+12)/5 = (24+x)/5 = 4.8 + (x/5)

So we want to know if 4.8 + (x/5) is greater than our median, which is some number between 3 and 8, inclusive. OK!

(1) tells us that x > 6. Let's try a few values. Since x is an integer, x is at least 7. If x is 7, the mean is 4.8 + (7/5), or 6.2, but the median is 7, so the mean is less than the median. If x = 1,000,000, however, the mean is clearly greater than the median. INSUFFICIENT

(2) tells us that x > 8. If x is greater than 8, x is at least 9, which means that the mean is at least 4.8 + (9/5), or 6.6. If the mean is 6.6, however, the median is 8, so the mean is less than the median. Of course, if x = 1,000,000, the mean is greater than the median. INSUFFICIENT

Combining the statements doesn't help, as (1) tells us 6 < x and (2) tells us 8 < x, so (1) is contained in (2), and (2) was insufficient. So the answer is E.
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by GMATGuruNY » Tue Nov 05, 2013 7:32 pm
prakashchandra wrote:x,3,1,12,8
If x is an integer, is the median of the 5 numbers shown greater than the average of the 5 numbers?

1. x > 6
2. x is greater than the median of 5 numbers
Put the integers in ascending order:
1, 3, 8, 12.

Test EXTREMES.

If x=9, both statements are satisfied:
Statement 1: 9>6
Statement 2: Since the set = {1, 3, 8, x=9, 12}, x is greater than the median of 8.
Here, the average = (1+3+8+9+12)/5 ≈ 6, so the median > the average.

If x=100, both statements are satisfied:
Statement 1: 100>6
Statement 2: Since the set = {1, 3, 8, 12, x=100}, x is greater than the median of 8.
Here, the average = (1+3+8+12+100)/5 ≈ 25, so the median < the average.

Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.
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by Brent@GMATPrepNow » Tue Nov 05, 2013 8:23 pm
prakashchandra wrote:x,3,1,12,8
If x is an integer, is the median of the 5 numbers shown greater than the average (AM) of 5 numbers?

1. x >6
2. x is greater than the median of 5 numbers
Target question: Is the median greater than the average?

Given: We have the set {1, 3, 8, 12, x}
Notice that there are 3 possible scenarios we need to consider:
Scenario #1: x is less than 3, in which case the median is 3
Scenario #2: 3 < x < 8, in which case the median is x
Scenario #3: x is greater than 8, in which case the median is 8

The average of this set will be (24+x)/5.

Okay, now onto the statements

Statement 1: x > 6
This rules our scenario #1, but we must still consider scenarios #2 and #3
Here are two possible values of x that yield conflicting answers to the target question:
Case a: x = 7 (median = 7 and average = 31/5), in which case, the median is greater than the average
Case b: x = 21 (median = 8 and average = 9), in which case, the median is NOT greater than the average
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x is greater than the median of the 5 numbers
This rules our scenarios #1 and #2, which leaves scenario #3
Here are two possible values of x that yield conflicting answers to the target question:
Case a: x = 11 (median = 8 and average = 7), in which case, the median is greater than the average
Case b: x = 21 (median = 8 and average = 9), in which case, the median is NOT greater than the average
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that the x-values we used to show that statement 2 is not sufficient ALSO satisfy statement 1. So, we know immediately that the combined statements are NOT SUFFICIENT.
To see what I mean, here are the two conflicting cases:
Case a: x = 11 (median = 8 and average = 7), in which case, the median is greater than the average
Case b: x = 21 (median = 8 and average = 9), in which case, the median is NOT greater than the average
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT.

Answer = E

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Brent
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