Set P above consists of 6 integers. For what value of x are the mode,

Question

P = {105, 106, 107, 108, 109, x}
Set P above consists of 6 integers. For what value of x are the mode, the median and the average equal?

A. 105
B. 106
C. 107
D. 108
E. 109

BrentGMATPrepNow · Accepted Answer

When we examine the set {105, 106, 107, 108, 109}, we can see that the 5 values are  equally spaced , which means  mean = median .
So, we can see that 107 is the mean and median of {105, 106, 107, 108, 109}.

So, if x = 107, the mean stays the same (107), the median stays the same (107), and the mode becomes 107

Answer: C

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Abhimanyu97 · Answer

Is the mean = median or mode? Sorry got a little confused here

BrentGMATPrepNow · Answer

Mean, median, and mode are three separate measurements.

However, if a set of numbers is equally spaced (e.g., 14, 24, 34, 44, 54}, then the mean = median.
So, for example, the mean (aka average) of (14, 24, 34, 44, 54} is 34, and the median of (14, 24, 34, 44, 54} is also 34.

ShahadatHJ · Answer

Suppose, the number is X.
Average =  \frac{535+X}{6} 
Since the number we pick will be a copy of another from the given 5 and we are assuming the unknown number to be X.
Median =  \frac{2X}{2} 
Now,  \frac{535+X}{6}  =  \frac{2X}{2} 
Solve for X (= 107 )

BLTN · Answer

Dear  BrentGMATPrepNow 
 Bunuel  can correct me, but I have doubt that in the case (e.g., 14, 24, 34, 44, 54} the mean = mode.
The same here,

Mode is the most frequent number. For instance, in case of bell curve 
1 2 3 4 4 5 6 7

Yet, in the case 1 2 3 4 5 6 7 there is no repetition. I would agree that all of the numbers could be the mode but I challenge that the Mean= Mode.

IanStewart · Answer

Brent - I think you have a typo here (and this addresses BLTN's question above). When a set is equally spaced, then mean = median is always true, and that's very useful on the GMAT. But when a set is equally spaced, it never has a mode at all (all the numbers will be different), so there are no useful facts to know about equally spaced sets and the mode. There's one exception to what I've just said: if you have an equally spaced list where every number is the same, say in the list 7, 7, 7, 7, 7, then, in that one situation, the numbers are not all different. Then it's still of course true that mean = median, and one might easily think mean = median = mode here, but actually by definition, this set also has no mode, because no value occurs more often than any other value. So an equally spaced list never has a mode. edit: I corrected that second paragraph; my initial post said '7' was the mode there too, and then remembered the technicalities around the definition of the mode (technicalities that never matter on the GMAT, nor in real life, so I tend to forget about them

).

IanStewart · Answer

What you're saying makes perfect logical sense, but mathematicians have defined the mode differently, perhaps a bit counterintuitively. By definition, a list only has a mode at all if some value in the list occurs more often than some other value in the list. So when every value in a list occurs once, or occurs equally often, the list is said to have no mode. So these lists have no mode:

1, 2, 3, 4, 5
100, 100, 100, 300, 300, 300, 500, 500, 500

because in each, no value appears more often than any other. These lists, on the other hand, have at least one mode:

1, 1, 1, 1, 4, 5 --> the mode is 1
15, 15, 25, 25, 35 --> this list has two modes, 15 and 25

edit: the GMAT is not at all interested in testing strange technicalities about mathematical definitions though, so if you didn't know about any of this, it would never affect your score at all.

BrentGMATPrepNow · Answer

Sorry, my bad!

The highlighted part above should read mean =   median .

I've edited my response accordingly.

Cheers,
Bread

Kimberly77 · Answer

Hi Brent  BrentGMATPrepNow , Given that Set P above consists of 6 integers. So not quite sure that we are only examing 5 values here? Wouldn't that make X is missing here or have I missed something? Thanks Brent

BrentGMATPrepNow · Answer

I start my solution by examining the 5 given values, 105, 106, 107, 108, 109
Once I make some observations about those 5 given values, I'm able to make some conclusions about the missing value, x

So I'm not examining only five values here

Kimberly77 · Answer

Thanks  BrentGMATPrepNow . I see what you mean now. Since 107 is both mean and median in the 5 values and the questio asked for what value of x are the mode, the median and the average equal? So final 6 values are {105, 106, 107, 107, 108, 109} ? Am I understand it correctly? Thanks Brent

BrentGMATPrepNow · Answer

That's right. 
If the mean and mean of {105, 106, 107, 108, 109} is 107, then adding a second 107 to the set will keep the mean and median at 107

Kimberly77 · Answer

Crystal clear now. Thanks Brent BrentGMATPrepNow for confirmation

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Set P above consists of 6 integers. For what value of x are the mode,

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Set P above consists of 6 integers. For what value of x are the mode,

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