The table shows the number of pages in each of 5 textbooks. What is th

Question

https://gmatclub.com/forum/download/file.php?id=45416 The table shows the number of pages in each of 5 textbooks. What is the greatest possible value of x for which the average (arithmetic mean) number of pages of the 5 textbooks is equal to the median number of pages of the 5 textbooks? A. 400 B. 450 C. 500 D. 550 E. 600 table.png

Archit3110 · Answer

the avg is = 2000+x/5

at value x=450 we get mean as 490 which would be equal to median 490 of the set ( 450,480,490,510,520).. least value would be 450 and greatest would be 550

( 480,490,510,520,550)
avg = 2000+550/5 = 510 
IMO D

AyushAgrawal1005 · Answer

Wouldnt it be better to take options and solve the question?
Or is there any equation for it

Rakshit25 · Answer

Here is a simple way to solve this without plugging in numbers -

The median has to be a value between 490 to 510(included). So you can just do (2000+x)/5=510, which will give x=550, the highest value of x

DonBosco7 · Answer

Hey  Bunuel   hr1212

In this question why doesn't the concept of AP apply.
The questions asks for what is the greatest possible value of x for which mean = median 
and we know that this is only possible if the series is in AP.
So from the table and from the answer choices, shouldn't the value be 500, so the series is in AP
480,490,500,510,520 --- giving x=500

Thanks in advance !

WhitEngagePrep · Answer

I'm not 100%, but it would appear that you're asking why Arithmetic Progression wouldn't apply here. It's not that it doesn't, necessarily, but that it doesn't have to because nothing in the problem stipulates that this is an evenly spaced set. The key is that the problem is asking for a maximum , meaning that multiple values of x are possible that would leave the median = mean, but only one will maximize the value of x. The median of a set of 5 numbers must be one of the numbers. So it will be 490 if x is 490 or under, 510 if x is 510 or above, and the median will actually equal x if x is between 490 and 510. Since i'm looking for a maximum value of x, I'll test the idea that x is greater than 510 first to see if that is possible, which means the median would be 510. The sum of the 4 given numbers is 2000 (somewhat easy math since 480+520 and 490+510 nicely complete each other to 1000). So if the average equals the median of 510, then (2000+x)/5 = 510 400 + x/5 = 510 x/5 = 110 x = 550. Since this answer fits the given constraints (I needed x to be greater than or equal to 510 and I needed the mean to equal the median), then 550 is the value of x when x is greater than 510 and this is the maximum value for x! Answer D Hope this helps!

Whit

WhitEngagePrep · Answer

Hi DonBosco7 I also just realized that you said something that we actually should point out, because a lot of test-takers get confused by this!

DonBosco7

The questions asks for what is the greatest possible value of x for which mean = median
and we know that this is only possible if the series is in AP.

I highlighted the text that is most important here! When it comes to sets where the mean=median, the AP connection is only in ONE direction. What I mean is this:

If you know that a set is evenly spaced, then you know that the mean=median.

However, the reverse is NOT necessarily true! Knowing that the mean=median of a set does NOT tell you whether the set is evenly space (or an AP). For example, check out the following sets:

For example, all of the following sets have mean=median and NONE are evenly spaced!

1, 4, 5, 6, 9 (mean=median=5)
1, 5, 5, 9 (mean=median=5)
1, 4, 5, 5, 6, 7, 7 (mean=median=5)
2, 8, 9, 11, 15, 15 (mean=median=10)

This is a common trap that people fall into, but just remember, the rule only works in ONE DIRECTION. If you know AP/evenly spaced, then you know mean=median. But honestly, there is no way to infer whether a set is evenly spaced - they pretty much have to tell you!

Sorry I missed this before in my response, but I hope it helps!

Whit

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