Help!

<Problem Text>


"Each Senior in a College Course wrote a Thesis. The lengths, in page, of those seniors' theses re summarized in the graph above.

a. What is the least possible number of seniors whose theses were within six pages of the median length?"
</Problem Text>

Having trouble with Part A here, even after reading through the answer a few times. The answer the book gives is 2 and I didn't see any errata for it.

I'm okay with finding the median, which is 24.5 (avg of middle 2 categories which are a both 20-29), but have no idea how to arrive at just 2 seniors.

The answer I get is 6 seniors, as it says "within 6 pages of median length", which I think makes the range of pages from 24.5 +/- 6, or 18.5 to 30.5. 20-29 pages sits within the range of 18.5-30.5, therefore all 6 seniors MUST be within 6 pages of the median.

But alas, it seems not to be so. Anyone able to toss me a clue over here?

I am also confused about the answer provided by MGGMAT. The answer contains atleast one typo error, which is not mentioned in the published errata. We will appreciate if any MGGMAT staff will explain the answer in a better way...

Hi - Even iam confused ...Instructors you reponse is appreciated.thanks.

The problem is on p140 of Word Translations BTW..

So we have a total of 20 students, and each of their page lengths fall within a fairly wide range. This gives us the freedom to manipulate their page lengths to create various medians and then to try to get as many students close to the median or far from it as we can. The median is the middle number, so if we line up our 20 students according to their page lengths the median will be the halfway point between the 10th and 11th students..

Let's designate the five page ranges as A, B, C, D, and E. Lining up the students in order, they look like this:

A B B B B C C C C C C D D D D D D D E E

The 10th and 11th students are both category C, so the median must be between 20 and 29. Note that the median can be literally anywhere in this range: if all the Cs are 20 pages, the median is 20 pages, but if all the Cs are 29 pages, the median is 29 pages..

A) We want the LEAST number who are within 6 of the median, so we try to push everyone as far from the median as possible. First, let the A be 0 pages and all the Bs be 10 pages; no matter what we make the median, these will all be at least 10 from the median. Now let all the Ds be 39 pages and the Es be 49 pages; these will all be at least 10 from the median. We are left to contend with the 6 Cs; note that the 10th and 11th papers are at the high end, so let's make them both 29 and make the others 20. Now that we've assigned page lengths to every paper, we get this distribution:

0 10 10 10 10 20 20 20 20 29 29 39 39 39 39 39 39 39 49 49

The median is 29, and only two papers are within six of that value..

B) This time we are trying to get everyone as close to the median as possible. We know the median must still be between 20 and 29, so call the A 9, the Bs 19, the Ds 30, and the Es 40. Now all we have to do is call all the Cs 24, and we have this distribution:

9 19 19 19 19 24 24 24 24 24 24 30 30 30 30 30 30 30 40 40

The median is 24, so anything from 18 to 30 is within six of the median. This gives us 17. If you don't like counting the 30s as within 6 of 24, just bump up the 11th paper to 25. Now the median is 24.5 and anything from 18.5 to 30.5 goes..

*Note that in both of these problems, it made no difference what we called the A and the Es, i just gave them the values i did for simplicity. There's also a little more analysis to be done to prove that the answer to the first question can't be less than 2, but that is explained quite well on page 143 of the WT strategy guide..

Now, would someone be so kind as to explain what you think the typo is in our book? :)

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Tim Sanders
Manhattan GMAT Instructor

Thanks Tim. Could you clarify the different medians below:




From the question stem the median is 29, which is also the case for part-A. However the median is changed to 24 for part-B. I'm not able to understand this particular change.

For B, median as 29 I have the following sequence:
9 19 19 19 19 29 29 29 29 29 29 30 30 30 30 30 30 30 40 40
which gives answer as 13.

Thanks,
Gitesh

Tim/SG sets the median at 29 in order to move as many as possible of the 20-29 page papers more than 6 pages away from the median.

Tim/SG sets the median at 24.5 in order to draw as many as possible of the 10-19 and 30-39 page papers within 6 pages.

Tim/SG does not suggest that the median can be 29 and 24.5 simultaneously. The former is part of a story that allows us to MINIMIZE a value, to get things as spread out as possible. The latter is part of a story that allows to MAXIMIZE a value, to get things as close together as possible.

Are you treating this as a DS problem? But it can't be. Questions A and B are just that, questions. They are not statements. They do not follow a question.

Can anyone please tell me how to calculate total number of students? I see that the solution in both the strategy guide and this thread says that the number of senior students is 20. I looked for this value in the problem and in the graph, but did not find it anywhere. Please help.

Sure. From the first two sentences of the problem, we know that each senior wrote a thesis, and that the theses' lengths are summarized in a chart.

The left bar of the chart shows the number of theses for each length grouping. Looking at this, I can see that the following theses were written:
0-9 pages 1 thesis
10-19 pages 4 theses
20-29 pages 6 theses
30-39 pages 7 theses
40-49 pages 2 theses

Adding these together, I can see that 20 theses were written; thus, there were 20 students in the class.

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Jamie Nelson
ManhattanGMAT Instructor

Thank you Jamie! It makes sense now. I did not interpret the question right.

Great! Thanks for letting me know.

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Jamie Nelson
ManhattanGMAT Instructor

Hi,

I found a typo at the end of the answer key option a ). The word mean is wrong ,it should be median .

Please confirm .

Thanks,
Pranab.

Hi Pranab,
In this case we know that we have twenty theses, and thus the median must be the average (or mean) of thesis 10 and 11. When the explanation refers to the "mean", it is in fact referring to the mean of theses 10 and 11, which is also the median for this problem. I grant that it is a little confusing. :-)

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Jamie Nelson
ManhattanGMAT Instructor

Great explanation, however in this question, how do we know that the y axis is students and not some other measure, like number of days?

Would be great if I could get a reponse.

Thanks in advance

Yes, on the real test, there would definitely be a label on both axes. These aren't GMAT-format questions - they're just meant to test you on the math concepts themselves. So it's okay just to use common sense here. :)

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Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT