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a very basic problem on sets

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a very basic problem on sets

by knight247 » Thu Jun 23, 2011 10:38 am
There are 400 students in Samuel Music School. Each and every student either learns to play
the violin or the piano, or both. If 50 students learn to play both the violin and the piano,
how many of students do not learn to play the piano?
(1)150 students learn to play piano in Samuel Music School.
(2)100 students do not learn to play violin in Samuel Music School.

(A)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
(B)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
(C)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
(D)EACH statement ALONE is sufficient to answer the question asked.
(e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Detailed explanations would be appreciated
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Source: — Data Sufficiency |
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by Ashley@VeritasPrep » Thu Jun 23, 2011 8:43 pm
See diagrams!

Note that this is at least one of two good ways to do the problem; it can also be done with a table (to follow).
Image

Image
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by Ashley@VeritasPrep » Thu Jun 23, 2011 9:05 pm
Method #2. Have both tools in your belt and pick your favorite :)


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by vaibhavdoshi » Mon Jun 27, 2011 10:01 am
N(TOTAL)=N(PIANO)+N(VIOLIN)-N(BOTH)+N(NEITHER)
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by mirantdon » Mon Jun 27, 2011 10:19 am
IMO C
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by Aditya57 » Wed Jun 29, 2011 5:59 am
Ashley@VeritasPrep wrote:Method #2. Have both tools in your belt and pick your favorite :)


Image
could this method be used for 3 different objects say piano,violin and guitar? do we have to use a 3*3 grid or do we use 3 or perhaps 6 2*2 grids ,I'm curious to know!
Thanks in advance.
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by Ashley@VeritasPrep » Wed Jun 29, 2011 9:10 am
Aditya57 wrote:
Ashley@VeritasPrep wrote:Method #2. Have both tools in your belt and pick your favorite :)


Image
could this method be used for 3 different objects say piano,violin and guitar? do we have to use a 3*3 grid or do we use 3 or perhaps 6 2*2 grids ,I'm curious to know!
Thanks in advance.
Ah, good question! You know, the problem with using a single grid for three yes/no categories is that there's no way in two dimensions to represent all the possible intersections. Let's use your example -- piano, violin, and guitar -- and let's use + and - to represent Yes and No, respectively. I've then got the possibilities of + and - (two possibilities) for each of three categories, so I really wind up with 8 (that is, 2^3) possible specific categories for people to fall into (for example, one possibility is P+,V-,G+. If I could "build" the chart in three dimensions, I could create those eight areas -- in fact, they could be conceived of exactly as you conceive of the octants in an xyz-space (for instance, in xyz-space I have an octant where x>0,y<0,z>0). But however many yes-no categories you had, that's how many dimensions you'd need to be able to draw in to accomplish this through one chart.

Your three 2x2 charts idea is clever and would certainly work. However, it wouldn't likely be the easiest way to do the problem, because in each chart you'd have to miss some portion of the information, so you'd wind up having ultimately to sort of splice all the information together coming from three different places. So while it would work, I'd suggest using a 3-circle Venn diagram for those problems, because that way you can represent all 8 categories (here, the universe outside all three circles represents G-,V-,P-).Image
You could certainly also try using the three 2x2 charts in conjunction with that Venn diagram -- using the diagram as the place to combine all the info from your charts.

In general, the relationship of a three-category problem to a two-category problem is the same as the relationship of a 3-variable system of equations to a 2-variable system of equations, i.e. you need an additional piece of information to solve for everything in the former. For instance, if there were only two instruments (violin and piano, say) and I knew that every student played at least one, and I knew that 20 out of 100 students didn't play violin, then I'd be able to conclude that 80 out of 100 did play violin and that those 20 who didn't definitely played piano. But if there were three instruments and I knew that every student played at least one and that 20 out of 100 didn't play violin, I'd still know that the other 80 did play violin, but I wouldn't know what to do with the 20 who didn't -- wouldn't know how many of them played guitar and how many played piano. These 3-category problems are rare on the GMAT, because of their increased complication, so you can safely rest easier on that front, I think :).
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