Set A, B, C have some elements in common. if 16 elements are in both A and B, 17 elements are in both A and C, and 18 elements are in both B and C, how many elements do all three of the sets A, B, and C have in common?

1). of the 16 elements that are in both A and B, 9 elements are also in C

2). A has 25 elements, B has 30 elements, and C has 35 elements

This is GMATPREP question. What is the best way to solve it?

i apologize for the screams of anguish that will result from doing this, but:

i can't draw a venn diagram on the page, and don't currently have the equipment to hand-draw one and scan it in. so: click here for a reference venn diagram.

* the upper left circle (pdr1-3) will stand for set a.
* the upper right circle (pdr1*gad) will stand for set b.
* the bottom circle (pdre) will stand for set c.

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(1) unless i'm reading this wrong, this statement is actually telling you the number you want. the 9 elements mentioned are in all three sets, and they're the only ones that are (think about this for a sec and it should sink in - if anything is in all three, then it's got to be one of those 16 in the first place, so you can be sure that you're not missing anything here). so that's definitely sufficient.

(2) because of statement (1) we know that 9 common elements will definitely work. if we put a 9 in the middle circle of the venn diagram above, then we have the following:
1, 7, 5 (imagine these numbers where you see '2, 1, 3' in the link)
8, 9, 9 (imagine these numbers where you see '0, 23, 4' in the link)
9 (where you see '191' in the link)

but, if you put an 8 in the middle circle, you get the following, which also works:
0, 8, 4
9, 8, 10
8

so, insufficient.

Anyone know any good references for Venn problems? I've gone over the strategy guide about solving from inside out and I still don't get it. Very frustrating.

hth:
http://www.regentsprep.org/Regents/math/venn/PracVenn.htm
http://www.math.tamu.edu/~kahlig/venn/venn.html

that ought to keep you busy for a little while.

we could generalize the idea of 'working from the inside out' to something like 'start by filling in the portions of the diagram that you actually can fill in with the given information'. it just so happens that the middle of the diagram happens to be one of those portions; hence our advice.

the reason you can't normally fill in the outer portions of a venn diagram until the end of a problem is that those portions, when expressed in words, are nitpicky and awful. for instance, if the three circles in a venn diagram are sets a, b, and c, then the outer part of circle a is 'those elements that are in set a, but not in set b or set c'. that's a bit too artificial to be given in most problems. on the other hand, the middle circle - 'elements that are in all three sets' - is a natural thing to cite.