Overlapping Sets

[kouranjelika]

Hi guys,

So these two questions are from: GMATCLUB (wasn't one of Stacey's banned sources, so I hope it's ok):


"Example 6:
When Professor Wang looked at the rosters for this term's classes, she saw that the roster for her economics class (E) had 26 names, the roster for her marketing class (M) had 28, and the roster for her statistics class (S) had 18. When she compared the rosters, she saw that E and M had 9 names in common, E and S had 7, and M and S had 10. She also saw that 4 names were on all 3 rosters. If the rosters for Professor Wang's 3 classes are combined with no student's name listed more than once, how many names will be on the combined roster?

Translating:
"E had 26 names, M had 28, and S had 18": E=26, M=28, and S=18;
"E and M had 9 names in common, E and S had 7, and M and S had 10": EnM=19 *I think they meant 9 here and 19 is a typo*, EnS=7, and MnS=10;
"4 names were on all 3 rosters": EnMnS=g=4;

Question:: Total=?

Apply first formula: Total = A + B + C - (sum \ of \ 2-group \ overlaps) + (all \ three) + Neither --> Total=26+28+18-(9+7+10)+4+0 --> Total=50.


Example 9 (hard):
In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

Translating:
"In a class of 50 students...": Total=50;
"20 play Hockey, 15 play Cricket and 11 play Football": H=20, C=15, and F=11;
"7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football": HnC=7, CnF=4, and HnF=5. Notice that "7 play both Hockey and Cricket" does not mean that out of those 7, some does not play Football too. The same for Cricket/Football and Hockey/Football;
"18 students do not play any of these given sports": Neither=18.

Question:: how many students play exactly two of these sports?

Apply first formula:

{Total}={Hockey}+{Cricket}+{Football}-{HC+CH+HF}+{All three}+{Neither}

50=20+15+11-(7+4+5)+{All three}+18 --> {All three}=2;

Those who play ONLY Hockey and Cricket are 7-2=5;
Those who play ONLY Cricket and Football are 4-2=2;
Those who play ONLY Hockey and Football are 5-2=3;

Hence, 5+2+3=10 students play exactly two of these sports.

Answer: 10. Discuss this question HERE."


For some reason they apply a different formula than the General:
T = G1 + G2 + G3 - (all in twos) - 2*(all in threes) + Neither

They add the threes (without doubling it). Can you please explain in which cases would we apply this formula and how to identify that the question would require it?

Thanks!

Source link: http://gmatclub.com/forum/advanced-over ... 44260.html

[jnelson0612]

Hi guys,

So these two questions are from: GMATCLUB (wasn't one of Stacey's banned sources, so I hope it's ok):


"Example 6:
When Professor Wang looked at the rosters for this term's classes, she saw that the roster for her economics class (E) had 26 names, the roster for her marketing class (M) had 28, and the roster for her statistics class (S) had 18. When she compared the rosters, she saw that E and M had 9 names in common, E and S had 7, and M and S had 10. She also saw that 4 names were on all 3 rosters. If the rosters for Professor Wang's 3 classes are combined with no student's name listed more than once, how many names will be on the combined roster?

Translating:
"E had 26 names, M had 28, and S had 18": E=26, M=28, and S=18;
"E and M had 9 names in common, E and S had 7, and M and S had 10": EnM=19 *I think they meant 9 here and 19 is a typo*, EnS=7, and MnS=10;
"4 names were on all 3 rosters": EnMnS=g=4;

Question:: Total=?

Apply first formula: Total = A + B + C - (sum \ of \ 2-group \ overlaps) + (all \ three) + Neither --> Total=26+28+18-(9+7+10)+4+0 --> Total=50.


Example 9 (hard):
In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

Translating:
"In a class of 50 students...": Total=50;
"20 play Hockey, 15 play Cricket and 11 play Football": H=20, C=15, and F=11;
"7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football": HnC=7, CnF=4, and HnF=5. Notice that "7 play both Hockey and Cricket" does not mean that out of those 7, some does not play Football too. The same for Cricket/Football and Hockey/Football;
"18 students do not play any of these given sports": Neither=18.

Question:: how many students play exactly two of these sports?

Apply first formula:

{Total}={Hockey}+{Cricket}+{Football}-{HC+CH+HF}+{All three}+{Neither}

50=20+15+11-(7+4+5)+{All three}+18 --> {All three}=2;

Those who play ONLY Hockey and Cricket are 7-2=5;
Those who play ONLY Cricket and Football are 4-2=2;
Those who play ONLY Hockey and Football are 5-2=3;

Hence, 5+2+3=10 students play exactly two of these sports.

Answer: 10. Discuss this question HERE."


For some reason they apply a different formula than the General:
T = G1 + G2 + G3 - (all in twos) - 2*(all in threes) + Neither

They add the threes (without doubling it). Can you please explain in which cases would we apply this formula and how to identify that the question would require it?

Thanks!

Source link: http://gmatclub.com/forum/advanced-over ... 44260.html

Hi,
I'll do the best I can to explain; please let us know if we can answer further questions.

The first thing I would do in this situation is write out the overlapping sets for three groups formula [Total = Group 1 + Group 2 + Group 3 - (number in both) - 2(number in all three) + (those in none)] and see what you information you have. The first problem you referenced is very simple because you are told every single one of these pieces of information except Total. So it is very easy to plug every number in and obtain the Total.

For the second one, you are given every component to the formula except for how many play all three. So it is easy to solve for how many play all three as a starting point.

Then, the question asks how many play exactly two? To help you conceptualize this, draw a three circle Venn diagram (two circles overlapping on top and a third circle on the bottom overlapping the other two). You can see that in the place in which two groups overlap, part of that overlap is taken up by the members who belong in all three groups. Thus, we would need to subtract those members out of every two overlap group.

The second one you posted is a little more complicated but need not be a big deal. Just start with the formula, figure out what you can figure out, and then take it from there.

Please let us know if you have more questions!

[kouranjelika]

Hi Jamie,

Thank you.
But in regards to the first one, if you actually did just plug in all that you have from the problem into the formula (as I did at first), you won't get the correct answer of 50. You would get a 28.
I've dived into it further and realized the subtle wording issue and how the original formula is not applicable here (at least without the manipulation of part of the data first or as the person who solved it in the source, manipulated the formula instead).
The issue is that the "doubles" are counting the "triples." So to use the formula, we would actually have to take out the number of people on all three roasters from the number given on two of each courses. Hence it would be a total of twos as follows: (26-3*4)
If such a manipulation is performed, we result in the correct answer of 50.
And again for the second one, we cannot simply solve for "2x" as the guys who play on all three, because we cannot forget that everyone who play on two are already counting those who play on all three. Therefore all the twos would have to have a -3x in the formula and then it can be solved for x (only the threes) and hence figure out those who are playing only ONLY two.

Anjelika

[RonPurewal]

Now, see, THAT reasoning ^^ is the kind of thinking that's going to let you win this game.

If your success depended on memorizing that formula, I wouldn't be teaching this test; there's no way I could memorize that formula. (For a bit of perspective, it took me several hours to memorize the quadratic formula in high school.)

What you're doing there"”thinking through the situation"”is what you want to be doing. If you have any formulas memorized, they should be Plan B.

[kouranjelika]

Yea, totally Ron!
Thank you for all your help and advice!

[RonPurewal]

You're welcome.