Overlapping Sets - idea is not clear

[priyankur.saha]

Kindly elucidate my doubt on overlapping set problems.

I am not able understand overlapping set problems particularly using venn diagram. Here are 2 problems and both use different formula to solve.

Question from MGMAT word translation:
Q1: Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

Solution: Marketing + sales + vision - (marketing and sales +marketing and vision + vision and sales) + (in all three)
= 20+30+40 - (5+6+9) + 4 = 74 (answer)

Question from MGMAT CAT test:
Q2: Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?
Solution:
Total = Poetry + History + writing - (poetry and history + poetry and writing + writing and history) - 2 (in all three)
59 = 22+27+28 - (6) - 2x
so, x = 6
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My question is why 2 problems are using different equations for same scenario? Is there any way to distinguish between two problems? How do I know which equation has to use?
I know the "inside out" approach but, in that way, total process is time consuming. Can't I use this quick formula to deal with 3-set problem Venn diagram?

[esledge]

For 3-set problems, my gentle advice is to learn to love the Venn diagram. There are tricky variations possible on these problems that make a one-size-fits-all formula dangerous. In fact, the the only way I was able to figure out the difference between these stories was to draw the circles.

Here's the tricky difference. In the first problem, when they say "(a number of) workers are on both the X and Y teams," that includes workers on X and Y only AND workers on X, Y, and Z. That's two different sections of the Venn diagram! So, of the 9 workers on both Marketing and Vision teams, 4 are on Marketing/Vision/Sales and 5 and on Marketing/Vision/notSales.

Q1: Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

On the second problem, the two-club only students are more clearly differentiated:

Q2: Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

I think the key difference is that between the words "on both" (which could mean "on these 2 teams only" or "on these 2 teams, plus another") and "exactly two" (which can only mean on 2 teams).

[priyankur.saha]

Thank you very much. Nice explanation Emily.

1. Because common sets include workers on X and Y only AND workers on X, Y, and Z, substraction of common sets is removing values of "All three" sets. Therefore "All three" part should be an addition (+ all three).
2. Because common sets include two-club only students, substraction of common sets is NOT removing duplicate values of "All three" sets. Therefore, twice of "All three" part should be substracted.

I think it's clear now...... :-)

[RonPurewal]

Thank you very much. Nice explanation Emily.

1. Because common sets include workers on X and Y only AND workers on X, Y, and Z, substraction of common sets is removing values of "All three" sets. Therefore "All three" part should be an addition (+ all three).
2. Because common sets include two-club only students, substraction of common sets is NOT removing duplicate values of "All three" sets. Therefore, twice of "All three" part should be substracted.

I think it's clear now...... :-)

yes.

[hardwick1010]

I follow the variances here between the two subtle wordings of each problem. However, I can't make sense of the -2x part of the second question.

Why is it 2x? Where does the 2 come from?

Thanks for clearing up whatever oversight I may be committing.

Question from MGMAT CAT test:
Q2: Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?
Solution:
Total = Poetry + History + writing - (poetry and history + poetry and writing + writing and history) - 2 (in all three)
59 = 22+27+28 - (6) - 2x
so, x = 6

[StaceyKoprince]

Let's start with this line of the explanation:

"Total = Poetry + History + writing - (poetry and history + poetry and writing + writing and history) - 2 (in all three)"

Are you aware of this formula? Do you understand why it is written the way it is? On the left side, we have the total number of students, right? And on the right side, we start with:
"Poetry + History + Writing"
which gives us a number larger than the total, because some students are in more than one club, right? eg, if there are 5 students total, I could have 2 in poetry, 2 in history and 4 in writing. Some of those students are in more than one club, so if I add Poetry + History + Writing, I end up with a number larger than 5, even though there are only 5 students total. Make sense so far?

So what I need to do is subtract out the instances of "double counting." For instance, if I know that Susie is in both poetry and history, then once I've counted her already as part of the poetry number, I don't want to count her a second time in the history group - she's only one person. For any students who have exactly two of these clubs, they will be counted twice, or one time too many. That's where the next part of the equation enters:
- (poetry and history + poetry and writing + writing and history)
Susie is a part of the "poetry and history" group, so I want to subtract her out once to account for the fact that I double-counted her to start with. And I do the same with any other students in the 2-club group.

Finally, some students are in all three clubs. A student in all three clubs would get counted three times in the opening part of the question: once for poetry, once for history, and once for writing. So I also have to subtract out the "extra" countings here, and in this case, one student is getting triple-counted, so I have to subtract 2 to get back down to counting that student only once. And that's the last part of the equation:
2 (in all three)
Each student who is in all three clubs has to be subtracted out twice in order to get back down to counting that student only once - so 2 times the number of students in all three.

Does that make more sense?

Then, in the next line of the explanation, the variable "x" is inserted. The "x" represents the unknown that the question asks us to find: the number of students in all three clubs. We want to solve for this value.

[manishverma90]

For 3-set problems, my gentle advice is to learn to love the Venn diagram. There are tricky variations possible on these problems that make a one-size-fits-all formula dangerous. In fact, the the only way I was able to figure out the difference between these stories was to draw the circles.

Here's the tricky difference. In the first problem, when they say "(a number of) workers are on both the X and Y teams," that includes workers on X and Y only AND workers on X, Y, and Z. That's two different sections of the Venn diagram! So, of the 9 workers on both Marketing and Vision teams, 4 are on Marketing/Vision/Sales and 5 and on Marketing/Vision/notSales.

Q1: Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

On the second problem, the two-club only students are more clearly differentiated:

Q2: Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

I think the key difference is that between the words "on both" (which could mean "on these 2 teams only" or "on these 2 teams, plus another") and "exactly two" (which can only mean on 2 teams).

Thanks esledge. I had exactly the same question as priyankur.saha after going through the exact questions.
Do you think it might be an idea to include this explanation with an example in the MGMAT Word Translation Strategy book? After going through the book, it was not clear to me either.

[manishverma90]

Also, here is question no. 7, Section 7 of the GMAT Test Paper Test code 31:

[deleted because problem is from a banned source - see below]

The OA is E - 325.

For this question, the second equation (as explained by esledge above) is used. But I am still not sure how this question is different from the first question (Q1: Workers are grouped) and similar to the second question (Q2: Each of the 59 members)?

Could someone please clarify? Have my GMAT appointment real soon. thanks

[tim]

in your first post, it sounds like you are asking us to make changes to our book. we appreciate such suggestions, but they are more likely to get to the right place if you send an email to Manhattan GMAT directly about it..

second, i'm very sorry to hear that you are using GMAT paper tests to study so close to the date of your exam. someone must have led you horribly astray if they told you those questions are adequate preparation for the GMAT. your primary focus for practice questions should be actual GMAT questions that are as recent as possible. you shouldn't even be using the 10th or 11th edition OG..

as for the question itself, we are unable to address it here or to allow it to be posted on our forums due to copyright concerns..

[manishverma90]

in your first post, it sounds like you are asking us to make changes to our book. we appreciate such suggestions, but they are more likely to get to the right place if you send an email to Manhattan GMAT directly about it..

Sure Tim, thanks. I will send an email to MGMAT about it.

second, i'm very sorry to hear that you are using GMAT paper tests to study so close to the date of your exam. someone must have led you horribly astray if they told you those questions are adequate preparation for the GMAT. your primary focus for practice questions should be actual GMAT questions that are as recent as possible. you shouldn't even be using the 10th or 11th edition OG..

as for the question itself, we are unable to address it here or to allow it to be posted on our forums due to copyright concerns..

I have actually done every single question of the OG12 at least once and majority of the questions in the QReview and VReview. Phew!!! I was just looking for some more official practice questions. So, I bought official GMAT Test papers from mba.com directly. I do take your point that I should keep in mind that these questions are older and retired questions. You would be glad to note that I have also finished 4 of the MGMAT CATs and I am due to finish the rest of the two over the next two days. :)

[tim]

i am the exact OPPOSITE of glad to hear that! you are wasting those practice tests - and more importantly, your time - by going through them so fast! combining this with the fact that you have "done" every question from the OG, it sounds like you are not taking your time to absorb the lessons from each question and each test and are focusing instead on getting through as much as you can in as little time as you can. this is one of the worst ways to prepare for the GMAT. DOING questions in rapid succession does not cause you to improve on the GMAT. taking your time to thoroughly understand each question and learn everything you can from it is what will get you better. this is why NO ONE ever needs to use paper tests to study, because you will not need more than the OG if you are doing things right. i have NEVER met a student who completed and thoroughly reviewed every question in the OG - every student i have ever met who has started this process and reviewed every question properly along the way realized LONG before they finished that book that they were so thoroughly prepared for the GMAT that there was no point in finishing the OG questions. ask yourself this question honestly: are you 100% sure that if i hand you a random question from the OG, you can answer it correctly and give a thorough explanation of every step required to get to the correct answer? if the answer is yes, stop working right now and go take your GMAT. if the answer is no, you are not done with the OG yet..

[manishverma90]

i am the exact OPPOSITE of glad to hear that! you are wasting those practice tests - and more importantly, your time - by going through them so fast! combining this with the fact that you have "done" every question from the OG, it sounds like you are not taking your time to absorb the lessons from each question and each test and are focusing instead on getting through as much as you can in as little time as you can. this is one of the worst ways to prepare for the GMAT. DOING questions in rapid succession does not cause you to improve on the GMAT. taking your time to thoroughly understand each question and learn everything you can from it is what will get you better. this is why NO ONE ever needs to use paper tests to study, because you will not need more than the OG if you are doing things right. i have NEVER met a student who completed and thoroughly reviewed every question in the OG - every student i have ever met who has started this process and reviewed every question properly along the way realized LONG before they finished that book that they were so thoroughly prepared for the GMAT that there was no point in finishing the OG questions. ask yourself this question honestly: are you 100% sure that if i hand you a random question from the OG, you can answer it correctly and give a thorough explanation of every step required to get to the correct answer? if the answer is yes, stop working right now and go take your GMAT. if the answer is no, you are not done with the OG yet..

I take your point on being sure about the OG questions if asked again, however, I am afraid, Tim, you might be largely making assumptions without knowing my hours, progress, style or situation. Thanks!

[tim]

yes i'm making assumptions. i don't know your situation, so until i have other information i assume it is more likely that your situation is like the thousands of other students i have dealt with than that those thousands of students fall into one category and you alone of all the students i have worked with fall into a separate category. if you believe your situation is completely unique and has never been encountered by any other student, please share the relevant information and i will be glad to suggest other alternatives for your study..

[manishverma90]

yes i'm making assumptions. i don't know your situation, so until i have other information i assume it is more likely that your situation is like the thousands of other students i have dealt with than that those thousands of students fall into one category and you alone of all the students i have worked with fall into a separate category. if you believe your situation is completely unique and has never been encountered by any other student, please share the relevant information and i will be glad to suggest other alternatives for your study..

Thanks but no thanks. I would much rather listen to an "instructor" with a little less negative, judgemental or egoistic attitude towards forum members.

While without knowing anything about my practise, you might have been a little too keen to give me "solutions" and have been attacking my approach, I am glad to report a 80% percentile in the Official GMAT. I might have something right going on there. :) Ciao!!!

[tim]

I'm glad to hear that you got a score you were pleased with. Your mode of studying, although it may not have been optimal, seems to have worked for you. I apologize if you interpreted my concern for your approach as anything other than a genuine attempt to be helpful. I am always very worried when I hear of a student using an approach that is not designed to achieve maximal success, and while I wish I could have steered you in a better direction, as long as you got a score that was sufficient for you I suppose that's what's really important.

I would certainly caution other readers on the forum that taking practice tests in rapid succession and studying from paper tests almost always results more in wasted time than in actual progress. You might get lucky with this approach - anomalies happen every day, and I'm sure you can all think of examples of this - but the safest approach for each of you is going to be to rely on the collected knowledge of our instructors across the multitudes of students who have gone through our program.