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Math Expert V
Joined: 02 Sep 2009
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asagraw wrote:
Hi Bunuel,

I am not able to understand first formula. As you mentioned "When we add three groups A, B, and C some sections are counted more than once. For instance: sections d, e, and f are counted twice and section g thrice." but aren't we counting g six times? Please explain as I am very much confused with this. Thanks.

Regards,
Ashish

Why is g counted 6 times?

Each of A, B, and C contains one g. When we sum, g is counted thrice.
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Great material. I guess it can solve one of the hardest 3 overlapping sets . I knew the Formula 2 but I never knew the detail application and tried to apply on the one which needs Formula 1 ...wasted lot of time on practice exam ... Now i guess I will be able to distinguish them .

Also I guess for problems with 2 sets , Matrix methods is best and can solve any problem. For 3 its become little problematic.

Regards.

Bunuel wrote:

Some 700+ GMAT quantitative questions will require you to know and understand the formulas for set theory, presenting three sets and asking various questions about them. There are two main formulas to solve questions involving three overlapping sets. Consider the diagram below: FIRST FORMULA

$$Total = A + B + C - (sum \ of \ 2-group \ overlaps) + (all \ three) + Neither$$.

Let's see how this formula is derived.

When we add three groups A, B, and C some sections are counted more than once. For instance: sections d, e, and f are counted twice and section g thrice. Hence we need to subtract sections d, e, and f ONCE (to count section g only once) and subtract section g TWICE (again to count section g only once).

In the formula above, $$sum \ of \ 2-group \ overlaps=AnB+AnC+BnC$$, where AnB means intersection of A and B (sections d, and g), AnC means intersection of A and C (sections e, and g), and BnC means intersection of B and C (sections f, and g).

Now, when we subtract $$AnB$$ (d, and g), $$AnC$$ (e, and g), and $$BnC$$ (f, and g) from $$A+B+C$$, we are subtract sections d, e, and f ONCE BUT section g THREE TIMES (and we need to subtract section g only twice), therefor we should add only section g, which is intersection of A, B and C (AnBnC) again to get $$Total = A + B + C - (sum \ of \ 2-group \ overlaps) + (all \ three) + Neither$$.

SECOND FORMULA

$$Total = A + B + C - (sum \ of \ EXACTLY \ 2-group \ overlaps) - 2*(all \ three) + Neither$$.

Notice that EXACTLY (only) 2-group overlaps is not the same as 2-group overlaps:
Elements which are common only for A and B are in section d (so elements which are common ONLY for A and B refer to the elements which are in A and B but not in C);
Elements which are common only for A and C are in section e;
Elements which are common only for B and C are in section f.

Let's see how this formula is derived.

Again: when we add three groups A, B, and C some sections are counted more than once. For instance: sections d, e, and f are counted twice and section g thrice. Hence we need to subtract sections d, e, and f ONCE (to count section g only once) and subtract section g TWICE (again to count section g only once).

When we subtract $$sum \ of \ EXACTLY \ 2-group \ overlaps$$ from A+B+C we subtract sections d, e, and f once (fine) and next we need to subtract ONLY section g ($$AnBnC$$) twice. That's it.

Now, how this concept can be represented in GMAT problem?

Example 1:
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?

Translating:
"are placed on at least one team": members of none =0;
"20 are on the marketing team": M=20;
"30 are on the Sales team": S=30;
"40 are on the Vision team": V=40;
"5 workers are on both the Marketing and Sales teams": MnS=5, note here that some from these 5 can be the members of Vision team as well, MnS is sections d an g on the diagram (assuming Marketing = A, Sales = B and Vision = C);
"6 workers are on both the Sales and Vision teams": SnV=6 (the same as above sections f an g);
"9 workers are on both the Marketing and Vision teams": MnV=9.
"4 workers are on all three teams": MnSnV=4, section 4.

Question: Total=?

Applying first formula as we have intersections of two groups and not the number of only (exactly) 2 group members:

$$Total=M+S+V-(MnS+MnV+SnV)+MnSnV+Neither=20+30+40-(5+6+9)+4+0=74$$.

Answer: 74. Discuss this question HERE.

Example 2:
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?

Translating:
"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": Total=59, Neither=0 (as members are required to sign up for a minimum of one);
"27 students for the history club": H=27;
"28 students for the writing club": W=28;
"6 students sign up for exactly two clubs": (sum of EXACTLY 2-group overlaps)=6, so the sum of sections d, e, and f is given to be 6, (among these 6 students there are no one who is a member of ALL 3 clubs)

Question:: "How many students sign up for all three clubs?" --> $$PnHnW=g=?$$

Apply second formula: $$Total=P+H+W -(sum \ of \ EXACTLY \ 2-group \ overlaps)-2*PnHnW + Neither$$ --> $$59=22+27+28-6-2*x+0$$ --> $$x=6$$.

Answer: 6. Discuss this question HERE.

Example 3:
Of 20 Adults, 5 belong to A, 7 belong to B, and 9 belong to C. If 2 belong to all three organizations and 3 belong to exactly 2 organizations, how many belong to none of these organizations?

Translating:
"5 belong to A, 7 belong to B, and 9 belong to C": A=5, B=7, and C=9;
"2 belong to all three organizations": AnBnC=g=2;
"3 belong to exactly 2 organizations": (sum of EXACTLY 2-group overlaps)=3, so the sum of sections d, e, and f is given to be 3, (among these 3 adults there are no one who is a member of ALL 3 clubs)

Question:: Neither=?

Apply second formula: $$Total=A+B+C-(sum \ of \ EXACTLY \ 2-group \ overlaps)-2*AnBnC + Neither$$ --> $$20=5+7+9-3-2*2+Neither$$ --> $$Neither=6$$.

Answer: 6. Discuss this question HERE.

Example 4:
This semester, each of the 90 students in a certain class took at least one course from A, B, and C. If 60 students took A, 40 students took B, 20 students took C, and 5 students took all the three, how many students took exactly two courses?

Translating:
"90 students": Total=90;
"of the 90 students in a certain class took at least one course from A, B, and C": Neither=0;
"60 students took A, 40 students took B, 20 students took C": A=60, B=40, and C=20;
"5 students took all the three courses": AnBnC=g=5;

Question:: (sum of EXACTLY 2-group overlaps)=?

Apply second formula: $$Total=A+B+C-(sum \ of \ EXACTLY \ 2-group \ overlaps)-2*AnBnC + Neither$$ --> $$90=60+40+20-x-2*5+0$$ --> $$x=20$$.

Answer: 20. Discuss this question HERE.

Example 5:
In the city of San Durango, 60 people own cats, dogs, or rabbits. If 30 people owned cats, 40 owned dogs, 10 owned rabbits, and 12 owned exactly two of the three types of pet, how many people owned all three?

Translating:
"60 people own cats, dogs, or rabbits": Total=60; and Neither=0;
"30 people owned cats, 40 owned dogs, 10 owned rabbits": A=30, B=40, and C=10;
"12 owned exactly two of the three types of pet": (sum of EXACTLY 2-group overlaps)=12;

Question:: AnBnC=g=?

Apply second formula: $$Total=A+B+C-(sum \ of \ EXACTLY \ 2-group \ overlaps)-2*AnBnC + Neither$$ --> $$60=30+40+10-12-2*x+0$$ --> $$x=4$$.

Answer: 4. Discuss this question HERE.

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 and M had 9 names in common, E and S had 7, and M and S had 10": EnM=19, EnS=7, and MnS=10;
"4 names were on all 3 rosters": EnMnS=g=4;

Question:: Total=?

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

Answer: 50. Discuss this question HERE.

Example 7:
There are 50 employees in the office of ABC Company. Of these, 22 have taken an accounting course, 15 have taken a course in finance and 14 have taken a marketing course. Nine of the employees have taken exactly two of the courses and 1 employee has taken all three of the courses. How many of the 50 employees have taken none of the courses?

Translating:
"There are 50 employees in the office of ABC Company": Total=50;
"22 have taken an accounting course, 15 have taken a course in finance and 14 have taken a marketing course"; A=22, B=15, and C=14;
"Nine of the employees have taken exactly two of the courses": (sum of EXACTLY 2-group overlaps)=9;
"1 employee has taken all three of the courses": AnBnC=g=1;

Question:: None=?

Apply second formula: $$Total=A+B+C-(sum \ of \ EXACTLY \ 2-group \ overlaps)-2*AnBnC + None$$ --> $$50=22+15+14-9-2*1+None$$ --> $$None=10$$.

Answer: 10. Discuss this question HERE.

Example 8 (hard):
In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

Translating:
"85% of those surveyed liked at least one of three products: 1, 2, and 3": Total=100%. Also, since 85% of those surveyed liked at least one of three products then 15% liked none of three products, thus None=15%;
"5% of the people in the survey liked all three of the products": AnBnC=g=5%;

Question:: what percentage of the survey participants liked more than one of the three products?

Apply second formula:
Total = {liked product 1} + {liked product 2} + {liked product 3} - {liked exactly two products} - 2*{liked exactly three product} + {liked none of three products}

$$100=50+30+20-x-2*5+15$$ --> $$x=5%$$, so 5% liked exactly two products. More than one product liked those who liked exactly two products, (5%) plus those who liked exactly three products (5%), so 5+5=10% liked more than one product.

Answer: 10%. Discuss this question HERE.

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.

Example 10 (hard DS question on three overlapping sets):
A student has decided to take GMAT and TOEFL examinations, for which he has allocated a certain number of days for preparation. On any given day, he does not prepare for both GMAT and TOEFL. How many days did he allocate for the preparation?

(1) He did not prepare for GMAT on 10 days and for TOEFL on 12 days.
(2) He prepared for either GMAT or TOEFL on 14 days

We have: {Total} = {GMAT } + {TOEFL} - {Both} + {Neither}. Since we are told that "on any given day, he does not prepare for both GMAT and TOEFL", then {Both} = 0, so {Total} = {GMAT } + {TOEFL} + {Neither}. We need to find {Total}

(1) He did not prepare for GMAT on 10 days and for TOEFL on 12 days --> {Total} - {GMAT } = 10 and {Total} - {TOEFL} =12. Not sufficient.
(2) He prepared for either GMAT or TOEFL on 14 days --> {GMAT } + {TOEFL} = 14. Not sufficient.

(1)+(2) We have three linear equations ({Total} - {GMAT } = 10, {Total} - {TOEFL} =12 and {GMAT } + {TOEFL} = 14) with three unknowns ({Total}, {GMAT }, and {TOEFL}), so we can solve for all of them. Sufficient.

Just to illustrate. Solving gives:
{Total} = 18 - he allocate total of 18 days for the preparation;
{GMAT } = 8 - he prepared for the GMAT on 8 days;
{TOEFL} = 6 - he prepared for the TOEFL on 6 days;
{Neither} = 4 - he prepared for neither of them on 4 days.

Answer: C. Discuss this question HERE.

Example 11 (disguised three overlapping sets problem):
Three people each took 5 tests. If the ranges of their scores in the 5 practice tests were 17, 28 and 35, what is the minimum possible range in scores of the three test-takers?
A. 17
B. 28
C. 35
D. 45
E. 80

Consider this problem to be an overlapping sets problem:
# of people in group A is 17;
# of people in group B is 28;
# of people in group C is 35;

What is the minimum # of total people possible in all 3 groups? Clearly if two smaller groups A and B are subsets of bigger group C (so if all people who are in A are also in C and all people who are in B are also in C), then total # of people in all 3 groups will be 35. Minimum # of total people can not possibly be less than 35 since there are already 35 people in group C.

P.S. Notice that max range for the original question is not limited when the max # of people in all 3 groups for revised question is 17+28+35 (in case there is 0 overlap between the 3 groups).

Answer: C. Discuss this question HERE.

____________________________________________________________________________________________________________
For more questions on overlapping sets check our Question Banks
Data Sufficiency Questions on Overlapping Sets
Problem Solving Questions on Overlapping Sets

Attachment:
Overlapping sets.png
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Great post! Thank You Bunuel.
A minor typo:
Q 6 has
"E and M had 9 names in common, E and S had 7, and M and S had 10": EnM=19, EnS=7, and MnS=10;

I believe, it should be
"E and M had 9 names in common, E and S had 7, and M and S had 10": EnM=9, EnS=7, and MnS=10;
Intern  Joined: 06 Nov 2014
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Bunuel wrote:
saidba wrote:
I'm sorry, can someone explain the distinction on when to use which Formula. What are the factors that would define the use of Formula 1 vs Formula 2, and vice versa?

Thanks in advance, and appreciate the great write up!

Bunuel I read the entire post and comments but I still have doubts. Here is what I understood regarding the difference between the two Formulas.

In the first formula we substitute the values of d,e and f given in the question and then get an answer.

While in the second formula Exactly-2 groups means = sum of d+e+f i.e in the question Exactly -2 groups means the sum of d+e+f is given rather than their individual values

Am I correct ??

Thanks and kudos !!
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Great post, thank you Bunuel, this will definately be my next stop. I only knew one of the two formulas and it didn't work on a question.
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In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

Translating:
"85% of those surveyed liked at least one of three products: 1, 2, and 3": Total=100%. Also, since 85% of those surveyed liked at least one of three products then 15% liked none of three products, thus None=15%;
"5% of the people in the survey liked all three of the products": AnBnC=g=5%;

Question:: what percentage of the survey participants liked more than one of the three products?

Apply second formula:
Total = {liked product 1} + {liked product 2} + {liked product 3} - {liked exactly two products} - 2*{liked exactly three product} + {liked none of three products}

$$100=50+30+20-x-2*5+15$$ --> $$x=5%$$, so 5% liked exactly two products. More than one product liked those who liked exactly two products, (5%) plus those who liked exactly three products (5%), so 5+5=10% liked more than one product.

Answer: 10%. Discuss this question HERE.

I understood everything else in this solution, except that I used the first equation, since the question doesn't say anything about liking exactly 2. It just asks how many liked more than 1. I used equation one to find the sum of 2-group overlaps, then added it to the 5% that liked all 3. If I'm missing something, could you please help clarify?

Thanks!
Math Expert V
Joined: 02 Sep 2009
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toby001 wrote:
In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

Translating:
"85% of those surveyed liked at least one of three products: 1, 2, and 3": Total=100%. Also, since 85% of those surveyed liked at least one of three products then 15% liked none of three products, thus None=15%;
"5% of the people in the survey liked all three of the products": AnBnC=g=5%;

Question:: what percentage of the survey participants liked more than one of the three products?

Apply second formula:
Total = {liked product 1} + {liked product 2} + {liked product 3} - {liked exactly two products} - 2*{liked exactly three product} + {liked none of three products}

$$100=50+30+20-x-2*5+15$$ --> $$x=5%$$, so 5% liked exactly two products. More than one product liked those who liked exactly two products, (5%) plus those who liked exactly three products (5%), so 5+5=10% liked more than one product.

Answer: 10%. Discuss this question HERE.

I understood everything else in this solution, except that I used the first equation, since the question doesn't say anything about liking exactly 2. It just asks how many liked more than 1. I used equation one to find the sum of 2-group overlaps, then added it to the 5% that liked all 3. If I'm missing something, could you please help clarify?

Thanks!

Check the discussion here: in-a-consumer-survey-85-of-those-surveyed-liked-at-least-98018.html

Hope it helps.
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Hi Bunuel,
Thanks a ton for this post. Really helped me clear my mind of useless assumptions.
Manager  B
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Apparently this is an excellent post although it didn't work so well due to notation. It took me great amount of time to decode the concept and eventually I think I cracked it down to this.

The notation I am reffering (sum of EXACTLY 2group overlaps) is misleading imho as the − can be intrpreted as " minus " and causes confusion ... Perhaps is better to find something more intuitive.
Intern  B
Joined: 23 Dec 2011
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Thanks Bunuel for sharing this. +1 Kudos!!!

The two formulas are really helpful and work each and every time. As mentioned by others, it took some time for me to understand both the formulas. So, I kind of tried to write the formula a way for my better understanding. Attaching a screenshot for the same. Hope this might be helpful for others.
Attachments overlappingSet.jpg [ 464.59 KiB | Viewed 4151 times ]

Intern  B
Joined: 28 Feb 2017
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Can someone please help me understand the difference between "sum of 2-group overlay" vs. "sum of EXACTLY 2-group overlay"

Exam is in a few days - is it possible to get by memorizing the 2 separate formulas and hoping that the question stem will explicitly state "EXACTLY 2"?

Just looking at the OG (2016, PS # 192) the hardest OS problem (the 300 subject virtual reality experiment) states "EXACTLY 2"
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Bunuel wrote:
3111987 wrote:
Dear Bunuel,
huge thanks for this post:
I'm trying to find difference between two given formulas in order to apply them correctly.

Am I right to say that formula #1 applicable to GENERAL CASES
FIRST FORMULA

$$Total = A + B + C - (sum \ of \ 2-group \ overlaps) + (all \ three) + Neither$$.

while formula#2 might be used ONLY when in premise says something as belong/ include ONLY to two groups/categories?
SECOND FORMULA

$$Total = A + B + C - (sum \ of \ EXACTLY \ 2-group \ overlaps) - 2*(all \ three) + Neither$$.

Thanks a lot for help-you are my hero! Yes, you can generalize this way. Check examples in my post to verify.

hi Bunuel,

I am still not clear on when to use which formula, Why can't we use the second formula to solve the first example can you please tell some specific detail in the question stem which can direct us to one of the those formulas .

Thanks
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Hello Bunuel!

Quick question: For overlapping sets questions, is there a rule for when to use this formula over using the table method (see attached image)?

I think I have a general idea of what method to use when but I can't seem to formulate this in words... for example, regarding a question on students in different classes I would use the formula you posted, but for a yes/no population question like the attached image, I would use a table.

Thanks!
Attachments Screen Shot 2017-07-25 at 12.11.42 PM.png [ 28.79 KiB | Viewed 3703 times ]

Intern  Joined: 29 Apr 2018
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Excellent. Sir will you please explain the formula for finding the members/things in exactly one of the category? Accordng to formula it is a + b + c Go to page   Previous    1   2   [ 34 posts ]

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