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Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Tuesday, 18 September 2007, 04:49 PM

For all the CAT aspirants taking CAT in 2009 or 2010, this chapter should provide some insight into Venn diagramsand methods for solving the problems. This chapter comes on the demand of some high octane TG users who areresponsible for my lack of sleep and excessive intake of caffeine last night. I hope this resolves many of theirproblems in Venn diagrams.

Venn diagrams are pictorial representations used to display mathematical or logical relationships between two or more

given sets (groups of things). The drawing consists of two or more circles, each representing a specific group. Each Venn diagram beginswith a rectangle representing the universal set. Then each set in the problem is represented by a circle. Any values that belong to morethan one set will be placed in the sections where the circles overlap. A typical venn diagram is shown in the figure below:

In the figure, set A contains the multiples of 2 which are less than 30 and set B contains multiples of 3 which are less than 25. Therefore, A ={2, 4, 6, 8, 10, 12... 26, 28} and B = {3, 6, 9, 12... 21, 24}. The various areas in the above diagram depict the following relationships:

Intersection (AB)- Denotes the set of elements that are shared by two or more given sets. In the figure

given below, the intersection of the two sets is shown.

A B = {6, 12, 18, 24}

Only A or Only B- The part of set A, or set B, which is not shared by any other set is known as "only A," or "only B." In the figuregiven below, the two parts are shown:

Only A = {2, 4, 8, 10, 14, 16, 20, 22, 26, 28}, only B = {3, 9, 15, 21}

Union (AUB)- Denotes all the elements of the given sets taken once.

A U B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28}

It can be seen that



The venn diagram for three sets is shown below:

It can be shown that

Problem-solving through Venn diagrams:

I use the following method to solve problems through Venn diagrams:

Solved Examples:

Of all the users on Totalgadha.com, 80% spend time in CAT Quant-DI forum whereas 60% spend time in CAT verbal forum. If only thoseusers will crack CAT who spend time in both the forums, what percentage of users of TotalGadha

will crack CAT? will not crack CAT?

Answer: n(AUB) = n(A) + n(B) - n(AB) 100% = 80% + 60% - n(AB) n(AB) = 40%

Therefore, 40% users of TG will crack CAT. And 60% of users (only A + only B) will not crack CAT.

NOTE: See that the surplus (superfluous part) can only be adjusted inside the area denoted for the intersection of the sets, a fact we will usein maxima- minima type of questions.

A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of the three popular options airconditioning, radio and power windows were already installed. The survey found:

15 had air conditioning2 had air conditioning and power windows but no radios12 had radio6 had air conditioning and radio but no power windows11 had power windows4 had radio and power windows3 had all three options.

What is the number of cars that had none of the options? (CAT 2003)1. 4 2. 3 3. 1 4. 2

Answer: We make the Venn diagram and start filling the areas as shown:

Total Number of cars according to the diagram = 2 + 6 + 3 + 1 + 5 + 2 + 4 = 23.Therefore, number of cars having none of the given options = 25 - 23 = 2.

New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhiindividually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyaniworks with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha. The totalnumber of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.



(CAT 2003- Leaked)

What is the number of projects in which Gyani alone is involved?1. 02. 1.3. 4.4. cannot be determined

What is the number of projects in which Medha alone is involved?1. 02. 1.3. 4.4. cannot be determined

Answer: The Venn diagram for the three consultants is shown below:

Total Number of projects = 2 x number of projects in which more than one consultant is involved - 1 = 2 x 19 - 1 = 37.Therefore, X + 8 + 6 + 3 + Y + 2 + X + Y - 16 = 37 X + Y = 17. The values of X or Y cannot be uniquely determined. Medha alone isinvolved in X + Y - 16 = 17 - 16 = 1 project.

Concept of Maxima and Minima:

1. When the total number of elements is fixed

Let's have a look at the Venn diagram of two sets again:

Imagine that in the beginning, the number of elements in all the areas is zero, as shown above. All the sets are empty right now.

Let's see what happens if I insert one element inside AB:

We can see that adding 1 element to AB increases the number of elements in both A and B by 1. The total number of elements in all

areas combined is 1 only (0 + 1 + 0) but if you add the number of elements in A and B (A + B), the addition will come up to 2. Therefore,adding 1 element to AB gives an extra 1 element. Hence, for every surplus of 1 element we can add 1 element to AB.

Lets see the Venn diagram for 3 sets:

In diagram 1, we have added 1 element to intersection of only two sets (A and B but not C). We can see that A and B both increase by 1 and



therefore we get a surplus of 1 element.

In diagram 2, we have added 1 element to intersection of all the three sets (A and B and C). We can see that A, B and C all three increaseby 1 element each and therefore we get a surplus of 2 elements.

Therefore, in case of three sets, we can accommodate the surplus by

adding elements to intersection of only two sets in which case a surplus of 1 element can be accommodated by increase of 1 elementin the intersection of only two sets.

adding elements to intersection of three sets in which case a surplus of 2 elements can be accommodated by increase of 1 element inthe intersection of three sets.

How is this related to maxima and minima?

Let's see:

According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is the minimumpercentage of people who like all three?

Answer: Let's first calculate the surplus:

percentage of people who like apples + percentage of people who like bananas + percentage of people who like cherries = 70% + 75% +80% = 225% a surplus of 125%.

Now this surplus can be accommodated by adding elements to either intersection of only two sets or to intersection of only three sets. As theintersection of only two sets can accommodate only a surplus of 100%, the surplus of 25% will still be left. This surplus of 25% can beaccommodated by adding elements to intersection of three sets. For that we have to take 25% out of the intersection of only two sets andadd it to intersection of three sets. Therefore, the minimum percentage of people who like all three = 25%

The question can be solved mathematically also. Let the elements added to intersection of only two sets and intersection of three sets be xand y, respectively. These elements will have to cover the surplus.

-->x + 2y = 125%, where x + y 100%. For minimum value of y, we need maximum value of x.--> x = 75%, y = 25%.

In a college, where every student follows at least one of the three activities- drama, sports, or arts- 65% follow drama, 86% follow sports,and 57% follow arts. What can be the maximum and minimum percentage of students who follow all three activities exactly two activities

Answer: Let us again see the surplus:

Percentage of students who follow drama + Percentage of students who follow sports + Percentage of students who follow arts = 65% +86% + 57% = 208% surplus = 108%. This surplus can be accommodated through adding elements either to intersection of only two setsor to intersection of only three sets. As the intersection of only two sets can accommodate only a surplus of 100%, the surplus of 8% will stillbe left. This surplus of 8% can be accommodated by adding elements to intersection of three sets. For that we have to take 8% out of theintersection of only two sets and add it to intersection of three sets. Therefore, the minimum percentage of people who like all three= 8%. In this case the percentage of students who follow exactly two activities will be maximum = 92%.

The surplus of 108% can also be accommodated through adding elements to only intersection of three sets. As adding 1 element tointersection of three sets give a surplus of 2 sets, adding 54% to intersection of three sets will give a surplus of 108%. Therefore, themaximum value of students who follow all three activities is 54%. In this case the percentage of students who followexactly two activities will be minimum = 0%.

We can also solve it mathematically x + 2y = 108%, where x + y 100%. The maximum value of x will give minimum value of y,whereas minimum value of x will give maximum value of y.

2. When the total number of elements is NOT fixed

In this case we assign the variables to every area of the Venn diagram and form the conditions keeping two things in mind: try to express the areas in the Venn diagram through least number of variables. all the numbers will be zero or positive. No number can be negative.

Out of 210 interviews of IIM- Ahmedabad, 105 CAT crackers were offered tea by the interview panel, 50 were offered biscuits, and 56 wereoffered toffees. 32 CAT crackers were offered tea and biscuits, 30 were offered biscuits and toffees, and 45 were offered toffees and tea.What is the maximum and minimum number of CAT crackers who were offered all three snacks? maximum and minimum number of CAT crackers who were offered at least one snack?

Answer: Lets make the Venn diagram for this question. Since we want to assume least number of variables, we can see that assuming avariable for the number of students who were offered all three snacks will help us express all the other areas. Let the number of studentswho were offered all three snacks = x.

In the above diagram, we have expressed all the areas in terms of x. To decide maximum value of x, we note that 32 - x, 45 - x and 30 - xwill be zero or positive. Therefore, the maximum value of x will be 30. (30 is the lowest among 30, 32 and 45). To decide minimum value of



x, we note than x - 19 and x 12 will be zero or positive. Therefore, x cannot be less than 19 (19 is the higher number between 19 and12).

Therefore, maximum and minimum number of CAT crackers who were offered all three snacks = 30 and 19.

The number of CAT crackers who were offered at least one snack = Total number of CAT crackers in the Venn diagram = x + 28 + 32 - x +x + 45 - x + x - 19 + 30 - x + x - 12 = 104 + x.As the maximum and minimum values of x are 30 and 19, respectively, the maximum and minimum value of 104 + x will be 134 and 123,respectively.

Maximum and minimum number of CAT crackers who were offered at least one snack = 134 and 123.

I am afraid I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems basedon this in the CBT Club this week.

Reply

Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Small Wonder - Tuesday, 18 September 2007, 05:02 PM

TG Sir ki jai ho

Thanks a ton. God Bless You!

Small Wonder!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Vaidyanathan Ganesan - Tuesday, 18 September 2007, 05:11 PM

TG,

I was always on the look for your articles on the homepage and this time i was not disappointed.. . A new article!!!!

Keep up your good work.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Gul Gul - Tuesday, 18 September 2007, 05:11 PM

YAhooooooooo!!!!! Thnx TG Muahhhhhhhhh.......

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby kishore ayyadevara - Tuesday, 18 September 2007, 05:24 PM

Thanks a lot TGi could not get the following point..

Percentage of students who follow drama + Percentage of students who follow sports + Percentage of students who follow arts = 65% +86% + 57% = 208% surplus = 108%. This surplus can be accommodated through adding elements either to intersection of only two setsor to intersection of only three sets. As the intersection of only two sets can accommodate only a surplus of 100%, the surplus of8% will still be left. This surplus of 8% can be accommodated by adding elements to intersection of three sets. For that we have to take 8%out of the intersection of only two sets and add it to intersection of three sets

but here we are considering 3 sets...so how come u took only 2?the other 2 intersection are 0 then?

Regards,Kishore.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Sri KLR - Tuesday, 18 September 2007, 05:36 PM

TG,



Good method to solve max and min.....

x+2y and x+y is too good...I like equations....no thinking needed from now on..

So far I used hit and trial.....but NO more ...feel relieved now....

Please give us some problems on four dimensionals also....

Also give some more prob on venns.. as exercise on Quant-DI forum...

I have kinda premonition that CAT might rear it's ugly head again....CAT(We ) is (are) going thru Testing times...

I have seen one prob in cat 2 years back....one tough venn prob..don remember...will get back with it..

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Software Engineer - Tuesday, 18 September 2007, 06:31 PM

Here, the quality of content is best, relally ! Thank You.

Software Engineer

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby King Kong - Tuesday, 18 September 2007, 07:01 PM

TG == Too Good !!?? [ ] Awesome stuff...

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Tuesday, 18 September 2007, 07:09 PM

Hi Kishore,

"intersection of only two sets" means area containing intersection of two sets but not the third set. In the figure given below, the area in red isthe intersection of only two sets.

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby padmaselvan lakshman - Tuesday, 18 September 2007, 07:38 PM

hi TG,

Thanx a lot for such a wonderful article.. U have made maxima and minima look a lot more easier through this article..

as suggested earlier, pls give us a quiz on Venn Diag for practice..

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hi TG,the thing is...u said that the intersection can take a maximum surplus of 100% but, why isnt it more than 100?

if the 3 shaded portions are a,b,c . then what if a+b = 100%(i.e., all the contents of the first circle are in the shaded part) and c>0?

Regards,Kishore.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby the underdog - Tuesday, 18 September 2007, 08:12 PM

"we are not worthy" "we are not worthy"*



* from the movie "Wayne's World"

hehe.. thanks TG!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Akon Convict - Tuesday, 18 September 2007, 08:32 PM

Thank U very much TG.....

God Bless You

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Tuesday, 18 September 2007, 08:34 PM

Hi Kishore,

How can something be more than 100%? If I have given you 10 boys at most can you have more than 10 boys combining all the areas?

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Crazy CAT - Tuesday, 18 September 2007, 08:49 PM

Thanx a tonne Tg ,

It was worth waiting for such a long time for this article.

one request Sir ,plz give some tips like this on functions and graph article also.

Regards

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a topic on cubes also please.....

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby the underdog - Tuesday, 18 September 2007, 09:25 PM

+1 Gullz Golu

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Help me in the following problem TG

All students in a class of 100 attended a summer camp. Each student had the option of enrolling for coaching in atmost 3 namely Football,Cricket and hockey.

a students had enrolled for hockey

b for football

c for cricket

Also d students had enrolled for eactly 1 sport, e for exactly 2, f for exactly 3 and g students had not enrolled for any of the three sports.

Q1 If d>e>f and c is less than a as well as b, what is the maximum possible value of c?

Q2 If a,b,c,d,e,f and g are all distinct the minimum possible no. of students who enrolled for atleast 1 sport is ?

Q2 If a is less than b as well as c, the no. of students who enrolled only for hockey is ?

Please Help....

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TG

seems i made some horrible interpretations thenthought A ,B,C are 100% each

what i find is the thrice intersected part(part common to all the circles) is counted thrice and (AnB) (BnC)(CnA)are counted twice

coming to the problem,

so the surplus 108% should be distributed in the following way

x+2y = 108% ( where x is the intersection of 2 circles n y is common to three)

my problem is , i could not get how x+y = 100%...do u mean to say that the elements belonging to ONLY A, B,C are all zero and every elementlies in the intersection parts?

Regards,Kishore.

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Kishore,

See if this is clear:

Say all students = 100 % in terms of equation a1+a2+a3=100 and we have a1+2a2+3a3=208. Subtract the above 2 equations we haavea2+2a3=108 now a3 should be minimum 8 and a2 maximum 92 i.e. 92+2*8=108 see now a1+a2+a3 = 100 is maintained. Also a3 max= 54 inthis case a2=0 and a1 will take 46. Is it a bit clear now?

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Top CAT - Tuesday, 18 September 2007, 11:09 PM

Exhaustive yet Simple

that,s ur trademark.....that's what we all like about this site...

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Abhi S - Wednesday, 19 September 2007, 02:21 AM

nw iv gone mad...i thot u r lost somewhere...bt u...bt u came wid a bang....thnx TG sir.....

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Abhi S - Wednesday, 19 September 2007, 02:23 AM

nw plz...u may hav a rest....let us hav a sip f d same caffiene....take care TG sir...

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby anand pani - Wednesday, 19 September 2007, 08:17 AM

great

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby kishore ayyadevara - Wednesday, 19 September 2007, 10:40 AM

Thank You Gullz

it's clear now...i was having problems with interpreting the assumptions

and thanks TG for this useful article

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Re: Venn Diagrams- Basics, Problems, Maxima and Minima



by Gul Gul - Wednesday, 19 September 2007, 10:46 AM

Thanks to TG....

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby dinesh munna - Wednesday, 19 September 2007, 12:41 PM

Sir...I am sorry but this article was not exactly what i expected.I was expecting you to put sum fundaes on solving problems like the one justposted here by sum guy.Pls luk into it and fit ur solution into that.thanq

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Wednesday, 19 September 2007, 01:22 PM

Hi Dinesh,

Unfortunately, I am not preparing you for a Math Olympiad but CAT only. Have a look at the Venn diagram problem that came in CAT 2006:

A survey was conducted of 100 people to find out whether they had read recent issues of Golmal, a monthly magazine. The summarizedinformation regarding readership in 3 months is given below: Only September: 18, September but not August: 23, September and July: 28,September: 28, July: 48, July and August: 10, None of the three months: 24.

What is the number of surveyed people who have read exactly two consecutive issues (out of the three)?(1) 7 (2) 9 (3) 12 (4) 14 (5) 17

The article already has question that came in previous CAT papers. There has been no mention of even maxima or minima in CAT.

I sincerely believe that this is all that you need to know to tackle Venn diagram problems in CAT.

As for the fundas for solving problems like the one posted here, I can solve it through the fundas mentioned in the article only.

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Tom Goel - Wednesday, 19 September 2007, 01:30 PM

I know what we can say in respect of TG falls very short of what he deserves...a word of thanks for TG not from the keyboard but from heart.

Thanks a lot and keep the good work going.......

+1 regrading article on cubes ;)

Amit Goel

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Wednesday, 19 September 2007, 01:46 PM

Hi Tom,

Let me try. I can certainly imagine the hours on 'paint' I will have to spend to draw images of cubes

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Gul Gul - Wednesday, 19 September 2007, 02:06 PM

TG

PLZ help out in the question posted i was able to crack half questions in that set but the rest half posted is posing some problem. Plz help sothat i can be completely confident in this topic..

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ravi kumar - Wednesday, 19 September 2007, 03:06 PM

TG

I feel ashamed to learn so much from you for free. Give me your address I want to send you some gurudakshina. Also feel sorry for ppl who

are not gadhas yet



best regards

Ravi Kumar

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby jitendra havaldar - Wednesday, 19 September 2007, 04:37 PM

Hi TG,

In the problem of cars having ACs,power windows(PW),radio(R) i didn't understand the way you have splitted the radio and the power windows

part by taking (4-3) = 1 in the area common to both R & PW ...

It would really be great if you take some time to explain this ..any help from anyone is also welcome on this....

Thanks.

Jitendra

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Abhishek Deb - Wednesday, 19 September 2007, 04:49 PM

Hi TG,

I wanted to know one thing that u said that "minimum values of the students following three activities will be 54% max indrama, arts ques.. I wanted to know then what will be the min of all the ppl having two activities interest.... will it be54%....?If not then how to solve it......

Abhishek.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Nitin Jain - Wednesday, 19 September 2007, 05:14 PM

Hello TG

Please clarify my doubt. You said that there are two cases for solving maxima and minima problems. In the former the total no of elementsare fixed and in the latter the total no of elements are not fixed. I guess total no of elements imply AUBUC? Because in the last question ofCAT interviewers , total no of elements are fixed i.e 210.Then how can we classify the question to fall under fixed elements or not?

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby sun is here - Wednesday, 19 September 2007, 09:04 PM

HI TG , A survey was conducted of 100 people to find out whether they had read recent issues of Golmal, a monthly magazine. Thesummarized information regarding readership in 3 months is given below: Only September: 18, September but not August: 23, September andJuly: 28, September: 28, July: 48, July and August: 10, None of the three months: 24.


iam not able to solve this problem please throw some light on it

Thanks Sun

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Thursday, 20 September 2007, 01:02 AM

Hi Gullz,

lease give me the options. I calculated highest value of c as 48% assuming some values but I do not know if you need answer in terms of a, b,d, e, f or otherwise.

Total Gadha



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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Mou Sukoshi - Thursday, 20 September 2007, 01:07 AM

A survey was conducted of 100 people to find out whether they had read recent issues of Golmal, a monthly magazine. The summarizedinformation regarding readership in 3 months is given below: Only September: 18, September but not August: 23, September and July: 28,September: 28, July: 48, July and August: 10, None of the three months: 24.


are these figures correct??

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby R V - Thursday, 20 September 2007, 07:47 AM

Hi Gullz how did you get the equation a1+2a2+3a3=108?? Could you please explain.

Thank you

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Gul Gul - Thursday, 20 September 2007, 09:37 AM

TG m so sorry i didnt write the options....I m in office and i forgot 2 bring the paper in which the question was given i will definitely post it in theevening... But i remember the options for following question

Q2 If a,b,c,d,e,f and g are all distinct the minimum possible no. of students who enrolled for atleast 1 sport is ? a) 9 b)12 c) 15 d) none

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Gul Gul - Thursday, 20 September 2007, 09:41 AM

R V

a1 only 1

a2 exactly 2

a3 exactly 3

when you add all a2 is added twice and a3 thrice see a venn diagram for 3 circles u will get it..

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Thursday, 20 September 2007, 10:17 AM

Hi Sun is here,

The options to this CAT 2006 question are NOT correct. This was one of the two questions in CAT 2006 that was wrong.

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Mohit Goyal - Thursday, 20 September 2007, 10:31 AM

Hi TG.

Wonderful Attempt to unearth the Concepts of venn Dig !! I have some very interesting sets on Venn dig and will shortly post them but only ifyou promise to reply me after solving cause i have posted 2 queries earlier n dint get ne reply....I am not a regular user of TG.com but i reallybelieve that you are doing a Noble Job Sir....Hats Off ...

Mohit

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby SS VV - Thursday, 20 September 2007, 10:57 AM

knew you were taking a while since u had to write it up to your standard which is excellent :>:>

you rock!!



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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby gourav kalra - Thursday, 20 September 2007, 01:24 PM

Hey..lovely article...any word of praise wud be less...

u really rock...am impressed... )

god bless u!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Thursday, 20 September 2007, 02:43 PM

Hi Mohit,

I will certainly reply to you. Do post your problems.

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Nitin Jain - Thursday, 20 September 2007, 03:12 PM

Hi TG

Pls reply to my post also given above.

Thanks

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Arun Prasad G - Thursday, 20 September 2007, 03:18 PM

TG, it is regarding this problem of apples, bananas and cherries.

x + 2y = 125%, where x + y 100%. For minimum value of y, we need maximum value of x.--> x = 75%, y = 25%.

I understand this. But after solving this, I wanted to fit this back into the venn diagrams and I am totally confused. For example,

I wrote AnBnC as 25, and AnB excluding C as 75, BnC excluding A as 75 and so on. Now I am trying to find only apples, only banana's andonly cherries. I am not able to get it fully as I am not getting 100% after writing individual percentages. Please explain this to me.

Also I have posted a query on your "How to find the Units Digit of a Number?" . Please respond tp that also.

THANK YOU VERY MUCH FOR YOUR WONDERFUL CONTRIBUTION TO ALL ASPIRANTS. ITS FANTASTIC TO READ EACH AND EVERY ARTICLEOF YOURS.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby rishi sawla - Thursday, 20 September 2007, 03:41 PM

Hey Arun,

I think Sum of AnB excluding C , BnC excluding A and AnC excluding B is 75.

Not individually.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ravi teja - Thursday, 20 September 2007, 05:03 PM

hai gulz,



i was not able to solve even a single question in the set u mentioned.can u pass the answer to me too if u get it please.if needed i can providethe options too.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby neelakanta siva - Thursday, 20 September 2007, 05:20 PM

for GOLU'S question ,

only hockey+only football+only cricket > both H & C and not F +both C & Fand not H+both F &

Hand not C > ALL THREE

means H only + F only + C only > intersection of 2 only > intersection of 3

we can take intersection of 3 as 0

now intersection of 2 games only can take a max value of 49, then only the other remaining part can be 51 and be greater.

49 can be alloted to the intersection of H and C only as we have to max cricket (C)

then intersection of F&C Xonly= 0

intersection of F&H Xonly=0

the remaining part can take 51 which should be divided as 16+17+18 and the 16 should be alloted to cricket(C) and 17 and 18 can be allotedto only F and only H. This is becz of the condition (C has to be less than A and B)

SO 0+0+49+16 = 65 IS THE MAX VALUE CRICKET CAN TAKE

PART 2 &3 OF THE Q IS NOT COMPLETE

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Thursday, 20 September 2007, 07:46 PM

Hi Nitin,

In the previous sets, it has been mentioned that every student follows at least one activity. This kind of statement is critical to determine thatAUBUC is fixed. Even if the total number of students/people etc. are given, if it is not given that every person belongs to at least one category,AUBUC is not fixed.

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby R V - Friday, 21 September 2007, 08:09 AM

hi Gullz, thanks for the explanation. i understood it.

RV

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Gul Gul - Friday, 21 September 2007, 09:04 AM

TG so sorry yesterday net was nt working at my place...,here r d options

If d>e>f and c is less than a as well as b, what is the maximum possible value of c? Options a) 63 b)64 c)65 d)66 e) none

If a,b,c,d,e,f and g are all distinct the minimum possible no. of students who enrolled for atleast 1 sport is ? a)9 b)10 c)11 d) 13 e) none

If a is less than b as well as c, the no. of students who enrolled only for hockey is ? a) 32 b) 48 c) 49 d) 50 e) none

Answers are c, e, c respectively......I hav 2 go 4 a full day training

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby sonal singh - Friday, 21 September 2007, 11:21 AM

hello Tg



Thanks for such a wonderful article.

Please explain how did u get 4-3 =1 for the following.and 15 had air conditioning specify all the elements in B or Only B

survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of the three popularoptions air conditioning, radio and power windows were already installed. The survey found:

15 had air conditioning2 had air conditioning and power windows but no radios12 had radio6 had air conditioning and radio but no power windows11 had power windows4 had radio and power windows3 had all three options.

What is the number of cars that had none of the options? (CAT 2003)1. 4 2. 3 3. 1 4. 2

Answer: We make the Venn diagram and start filling the areas as shown:

Total Number of cars according to the diagram = 2 + 6 + 3 + 1 + 5 + 2 + 4 = 23.Therefore, number of cars having none of the given options = 25 - 23 = 2. thanks

sonal

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Sumit Singla - Friday, 21 September 2007, 09:04 PM

Thanks for the brilliant article TG... Something that I solved mostly by hit and trial has been made simple enough to be solved verbally! YOUROCK!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Saurabh Goyal - Friday, 21 September 2007, 11:46 PM

Hi Tg,Found ur Venn diagram topic very helpful, but still getting ques with more difficulty...I am facing problem in getting the logic of the question mentioned below..

Q. In a town 70% of the persons suffer from disease A, 80% from disease B, 75% from C and 85% from disease D and "P%" from all the fourdiseases. Find the minimum value of P.

i tried this ques by the above mentioned funda in your post, but not getting any where with the logic. will it be 200% of surplus now?as thecommon 1 between all four fields will be increased by 1 in comparison to 3 fields???

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Saturday, 22 September 2007, 11:50 AM

Hi Saurabh,

A + B + C + D = 310%. Therefore, surplus = 210%. Now, the intersection of 4 sets can accommodate a surplus of 3 per element. Theintersection of 3 sets can accommodate a surplus of 2 per element. The maximum surplus it can accommodate is 2 100% = 200%. Thesurplus of 10% would be left. Now you can decrease intersection of two sets and shift the decreased amount in the intersection of 3 sets. Goingby this manner, you can quickly obtain that if you keep 90% in intersection of 2 sets and 10% in the intersection of 3 sets, the surplusaccommodated = 90 2 + 10 3 = 210%.

Therefore, the minimum value of P = 10%. The maximum will be when P accommodates all the surplus. Therefore, maximum value = 210/3 =70%

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby nitin verma - Monday, 24 September 2007, 03:39 PM

Hi TG,

I like this site very much. I thank you for entertaining my request of explaining my doubt. I am very much thankful to you. YOur explanation isfantastic and easy to grasp. You wont believe before i struggled hard to understand the concept but could not, but your docile explanation is soclear that any sane person can grasp it. Thank you very much. I will come back to you with more such request. You are helping cat aspirants ina big way.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Gul Gul - Tuesday, 25 September 2007, 08:52 AM

TG options r given now.........

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Tuesday, 25 September 2007, 11:31 AM

Hi Gullz,

Have taken the printout of your question. Let me have a go at it.

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Abhra Chatterjee - Wednesday, 26 September 2007, 06:14 PM

A too good article........

Thanks a lot TG

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby amit sharma - Thursday, 27 September 2007, 02:28 AM

Hi TG

It was a goog article but i have one query

In the above que we escape the intersection of 2 set which can accomodate 1 surplus per element and if we considder it then min value of P=0.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby abc abc - Thursday, 27 September 2007, 01:56 PM

Thanks a ton for this one !!!!!!

This topic has not figured at length in most of the notes available to students !!!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby shinjini mondal - Saturday, 29 September 2007, 11:09 PM

hi TGnice article,esp liked maxima minima application in it.and please do add for combination of four sets.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby sweta x - Sunday, 30 September 2007, 08:42 PM

Cities.doc

Hi TG , I've been following your forum since long now ....And I must say thank you for all the sincere effort put into making MBA prep easier for all ofus.

I have a doubt regarding the venn diagrams article .....Had a problem on similar lines in Mock CAT 6 of time and tried using the funda ,( it was a problem with 5 sets of data overlapping) , so I thinkI might not have understood the problem properly.



Attached is the problem and my approach ....

I'd be very gr8ful if u could tell me whats wrong with my approach !!!!!!!!!!!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby pranav tendolkar - Sunday, 30 September 2007, 11:13 PM

reeeeeeeeeeaaaallly very goood,

can u explain same things in case of 4 or 5 circle venn diagrams... is there any theoretical way other than drawing venn diagram as higherdiagrams are complex

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Alpa dedhia - Monday, 1 October 2007, 10:23 AM

Hi sweta,for the first question i have simply taken the least number of people in each cities who can opt for all the qualities .so the no is 26+25+26+18+15+29 = 139

for the second question .. is 37 the answer .. please let me knw..

regardsAlpa

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ran for_cat - Tuesday, 2 October 2007, 12:35 PM

for the 1st one

a+b+c-e-2f+g=100

also d+e+f+g =100

now a+b+c = 100+e+2f-g

so e+2f+-g should be max...

now d+e+f+g =100 and d>e>f ==> for e and f to be as max as possible e should be 32 and f is 33...(g=0 and f=35)

so a+b+c =100+32+66=198

so for a>c and b>c c can have a max value of 65 here.

for the 2nd question:

a+b+c-e-2f+g=100

d+e+f+g =100

keep d=0 ==> we can check for any arbitary values of e f and g that the distinct values are possible

e+f+g =100 or g=1 e=50 f=49

a+b+c =100 -1 +50+49=198

a b c can have different values here as well.

for the 3rd question b>a and c>a

==> let say b and c constitutes the same elements in that case the max possible value of a is 49.where a contains all the elements distinctfrom b and c

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Uthay Mathi - Tuesday, 2 October 2007, 05:21 PM

This is a cool stuff dear TG.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby lavika gupta - Wednesday, 3 October 2007, 04:26 PM



hi tg.pls post some text on functions,maxima and minima

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby sweta x - Wednesday, 3 October 2007, 11:34 PM

Hey Alpha, U got it right how did u solve the second one (37)

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Kumar M - Wednesday, 3 October 2007, 11:51 PM

Hi TG

In the first exampe problem of maxima & minima u have mentioned "As the intersection of only two sets can accommodate only a surplus of100%".How it can be only 100%

why cant we accommadate the suplus of 110% as follows?

only Apple and Banana = 30%;only apple and cherry = 40%; only Banana and Cherry = 40% remaining 5 can be only Banana.

Can we minimise the only three part as i explained. Please help me....

Thanks & Regards,

CM

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Varun Choudhary - Thursday, 4 October 2007, 11:59 AM

Kudos Sir.............

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby sgx100 Online - Saturday, 6 October 2007, 07:04 AM

TG Sir

Mindblowing article Absolutely terrific !!!

Thnx a lot

Sirif u can help with one article on Graphs (linear,quadratic,modulus...)it wud be really helpful to all of us

Thanking You

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ashish tyagi - Saturday, 6 October 2007, 11:20 AM

really a very gud article t g sir, i have a problem pls solve it........

1 A survey was conducted on the eating habits of a group of 1000 people .results show that 92% of the people surveyed eat south Indianfood, 91% eat north Indian food,82% eat American food , 78% eat Chinese food, 79% eat italin food and 80% eat continental food. What mustbe the minimum no. of people who eat all the 6 type of food, if 7 people do not eat any of the 6 types of food?

A:0 B:13 c:27 D:55

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby shriram samant - Saturday, 6 October 2007, 05:10 PM

fantastic...

mindblowing...

superb....



thanks TG.... Total Genius.....

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Ankush Keshwani - Sunday, 7 October 2007, 05:52 PM

A very good explanation.Thanks a lot!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Abhra Chatterjee - Thursday, 11 October 2007, 10:19 AM

Hi TG,

I am not able to solve this question properly: A survey about Tv viewership was conducted on 100 respondents. The results are, 93 liked sony,89 liked Zee, 81 liked Stratv, 75 liked zee cinema, 78 like Mtv. 1 did not like any of the above.Find the minimum number of people liking all the the 5 channels.I will tell you the approach that I followed. Please correct me where I am going wrong.The total number of people watching any of these channels = (100-1)=99(I don't watch any).Now the surplus = [(93+89+81+75+78) - 99] = 317. Now for every increase in the number of viewers watching two channels, there is actually a20% increase from the initial (I am not sure whether we have to consider all the 5 combinations or not). Then for every increase in the numberof people watching 3 channels, there is a resultant increase of [8-5]/5 = 60% (Here also are we to consider the total number of combinationspossible). Again for every increase in the number of people watching 4 channels, the resultant increase = [9-5]/5 = 80%.Now here I consider the combinations of groups possible for each of these groups:2 channel groups = 5.3 channel groups = 3.4 channel groups = 2.So the surplus we have been able to cover till now = (100+180+160) = 340. So there need not be any people watching all the five channels.

I am not at all confident of the answer arrived at.Somebody Please help !!!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Prasad L - Saturday, 13 October 2007, 12:01 PM

TG Sir,

I have no words to describe the amazing work you are doing ......

BTW, the TIME material also has some quick methods to crack the venn diagrams. Hope the TG users make use of that as well... ATB!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Mou Sukoshi - Monday, 15 October 2007, 01:05 PM

hi ashish.....

A survey was conducted on the eating habits of a group of 1000 people .results show that 92% of the people surveyed eat south Indian food,91% eat north Indian food,82% eat American food , 78% eat Chinese food, 79% eat italin food and 80% eat continental food. What must bethe minimum no. of people who eat all the 6 type of food, if 7 people do not eat any of the 6 types of food?

A:0 B:13 c:27 D:55

ppl having south indian food = 920

ppl having north indian food = 910

ppl having American food = 820

ppl having Chinese food = 780

ppl having Italian food = 790

ppl having Continental food = 800

hence, total= 920 + 910 + 820 + 780 + 790 + 800 = 5020

Of the 1000 ppl, 7 dont like any of the 6 types.

Remaining = 1000 -7 = 993

Surplus = 5020 - 993 (Becoz 7 ppl dont like any of the 6 types of food)

= 4027

Accomodating the surplus in the portions of 5 intersecting groups, we can have at most 993 * 4 =

3972 ppl surplus.

Remaining surplus to be accomodated in the portion where all 6 groups intersect = 4027 -3972 =



55

Hence, answer (D)55..........

Plz let me know if i am correct.......

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Rishi Kapoor - Wednesday, 17 October 2007, 07:02 AM

Hi Abhra...

The answer is atleast 20 people watch all 5 channels...

For solutiuon, see the above post

...RK...

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby manjari malpani - Friday, 19 October 2007, 01:37 PM

hi

the article is awesome !!!!! thanks ....n ashish is the answer to ur question 27 ?option c

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Ajay ..... - Thursday, 8 November 2007, 11:49 PM

hi sompornacan u explain why did u accomadated persons in 5 intersecting group and why not in 1 2 3 4 one's means u r assuming those to be Zero ???

Plz do explain.otherwise u are absolutely correct..

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby the terminator - Monday, 12 November 2007, 05:57 PM

@ dinesh munna hey if u urself cant post somethg useful let others do it and by d way donno y u r so critical abt the article. and can u u plz explain wat u meanby ur so called "FUNDAS". think before u write. before this how many useful posts u hav put up???nothing personal but be careful wat u write.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby the underdog - Tuesday, 13 November 2007, 10:56 AM

Hi TG,

I don't think the CAT 2006 question (July-Sept-August) is wrong. Its asking for those who read consecutive issues: July-August and August-September

July-August: 7August-September: 2

So answer is 9.

Or else I have misunderstood the question.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Bharat Suri - Tuesday, 13 November 2007, 09:38 PM

sir shouldnt maximum value ,regarding the example given by u about the college students, be 57% and NOT 54% ie the maximum valuefor the students to take all the 3 fields as all the arts ppl can take the other two as well .

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Mou Sukoshi - Friday, 16 November 2007, 12:13 PM

hi ajay



look closely thru TG's article......

we move to 2 intersecting grps only when we are unable to accomodate within non-intersecting portions......

similarly, we move to 3 intersecting grps only when we are unable to accomodate within 2 intersecting grps........

like in the prob, we move the surplus to gps where all intersect only when we cannot accomodate them within 5 intersecting grps........

so we have already considered smaller intersecting grps before moving on to the higher number of intersecting grps......

i suggest u draw a diag as it will help u to understand the accomodation of the surplus better........

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Mou Sukoshi - Friday, 16 November 2007, 12:15 PM

hi ajay

look closely thru TG's article......

we move to 2 intersecting grps only when we are unable to accomodate within non-intersecting portions......

similarly, we move to 3 intersecting grps only when we are unable to accomodate within 2 intersecting grps........

like in the prob, we move the surplus to gps where all intersect only when we cannot accomodate them within 5 intersecting grps........

so we have already considered smaller intersecting grps before moving on to the higher number of intersecting grps......

i suggest u draw a diag as it will help u to understand the accomodation of the surplus better........

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby yash modi - Wednesday, 4 June 2008, 10:57 PM

a big thank u to tg sir,

brilliantly explained concepts ...thanks a ton sir...better than any coaching you can get..thanks again;;

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Divyasha Ray - Tuesday, 24 June 2008, 05:26 PM

Hi TG sirCan u please help me with this particular question???A survey was conducted in a community of 350 people regarding three games - Chess ,Carrom and Chinese Checkers. The Followinginformation is obtained in the survey.(i) Thrice the no. of people who play all the three games is equal to the no. of people who Chinese Checkers .

(ii) The no. of people who play Chinese Checkers and Carrom is equal to the no. of people who play Chess only

(iii) In every three people who play Chess and Chinese Checkers only, there are five people who play none of the three games.

(iv) In every seven people who play Chinese Checkers, four people play Carrom also.

(v) For every four people who play exactly two games,there is one person who plays Carrom and Chinese Checkers only and two personswho play none of the three games

Questions:1. How many people play exactly two games?2. How many people play Chess but not Carrom?3. How many people do not play Chinese Checkers?4.How many people play Chess or Carrom

DivyashaEdit | De

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby abhinav tripathi - Thursday, 26 June 2008, 11:21 AM

TG you rock man.....!!!!



venn diagrams were always a matter of concern for me........you freed me ............!!!thanks a lot....waiting for an article on graphs.......

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby shubham singh - Thursday, 3 July 2008, 02:26 PM

tg u just great!!!

initially i was so scared of venn diagram n maxima n minima but now after going thru ur article ,m much relieved...thanku

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Mou Sukoshi - Friday, 4 July 2008, 11:13 AM

@ Divyasha

There shd be one more condition for the Carrom Only portion. All the other sections can be expressed in terms of a single variable. Plz checkthe prob.

Somparna

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby A S - Monday, 7 July 2008, 09:48 AM

hi tg,

This is the first time I am writing in this forum. And before everything else i want to say that you are simply great!! I got to find your website 2months back, and you know what, within this short timespan my confidence and grip over quant. part have increased tremendously. I wrote CAT

in 2006, and needless to say it was pathetic . This year I am going to write it more confidently. And one day I'll join you to help students...atleast thats what I dream today..Anyway, lets come to the business!!! I understand the way you have presented the maxima and minima using Venn diagram..but I think it canbe simplified to further extent. I discovered this when I was at Class 12 and used it confidently without any hiccup. Hope this will help otherstoo. Please let me know your valuable input on the same.

Let X = Elements having only A or only B or only C Y = Elements having two elements exactly. Z = Elements common with all three elements (intersection of all three sets)

So, we'll always be given, (X + 2Y + 3Z) (X + Y + Z) = 100 % We generally have to find min or max of Z.That can easily be done by adjusting X or Y.

Let's take this example:"In a college, where every student follows at least one of the three activities- drama, sports, or arts- 65% follow drama, 86% follow sports, and57% follow arts. What can be the maximum and minimum percentage of students who follow all three activities exactly two activities"

(X + 2Y + 3Z) = 208%(X + Y + Z) = 100%

[If Z has to be fixed (be min or max) either X or Y has to be zero (they can not be negative, remember). Now who will correspond to max andwho to min - only question. If X = 0, Y + Z will 'eat away' maximum Z, leaving small Z. So it will be used to calculate min Z. Vice-versa.]

In this case,for min Z, X = 0%=> Z = 8%, X = 0%, Y = 92%for max Z, Y = 0%=> Z = 54%, X = 46% and Y = 0%

It may seem to very lengthy process, but practically very small and easy to understand.Sorry for posting such a long and boring mail. But I got the chance to communicate with you and could not resist myself...sorry again!!

Abhishek

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Monday, 7 July 2008, 12:42 PM

Hi Abhishek,

Welcome to the gadha gang.

Thank you for the method. I will try it out and check. And good luck to you or your CAT. As for helping students, you are more than welcome.

We can certainly do with some more hands.

Total Gadha



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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby dibyaranjan mallick - Tuesday, 8 July 2008, 01:08 AM

416 will be the total...416 - 99 = 317 be the surplus3x + 4y = 317, x + y = 100% max i.e. 99so, y = 317 - 99*3 = 20. (ans)

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby dibyaranjan mallick - Tuesday, 8 July 2008, 01:20 AM

yaar it's the same thing as written in this article.

(X + 2Y + 3Z) = 208%(X + Y + Z) = 100%---------------------- deducty + z = 108%

and that's what this article explains

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Prashant Kumar - Tuesday, 8 July 2008, 08:36 AM

Hello TG,

It really very tailor made informaion for CAT.But tell me one thing

In the section "When the total number of elements is NOT fixed " .We have 210 interviews in IIM A.So cant we takethis problem in the same way as calculating the surplus (105+56+50-210=1).But the solution comes out to bespurious using this.Please help me

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby A S - Tuesday, 8 July 2008, 10:59 AM

Hey dibya, of course it will give the same answer and will eventually form same equations..what i mean to say that in stead of the process'injecting' one element in common area, i find it easier to calculate them in this algebraic method..I am not giving a brand new theory..just

another perspective of looking at this kind of problem.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Rajat Vashishta - Sunday, 13 July 2008, 12:34 AM

Thanks TG for a great article. it was really useful. this is a question from one of the AIMCATs. I found it really tough to solve..please guide.

Q. In a group of 100 students, each student has to opt for one or more of the 3 subjects among Physics, Chem and Maths. The number ofstudents opting for maths is more than that of physics which in turn is more than that of chem, which in turn, is more than the no of studentsopting for exactly 2 of the 3 subjects, which in turn, is more than the no of students opting for all the 3 subjects. It is also said that at least onestudent opted for all the 3 subjects.

1. Max no of students opting for chem:?a)72 b)79 c)80 d)81 e)None of these

2. Min no of students opting for maths:?a)38 b)37 c)36 d)35 e)34

3. If exactly half the students opted for maths, what is the max no of students who opted for all the three subjects?a)15 b)16 c)17 d)18 e)19

4. Max no of students who opted for only physics:?a)33 b)50 c)49 d)48 e)52

5. Max no of students who opted for physics and chem but not maths:?a)47 b)48 c)49 d)50 e)51

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Rajat Vashishta - Sunday, 13 July 2008, 12:43 AM

"49 can be alloted to the intersection of H and C only as we have to max cricket (C)"..how?if we do that won't c become greater than b? the q says c

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby mohan chandra - Wednesday, 16 July 2008, 12:07 AM

Hi,

Could you please guide me to solve this problem of Chinese checkers.

Regards,Mohan

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Amey Panke - Wednesday, 16 July 2008, 01:49 AM

thank you TG sir..........the concepts are of great help.....u rock sir!!!!!!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ashish jain - Thursday, 17 July 2008, 03:09 AM

Hello TG sir,

Following the same question which Nitin Jain has asked above, I still have doubt regarding the classification of problems into Fixed and Non-Fixed category.

In your answer above, did you mean that if it is not given that every person belongs to at least one category , then there may be some personswho doesn't belong to any category and hence AUBUC is not fixed ?

Can you explain this classification in detail please ?

Thanks and Regards,Ashish

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby inderpreet makkar - Thursday, 17 July 2008, 03:11 AM

Hello TG Sir

I dint get the concept of x+y and x+2y clearly...can u pls help me wit hthe same...

Regards

Inder

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby inderpreet makkar - Thursday, 17 July 2008, 03:29 AM

Hi

Understood the logic u gave to Kishore...thnx for the same..dint get the last part though, how could u say that themaximum could be 92 abd minm beibg 8...its beyond my comprehension presently..would appreciate if u can help me widthat..

Rgds

Inder

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Riyaz Iqbal - Tuesday, 22 July 2008, 09:30 AM

Hi TG, I'm impressed ! Again !! Thanks for your quality conceptual lesssons. I've a question about the Drama,Arts,Sports question. Since theleast percentage given is 57% why can't we just include 57 to the intersection of all three areas and adjust the remaining values? Leave theareas of intersection of exactly two sets empty(which gives the minima of zero for that area,as you've already proved) and put 57 in theintersection of all three areas.Again zero for 'only arts', 8 for 'only drama' and 29 for 'only sports'. It seems this satisfies all the conditions(Am Imissing something?). So, shouldn't the max value be 57 instead of 54? Please clarify.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Riyaz Iqbal - Tuesday, 22 July 2008, 05:46 PM



Sorry, TG.I found the fallacy in my argument. The total percentage doesn't add up to 100 in the case proposed by me.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Adithya G - Wednesday, 23 July 2008, 09:03 AM

Hi TG, Nice stuff..very much impressed with the quality and simplicity.I would like add one more technique about minima when total number is fixed or when total number can be found out.. I ll explain throughthe same examples illustrated above.

1.According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is the minimum percentage of people who like all three?

Ans: for min of all three= 100-((100-70)+(100-75)+(100-80)) =>100-(30+25+20)=100-75=252.In a college, where every student follows at least one of the three activities- drama, sports, or arts- 65% follow drama, 86% follow sports, and 57% follow arts. What can be the maximum and minimum percentage of students who follow all three activities exactly two activities

Ans:min For all three activities= 100-((100-65)+(100-86)+(100-57)) =>100-(35+14+43)=100-92=8%

max For exactly two=100-8=92%(since x+2y=100 and min of all three can be used)

Regards Adi

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4 Set or Greater Venn diagramsby devi prakash - Tuesday, 5 August 2008, 03:30 AM

Hi TG plz give some easier ways to solve 4 set or 6 set venn diagrams

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Re: 4 Set or Greater Venn diagramsby Junglee Gadha - Friday, 12 September 2008, 05:34 PM

hi devi da...

please post your 4 set or 6 set venn diagrams problem...

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby amit amit - Wednesday, 17 September 2008, 08:34 PM

I am a avid reader of TG site. An I have to admit that the quality of stuff on this site is simply great. In today's world where everything iscommercialized, the way u r spreading knowledge is simply unheard of. I work on Linux and cant help noticing the similarity between theunderlying theme of Linux and your endeavor. Knowledge is free and should be shared. You are on something great TG. Keep up the good

work. God Bless.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Total Gadha - Friday, 19 September 2008, 07:58 PM

Hi Amit,

Thank you.

Total Gadha

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Saurabh Mishra - Saturday, 20 September 2008, 11:54 AM

Venn diagrams involving maxima and minima have always been my nemesis.This post really helped a lot.Would be really gre8 if we could get some more practice questions or samples.

Nevertheless, a great post

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby srini vaas - Saturday, 20 September 2008, 04:47 PM

Hey TG,



Amazin article...

Jus one doubt..

wats ur real name???

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby sathyan sadagopan - Monday, 22 September 2008, 04:34 AM

hi TG,A small clarification,in d above explanation,

Going by this manner, you can quickly obtain that if you keep 90% in intersection of 2 sets and 10% in the intersection of 3 sets, the surplusaccommodated = 90 2 + 10 3 = 210%.

shouldn this be

Going by this manner, you can quickly obtain that if you keep 90% in intersection of 3 sets and 10% in the intersection of 4 sets, the surplusaccommodated = 90 2 + 10 3 = 210%.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby karan sobti - Tuesday, 23 September 2008, 10:56 PM

a very good article indeed.....but i need more problems & examples like these....

thanx

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby karan sobti - Tuesday, 23 September 2008, 10:57 PM

a very good article indeed.....but i need more problems & examples like these....

thanx

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby SHRESHTH ANAND - Wednesday, 24 September 2008, 02:13 AM

Hi TG,Indeed it's a very nice article but I have some problems in handling Maxima and Minima problems.Say, If i have to find out the following things :

In a town people read five kinds of newspapers say A, B ,C,Dand E.Say, 90% read paper A , 74% paper B , 80% paper C,65% paper D and 40% paper E.Now if I have to find at most how many read exactly four out of five papers and exactly 3 out of five.

And

at least how many read exactly four out of five papers and exactly 3 out of five.

How should I go about it?????

Kindly help

RegardsShreshth Anand

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Sreedevi PV - Wednesday, 24 September 2008, 06:58 PM

Hi TG,

Again Kudoooos. U rock.

Max&min funda was a bit confusing when i first read it. . Now its clear....and seems like a real gem.

Question:According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is theminimum percentage of people who like all three?

To get the answer, I use a one-liner funda.Seems it works everytime. Hope its useful.

Minimum Percentage of people who like all three= 80 (100-75)-(100-70) =25



Cheers!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Rohit Pathak - Thursday, 25 September 2008, 10:54 AM

Hi Shreshth,

For at most how many read exactly 4 out of 5 papers the answer that i am getting is 83% and for at least how many read exactly 4 out of 5 iam getting 49% please let me know if these are correct.

Thanks

Rohit

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Aastha Chalana - Thursday, 25 September 2008, 03:20 PM

Hi,

This lesson was of great help, though small but important fundas.

In all i throughly enjoyed working on it!

Thanks.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Kosher InBlues - Thursday, 25 September 2008, 09:58 PM

HI TG I am a new fan of yours , and i really like the way you present a topic "to the point ",,,,,,Thanks

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby LALIT MALIK - Wednesday, 8 October 2008, 05:36 AM

Hi

Can you tell why before adding the surplus to the intersection of the three sets we are subracting the same amount from the intersection of onlytwo sets like if the surplus is 108% then can't we represent the surplus in this manner - 100% is from intersection of two sets only and 4%(which means surplus of 8%) from the intersection of three sets. why we have take 8% out from the intersection of 2 sets and then add 8% tointersection of three sets.

Kindly do reply

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Neelesh Sethi - Monday, 27 October 2008, 12:58 AM

thnx sir for great topic,intersection of only two sets can accommodate only a surplus of 100%,is it because intersection of 2 eta gives surplusof 1 element.So by that logic is it tht intersection of 3 sets can accomodate 200%.by that logic intersection of n sets can accomodate n-1 %surplus.and also if any 1 knows line technique for such problems, plz explain.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby shantanu rangaswami - Monday, 10 November 2008, 11:08 AM

hey ,,,,wass up,,,can u check sixth line from top of ur reply

.......d>e>f,,,nd g is 0shouldn't d=35 e=33 &f=32

....if m not wrong...



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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Sukanya Chowdhury - Friday, 19 December 2008, 12:32 AM

Hi Shreshth,

I'v also got the same answer..

At least 49% and at most 83% for those who read exactly four out of 5 papers .

@TG ,

Can yu please check whether it's right .n solve the other part of the qsn.

THanks .

Sukanya.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby vikas sharma - Saturday, 21 February 2009, 11:35 AM

In a college, where every student follows at least one of the three activities- drama, sports, or arts- 65% follow drama, 86% follow sports,and 57% follow arts. What can be the maximum and minimum percentage of students who follow all three activities exactly two activities

Answer: Let us again see the surplus:

Percentage of students who follow drama + Percentage of students who follow sports + Percentage of students who follow arts = 65% +86% + 57% = 208% surplus = 108%. This surplus can be accommodated through adding elements either to intersection of only two setsor to intersection of only three sets. As the intersection of only two sets can accommodate only a surplus of 100%, the surplus of 8% will stillbe left. This surplus of 8% can be accommodated by adding elements to intersection of three sets. For that we have to take 8% out of theintersection of only two sets and add it to intersection of three sets. Therefore, the minimum percentage of people who like all three= 8%. In this case the percentage of students who follow exactly two activities will be maximum = 92%.

The surplus of 108% can also be accommodated through adding elements to only intersection of three sets. As adding 1 element tointersection of three sets give a surplus of 2 sets, adding 54% to intersection of three sets will give a surplus of 108%. Therefore, themaximum value of students who follow all three activities is 54%. In this case the percentage of students who followexactly two activities will be minimum = 0%.

We can also solve it mathematically x + 2y = 108%, where x + y 100%. The maximum value of x will give minimum value of y,whereas minimum value of x will give maximum value of y.

HI SRI KLR CAN U PLEASE EXPLAIN ME THIS AGAIN I DIDNT GET WHY 54% AND HOW ADDING IN ONE ADD IN ALL .WAITING FOR REPLY

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby vikas sharma - Saturday, 21 February 2009, 11:46 AM

hi TG SIR

CAN U ELABORATE BELOW GIVEN AGAIN

In diagram 1, we have added 1 element to intersection of only two sets (A and B but not C). We can see that A and B both increase by 1 andtherefore we get a surplus of 1 element.

In diagram 2, we have added 1 element to intersection of all the three sets (A and B and C). We can see that A, B and C all three increaseby 1 element each and therefore we get a surplus of 2 elements.

Therefore, in case of three sets, we can accommodate the surplus by

adding elements to intersection of only two sets in which case a surplus of 1 element can be accommodated by increase of 1 elementin the intersection of only two sets.

adding elements to intersection of three sets in which case a surplus of 2 elements can be accommodated by increase of 1 elementin the intersection of three sets.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Ashish Sharma - Saturday, 25 April 2009, 09:26 AM

supercool



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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby vakati babu - Sunday, 31 May 2009, 11:19 PM

There are 10 people learning A, 11 people learning B and 14 people learning C. The number of people learning just one is 20. The number ofpeople learning all three is 3. How many people only learn two of three?

how to answer this /

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby William Wallace - Wednesday, 3 June 2009, 05:52 PM

Hi TG,

Kudos to u as it's really a wonderful article but i have some doubts related to that.

in 1st problem "According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What isthe minimum percentage of people who like all three?" u said surplus is 125 % and hence the minimum% of people who like all three is 25%

But in second problem "In a college, where every student follows at least one of the three activities- drama, sports, or arts- 65% followdrama, 86% follow sports, and 57% follow arts. What can be the maximum and minimum percentage of students who follow all three activities exactly two activities"

The surplus of 108% can also be accommodated through adding elements to only intersection of three sets. As adding 1 element tointersection of three sets give a surplus of 2 sets, adding 54% to intersection of three sets will give a surplus of 108%. Therefore, themaximum value of students who follow all three activities is 54%

My question is that shouldn't the answer in the 1st problem be 12.5 % rather 25% as is the case in the second question?? I hope u get my

doubt

Please answer this and clear this doubt.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby William Wallace - Wednesday, 3 June 2009, 06:05 PM

Hi Vakati,

draw diagram , formulate equations and done ...Is ans 3???

a+d+f=7b+d+e=8f+e+c=11a+b+c=20

to find d+f+c=(26-20)/2=3.

Sorry could't draw the diagram.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby vikas yadav - Friday, 19 June 2009, 03:18 PM

@TargetIIM

Hi dude,

Please read the statements below carefully as mentioned by TG sir while explaining the same problem.

This surplus of 25% can be accommodated by adding elements to intersection of three sets. For that we have to take 25% out of theintersection of only two sets and add it to intersection of three sets. Therefore, the minimum percentage of people who like all three = 25%.

So,actually v are taking out the elements from the intersections of two sets and putting them in all the 3 sets intersection space.This wouldmean covering a surplus of 1 per element taken out from the sets of twin intersection.Hence,it would be 25% and not 12.5% dear.

I hope you got my explaination well.

Thanks

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby William Wallace - Friday, 3 July 2009, 02:28 PM

Thanks Vikas

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Pragya dasgupta - Sunday, 12 July 2009, 07:30 PM

HEllo

Can Anyone help me ith this problem

Is there any Fixed process to solve this ype of problems.

in a job fair among 100 participants who attended the interview,the number of participants who attended for company A is more than that of co.B is more than that of C.The no of participants who attended interview of exactly one co. is more than that of 2,which in turn is more than thatof 3.Among these 100 participants at least one person attended the interview of all the 3 companies.Q:What is the minimun possible no. of participants who attended the interview of co. A?a.36 b.38 c.37 d.40 e.48Q.What is the max possible no of participants who attended the interview of co. C?a.56 b.45 c.65 d.67 e.64

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby vivek ghiria - Sunday, 26 July 2009, 06:25 PM

Hi TG

I am going through lessons in this forum one by one. I have one concern. After going through a particular lesson, If few problems pertaining to that chapter will be available, then it will help us a lot. As of now, I have to search the problems related to that chapter in Quant-DI forum which take bit extra time.

Thanks a lot for the lessons - Vivek

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Nitin Kumar - Thursday, 13 August 2009, 11:08 PM

Hi TG,

Que: In a class of 120 students, each student studies at least one of the subjects from H, E and M. 59 study H, 67 study E and 73 study M. 34 studyM+H, 26 study E+M and 33 study H+E. Then find the max and min number of students who study all the three subjects?

If I solve the question by drawing the venn diagram the max value I am getting as 26.

While if I solve this question in this way then I m getting rt ans.

total students - 120

surplus = 59+67+73-120=79

Total number of students study more than one sub = 34+33+26=93

b'coz surplus is 79 but being accomodated is 93, that's why max no. of students study more than 3 subjects is 93-79=14

Please let me know is this correct way of sloving this que.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ashu jain - Thursday, 17 September 2009, 01:04 PM

The problem that I am facing is that the lesson is given on solving the question based on percentage but most of the time the question doesnot contain percentage but real numbers where converting into percentage can be cumbersome.Like this question given below:

Q)In a survey report, which was conducted in a club of 300 men, it was found that 100 men use a products of brand A, 75 use the sameproducts of brand B and 175 use the same products of brand C. What is the maximum possible number of men using all the three brands?

Here x+2y = 50.What now???

I am not getting the hang of this thing!!!!!!!!!!!!!!!!!!!!!!

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ROHIT K - Thursday, 17 September 2009, 03:05 PM

Hi ashu,




is the answer 200..?

Please check and revert back..

Rohit

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Pallav Jain - Thursday, 17 September 2009, 05:08 PM

Hi All,

Can anyone please explain me how to approach this kind of questions?

In a class of 30 students, there are 2 categories of students: Liars: who always lie.Truth Tellers: who always speak the truth.These students make three different statements as follows :Statement I :Seven of them say There are exactly 7 Liars in this classStatement II :Eight of them say There are exactly 8 Liars in this classStatement III :The rest of them say There are exactly 15 Liars in this class.How many Liars are there in the class ?

Options :-

a) 7 b) 8 c) 23 d) 22 e) 15

Thanks

Pallav

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The OA is 25.

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I think answer should be option e) 15.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ROHIT K - Thursday, 17 September 2009, 11:33 PM

Hi Ashu,

Sorry mate i made a big goof up ..

Anyways coming back to your question,


Ans:

Total Men=300A=100, B=75, C=175

A+B+C=350

Surplus=350-300=50

Now how do we allocate this surplus 50 to maximize A^B^C..?

For each men allocated to A^B^C, we get surplus of 2.

Therefore for surplus of 50 we allocate 50/2=25 men to A^B^C.

Hence 25 is the Answer..

Please check if I am going wrong anywhere..

Hope this helps..

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby ashu jain - Saturday, 19 September 2009, 02:06 PM



thanks Rohit. you are right bro.

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Suresh S - Monday, 21 September 2009, 12:04 PM

Hi sir .. its really helped me much sir

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Rajat Vashishta - Tuesday, 20 October 2009, 11:02 AM

In the problem"According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is the minimumpercentage of people who like all three? "

why do we have to include 25% in the intersection of 3 sets? Cant this no be 12.5%?? In that case too, the intersection of 3 sets would consumea surplus of 25%, while the intersection of 2 would consume 100%. So the whole surplus of 125% would be accommodated.Please tell me where i am wrong?

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Re: Venn Diagrams- Basics, Problems, Maxima and Minimaby Rajat Vashishta - Tuesday, 20 October 2009, 03:05 PM

and vikas's explaination is not sufficient. why we are taking 25% from the twin set intersection is because putting 25% in 3 set intersectionwould give us a surplus of 150% so we need to lessen it from the 2 set intersection

total gadha-venn diagram

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