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presents

QUANT FOR

CAT 2009

VENN DIAGRAMS

Venn Diagrams with 2 groups

a b c

A B

Only A Only B

Both A & B

n

Not A or B

Problem

Of all CAT aspirants, 80% spend time for Quant whereas 60% spend time for verbal. If only those users will crack CAT who spend time for both, what percentage of CAT aspirants

will crack CAT? will not crack CAT?

Solution

a b

c

QA = 80 VA = 60

Only QA Only VA

Both QA & VA

n= 0

Not QA or VA = 0

QA = 80 => a+c = 80 -> (1)

VA = 60 => b+c = 60 -> (2)

Now, a+b+c+n = 100

As n = 0 , a+b+c = 100 -> (3)

Add (1) & (2)

a+b+2c = 140

Now, from (3) - (4)

a+b+2c = 140

a+b+c = 100

C = 40, These 40% qualify CAT

So 60% don’t qualify CAT.

a b

c

Venn Diagrams with 3 groups

A B

Only A

Only B

Both A & B

n Both B & CC

Only CBoth A & CAll A, B & C

None of A,B,Cd

e fg

Problem – CAT 2003

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 - air conditioning, radio and power windows were already installed. The survey found: 

15 had air conditioning 2 had air conditioning and power windows but no radios 12 had radio 6 had air conditioning and radio but no power windows 11 had power windows 4 had radio and power windows 3 had all three options. 

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

Solution Lets fill in the Venn Diagram with the info given :

4 2

5

ACRadio

Only A

Only R

Both A & R

n = ? PW

Only PBoth A & P

None of A,R,P

6

2 13

All A, R & P

Both R & P

Solution contd…

As we know total number of cars = 25

a+b+c+d+e+f+g+n = 25

a+b+c+d+e+f+g = 23

Filling in the required values we get

n = 2

Problem : (CAT 2003- Leaked)  New Age Consultants have three consultants Gyani, Medha and

Buddhi.

The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved.

All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects.

Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha.

The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved. 

Problem contd…

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 

Solution:

Filling in the values given:

a c

b

GyaniMedha

Only G

Only M

Both G & M

n = 0Buddhi

Only BBoth G & B

None of G,M,B

8

3 26

All G, M & B

Both M & B

Solution:contd…

We are left with the last line :

The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.

Total = a+b+c+d+e+f+g+n=2(d+e+f+g) - 1

Now, n = 0, d = 8, e = 6, g = 2

a+c = b + 16

Also, Total no. = 2 (8+6+3+2) – 1 = 37

We cannot determine a and c individually. So, answer to Q1 is Option 4. – Cannot be determined

Solution:contd…

But, from Total = a+b+c+d+e+f+g = 37, we get

a+b+c = 37 – (d+e+f+g) = 37-(8+6+2+3) = 19

So, a+b+c = 19 and a+c = b – 16,

Using both the above equations, we get b = 1

Answer to Q2 is Option 2 ie., 1,

Maxima and Minima

0 0

0

A = 1 B = 1

Only A

Only B

n Both B & CC = 0

Only CBoth A & CAll A, B & C

None of A,B,C1

0 00

Both A & B

Maxima and Minima contd…

0 0

0

A = 1 B = 1Only B

n Both B & CC = 1

Only CBoth A & CAll A, B & C

None of A,B,C0

0 01

Both A & B

Only A

Maxima and Minima contd…

Adding 1 element to intersection of 2 sets gives surplus of 1 element.

Adding 1 element to intersection of 3 sets gives surplus of 2 elements.

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 minimum percentage of people who like all three? 

Solution

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%. 

Solutioncontd…

Now this surplus can be accommodated by adding elements to either intersection of only two sets or to intersection of only three sets.

For minima of intersection of 3 sets => Put 100% in intersection of 2 sets (this is max. for this area) Left over 25% has to be accommodated in intersection of 3

sets. So Minima of intersection of 3 sets = minimum %age of

people who like all 3 fruits = 25%

Problem

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

1·          all three activities2·          exactly two activities 

Solution

Calculate surplus : Drama + Sports + Arts = 65+86+57 = 208% Surplus = 208 – 100 = 108%

Now, surplus can be accommodated in 2 ways : Maximum in 2 only and minimum in all 3 Maximum in All 3 and minimum in 2 only

Solutioncontd…

Maximum in 2 only and minimum in all 3: 100% can be accommodated in 2 only 8% will have to be accommodated in all 3. Maximum percentage who follow 2 events = 100

Some Answers : Minimum percentage who follow all 3 = 8 Max. percentage who follow exactly 2 = 100 – 8 = 92%

Solutioncontd…

Maximum in All 3 and minimum in 2 only: As 1 in All 3 area causes increase of surplus as 2. If we put 54% in all 3 -> surplus = 54 x 2 = 108.

Some Answers : Max. who follow all 3 = 54% Min. no. who follow exactly 2 = 0%

Exercises

Do Check out the exercise given along with the lesson for more practice problems of this type.

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