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 presents

QUANTFOR  

CAT 2009

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 VENN DIAGRAMS

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 Venn Diagrams with 2 groups

a bc

A B

Only A Only B

Both A & B

n

Not A or B

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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?

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Solution

a bc 

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. 

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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,C

d

e fg

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Problem  –  CAT 2003

A survey on a sample of 25 new cars being sold at a localauto dealer was conducted to see which of the three popularoptions - air conditioning, radio and power windows werealready 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

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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

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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 getn = 2

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Problem : (CAT 2003- Leaked)

New Age Consultants have three consultants Gyani, Medha andBuddhi.

The sum of the number of projects handled by Gyani and Buddhiindividually is equal to the number of projects in which Medha isinvolved.

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 3projects with Gyani but without Medha.

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

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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

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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

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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,

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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,C

1

0 00

Both A & B

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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,C

0

0 01

Both A & B

Only A

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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. 

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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?

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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%.

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Solution contd… 

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 3sets.

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

people who like all 3 fruits = 25%

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Problem

In a college, where every student follows at leastone of the three activities- drama, sports, or arts: 65% follow drama,

86% follow sports, and

57% follow arts.

What can be the maximum and minimumpercentage of students who follow

1·  all three activities2·  exactly two activities

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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

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Solution contd… 

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%

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Exercises

Do Check out the exercise given along

with the lesson for more practice

problems of this type.

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 THANK YOU!

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