class 10 sets

In a survey, it was found that 80% people
liked oranges, 85% likes mangoes and 75%
liked both. But 45 people liked none of them.
By drawing a Venn-diagram, find the number
of people who were in the survey.
Solution
Let O and M denote the set of people who like
Oranges and Mangoes respectively.
By question,

n(O)=80%, n(M)=85%, n(O⋃ )=45,n(O⋂M)=75%
Now showing information in Venn-diagram we get,

OM U

5% 75% 10%
10%

From Venn-diagram it is clear that

10%=45
Or, 1%=

=4.5
∴ 100%=100*4.5

= 450
Hence there were 450 peoples in the survey.
Alternative method
Let the number of people who were in the
survey be x
Then,

10% of x=45

or, ∗ =45


or, 10x=45*100

or, 10x=4500

or, x=
= 450

Hence there were 450 people in the survey.

In a group of 95 students, the ratio of students who
like mathematics and science is 4:5. If 10 of them
like both the subjects and 15 of them like none of
the subjects then by drawing a Venn-diagram find
(i) How many of them like mathematics only?
(ii) How many of them like science only?
Solution
Let M and S denote the sets of students who like
Mathematics and science respectively.
Then according to question

n(U)=95, n(M)=4x, n(S)=5x, n(M⋂S)=10, n(O⋃ )=15

Now showing above information in Venn-diagram

MS U

4x-10 10 5x-10

15

From Venn-diagram
4x-10+10+5x-10+15=95
x=?

x=10

(i) (M)=?

= 4x-10
= 4*10-10
=30
∴ 30 students like mathematics only.
(ii) (S)=?
= 5x-10
= 5*10-10
= 40
∴ 40 students like Science only.

Hence 30,40 students like Mathematics and Science

respectively.

In a group of 58 people, 17 play football only and
28 play volleyball only. If 8 play neither of two
games, then
(i) Represent above information in Venn-diagram.
(ii) Find the ratio of people who play football to the
people who play volleyball.
Solution
Let F and V represents the set of people who like
football and volleyball respectively.
By question
n(U)=58, (F)=17, (V)=28 and n(F⋃ )=8

(i) Representation on Venn-diagram U

FV

17 x 28

8

From Venn-diagram ,
17+x+28+8=58
x=?
x=5

Now

n(F)=?

= 17+x

= 17+5

=22

n(V)=?

=28+x

= 28+5

=33

(ii)Required ratio= ( )= =


Hence the no. of people who play football to volleyball

is 2:3

DAY 11

In a survey , one third children like only mango

and 22 don’t like mango at all. Also children like


Orange but 12 like none of them.

(i) Show the above data in a Venn-diagram.

(ii) How many children like both types of fruit?

Solution

Let M and O be the set of students who like mango

and orange respectively.

Then by question,

n(U)=x, (M)= , n( )=22, n(O)= ,n(M⋃ )=12

Now showing in Venn-diagram we get U

MO

n(O)=


12

From Venn-diagram

+ +12=x


or, 12=x- -


or, 12=

∴ x=45

∴ n(U)=45

We know that

n(U)= (M)+n( )

or, 45= (M)+n(M⋂O)+22
or, 45= +n(M⋂O)+22

n(M⋂O)=?

Hence 8 children like both type of fruit.

If n(A)=20 and n(B)=15 then find the minimum and
maximum value of
(i) n(A⋃B)
(ii) n(A⋂B)
Solution
We have given
n(A)=20 , n(B)=15
Case I
Disjoint case
Case II
Overlapping Case

Case I: Disjoint B
15
A

20

Maximum n(A⋃B)= ?
=20+15
= 35

Minimum n(A⋂B)=?
=0

Case II: Overlapping Case

AB

5 115 5

Minimum n(A⋃B)= ?
=15+5
= 20

Maximum n(A⋂B)=?
=15