Question

9. In a village of 120 families, 93 families use firewood for cooking, 63 families use
kerosene, 45 families use cooking gas, 45 families use firewood and kerosene, 24
families use kerosene and cooking gas, 27 families use cooking gas and firewood.
Find how many use firewood, kerosene and cooking gas.9. In a village of 120 families, 93 families use firewood for cooking, 63 families use
kerosene, 45 families use cooking gas, 45 families use firewood and kerosene, 24
families use kerosene and cooking gas, 27 families use cooking gas and firewood.
Find how many use firewood, kerosene and cooking gas.

Answer 1

CARDINAL NUMBER OF A FINITE SET :
The Cardinal number of a finite set A is the number of distinct elements in the set A. It is denoted by n(A).
•It is not possible to define the Cardinal number of an infinite set.
•The Cardinal number of the empty set is zero.
•The Cardinal number of a singleton set is 1.

Intersection of two sets :
The intersection of the sets a and b is the set of all the elements which belong to both A and B. It is denoted by A ∩ B (“ A intersection B”).
•If A and B do not have any element in common then A ∩ B= a null set = Ø.

SOLUTION :
Let U = total number of families, F= families use firewood, K= families use kerosene , G = families use cooking gas n( F ∩ K)= families use both firewood & Kerosene, n( K ∩ G)= families use both kerosene & cooking gas, n( G ∩ F)= families use both cooking gas & firewood .

GIVEN :
n(U) = 120
n(F) = 93
n(K) = 63
n(G) = 45
n( F ∩ K)= 45
n( K ∩ G)= 24
n( G ∩ F)= 27

NUMBER OF FAMILIES WHO USE FIRE WOOD, KEROSENE AND COOKING GAS= n(U) - { n(F) + n(K) + n(G) - n( F ∩ K) - n( K ∩ G) - n( G ∩ F)

= 120 - { 93 + 63 + 45 -( 45 + 27 + 24)}
= 120 - 105 = 15

Number of families who use firewood, kerosene and cooking gas = 15

HOPE THIS WILL HELP YOU…

Answer 2

Hi,

According to the problem given ,

Let the number of families which use ,

i ) Firewood = n( F ) = 93 ,

ii ) Kerosene = n( K ) = 63 ,

iii ) Gas = n( G ) = 45 ,

iv ) Firewood and Kerosene = n( F n K ) = 45 ,

v ) Kerosene and Gas = n( K n G ) = 24 ,

vi ) Gas and Kerosene = n( G n F ) = 27 ,

vii ) Let Firewood and Kerosene and Gas

= n( F n K n G ) = x

Total families in the village = n( F U K U G )

= 120

We know that ,

n(FUKUG) = n(F)+n(K)+n(G)-n(FnK)-n(KnG)-

n(GnF) + n( FnKnG )

120 = 93 + 63 + 45 - 45 - 24 - 27 + x

120 = 201 - 96 + x

120 + 96 - 201 = x

216 - 201 = x

x = 15

Therefore ,

n( F n K n G ) = x = 15

I hope this helps you.

: )