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.

Here let’s consider that F, G, K be the families using Firewood, Gas, Kerosene for cooking respectively and no of families be U

It’s given that

No of families = n(U) = 120

Families using firewood for cooking = n(F) = 93

Families using Gas for cooking = n(G) = 45

Families using kerosene for cooking = n(K) = 63

Families using firewood and kerosene for cooking = n(F ⋂ K) = 45

Families using kerosene and gas for cooking = n(K⋂G) = 24

Families using gas and firewood for cooking = n(G⋂F) = 27

Families using gas,kerosene and firewood for cooking = n(G⋂K⋂F)

Let Families using gas, kerosene and firewood for cooking = X

So from the above Venn diagram we get the following data,

People using only Gas, G’ = X – 6

People using only Kerosene K’ = X – 6

People using only firewood, F’ = 21 + X

People using only firewood and gas = (F ⋂ G) – X

People using only gas and kerosene = (G ⋂ K) – X = (24 – X)

People using only firewood and kerosene = (F ⋂ K) – X = (45 – X)

We know that adding all the region of the Venn diagram will give the value of total number of element involved.

So, using all the data we have we can find the following

No of families = no of families using one fuel + no of families using two

Fuels + no of families using all the three fuels

Using the data obtained in the above expression we will find,

U = F’ + G’ + K’ + (F ⋂ G ⋂ K’) + (G ⋂ K ⋂ F’) + (F ⋂ K ⋂ G’) + X

U = F’ + G’ + K’ + (F ⋂ G) – X + (G ⋂ K) – X + (F ⋂ K) – X + X

PUTTING ALL THE VALUES

⇒ U = 21 + X + X – 6 + X – 6 + 27 – X + 24 – X + 45 – X + X

⇒ 120 = 105 – X

⇒ X = 15

Hence 15 people use all the three kind of fuel.