In a town 85% of the people speak Tamil, 40% speak English and 20% speak Hindi. Also, 32% speak English and Tamil, 13% speak Tamil and Hindi and 10% speak English and Hindi, find the percentage of people who can speak all the three languages.

Let T, E and H be the people speaking Tamil, English and Hindi respectively.

so we are given

Total percentage of people = 100%

Percentage of people speaking Tamil = n(T) = 85%

Percentage of people speaking English = n(E) = 40%

Percentage of people speaking Hindi = n(H) = 20 %

Percentage of people speaking Tamil and Hindi = n(T ⋂ H) = 13%

Percentage of people speaking English and Hindi = n(E ⋂ H) = 10%

Percentage of people speaking English and Tamil = n(E ⋂ T) = 32%

Percentage of people speaking Tamil, English and Hindi = n( T ⋂ E ⋂ H)

Assume that Percentage of people speaking Tamil, English and Hindi be P.

We can solve this by using Venn diagram,

The above Venn diagram gives the following,

People speaking all the three language = P

People speaking English and Hindi and no tamil = (E ⋂ H ⋂ T’) = 10 – P

People speaking Tamil and Hindi and no English = ( T ⋂ H ⋂ E’) = 13 – P

People speaking English and Tamil and no Hindi = ( T ⋂ E ⋂ H’) = 32 – P

Only Hindi speaking people = H’ = P – 3

Only English speaking people = E’ = P – 2

Only Tamil speaking people = T’ = 40 + P

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

Total people = T’ + H’ + E’ + (E ⋂ H ⋂ T’) + (T ⋂ H ⋂ E’) + (T ⋂ E ⋂ H’) + P

⇒100 = (40 + P) + (P – 3) + (P – 2) + (10 – P) + (13 – P) + (32 – P) + P

⇒ 100 = 40 + P + P – 3 + P – 2 + 10 – P + 13 – P + P

Eliminating elements of opposite signs

⇒100 = 90 + P

⇒P = 100 – 90

⇒P = 10

∴ From the above its clear that only 10% people can speak all the three language.