Question

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.

Answer 1

Hint: To solve this type of question, you have to use the concept of set theory and Venn diagram. According to set theory the union of three-set (A, B, C) is defined as$n\left( A\cup B\cup C \right)=n\left( A \right)+n\left( B \right)+n\left( C \right)-n\left( A\cap B \right)-n\left( B\cap C \right)-n\left( C\cap A \right)+n\left( A\cap B\cap C \right)$ .

Complete step-by-step answer:
Venn diagram for this question is shown in the fig.1

                           Fig.1
Where
n(U) represents the total number of people in town. Let’s assume this value is equal to 100.
n(A) represents the number of people in town who can speak Tamil.
n(B) represents the number of people in town who can speak English.
n(C) represents the number of people in town who can speak Hindi.
n(X)= $n\left( A\cap B \right)$ represents the number of people in town who can speak Tamil and English.
n(Y)= $n\left( B\cap C \right)$ represents the number of people in town who can speak English and Hindi.
n(Z)= $n\left( C\cap A \right)$ represents the number of people in town who can speak Hindi and Tamil.
n(P)= $n\left( A\cap B\cap C \right)$ represents the number of people in town who speak all three languages ( Tamil, Hindi, English).
According to the question, we have
% of people speaking any language = $\dfrac{N}{T}\times 100-(1)$
Where N= Number of people speaking that language.
            T= Total number of people in town.

Using the above formula, we get
% of people speak Tamil = 85% (given)

      $\begin{align}
  & n\left( A \right)=\dfrac{85}{100}\times T \\
 & \\
\end{align}$
              \[\begin{align}
  & =\dfrac{85}{100}\times 100 \\
 & =\dfrac{85}{100}\times 100 \\
\end{align}\]
Now cancelling 100 from numerator and denominator, we get
                = 85


Similarly, we can say that

$\begin{align}
  & n\left( B \right)=40 \\
 & n\left( C \right)=20 \\
 & n\left( A\cap B \right)=32 \\
 & n\left( C\cap A \right)=13 \\
 & n\left( B\cap C \right)=10 \\
\end{align}$

According to the union of three sets, we know that

$n\left( A\cup B\cup C \right)=n\left( A \right)+n\left( B \right)+n\left( C \right)-n\left( A\cap B \right)-n\left( B\cap C \right)-n\left( C\cap A \right)+n\left( A\cap B\cap C \right)$

In this question n(U)= $n\left( A\cup B\cup C \right)$ =100 because every person in town at least speaking one language (given)

So
$n\left( A\cup B\cup C \right)=n\left( A \right)+n\left( B \right)+n\left( C \right)-n\left( A\cap B \right)-n\left( B\cap C \right)-n\left( C\cap A \right)+n\left( A\cap B\cap C \right)$
Now putting the value of $n\left( A\cup B\cup C \right),n\left( A \right),n\left( B \right),n\left( C \right),n\left( A\cap B \right),n\left( B\cap C \right),n\left( C\cap A \right)$
$\begin{align}
  & \Rightarrow 100=85+40+20-32-13-10+n\left( A\cap B\cap C \right) \\
 & \Rightarrow n\left( A\cap B\cap C \right)+145-55=100 \\
 & \Rightarrow n\left( A\cap B\cap C \right)+90=100 \\
 & \Rightarrow n\left( A\cap B\cap C \right)=100-90 \\
 & \Rightarrow n\left( A\cap B\cap C \right)=10 \\
\end{align}$

So, Number of people in town who can speak all three languages = 10

% of people in town who can speak all three languages
 \[\begin{align}
  & =\dfrac{10}{100}\times 100--using(1) \\
 & =10\% \\
\end{align}\]

Hence, the percentage of people who can speak all three languages = 10 %

Note: The union of the three-sets formula is also known as the inclusion/exclusion formula. The inclusion/exclusion formula for the union of n finite sets is
$n\left( {{A}_{1}}\cup {{A}_{2}}\cup {{A}_{3}}\cup --\cup {{A}_{n}} \right)=\sum\limits_{1\le i\le n}{n\left( {{A}_{i}} \right)}-\sum\limits_{1\le i < j\le n}{n\left( {{A}_{i}}\cap {{A}_{j}} \right)}+\sum\limits_{1\le i < j < k\le n}{n\left( {{A}_{i}}\cap {{A}_{j}}\cap {{A}_{k}} \right)}+-$
                          \[--+{{\left( -1 \right)}^{n+1}}n\left( {{A}_{1}}\cap {{A}_{2}}\cap {{A}_{3}}\cap --\cap {{A}_{n}} \right)\]