Question

In a class of 40 students, 25 speak Hindi and 15 speak both English and Hindi. All students
speak atleast one of the two languages. How many students speak only English.

Please don't give spam answer.

Answer 1

$\large\underline{\sf{Given- }}$

In a class of 40 students,

• 25 speak Hindi

• 15 speak both English and Hindi.

• All students speak atleast one of the two languages.

$\large\underline{\sf{To\:Find - }}$

How many students speak only English.

$\large\underline{\sf{Solution-}}$

Given that

In a class of 40 students, 25 speak Hindi and 15 speak both English and Hindi. All students speak atleast one of the two languages.

Let assume that,

A denotes the set of students who speak Hindi

and

B denotes the set of students who speak English.

According to statement,

$\green{\bf :\longmapsto\:n(A) = 25}$

$\green{\bf :\longmapsto\:n(A \: \cap \: B) = 15}$

$\green{\bf :\longmapsto\:n(A \: \cup \: B) = 40}$

Now, we know that,

$\rm :\longmapsto\:n(A\cup B) = n(A) + n(B) - n(A\cap B)$

On substituting the values, we get

$\rm :\longmapsto\:40 = 25 + n(B) - 15$

$\rm :\longmapsto\:40 = 10 + n(B)$

$\rm :\longmapsto\:40 - 10 = n(B)$

$\bf\implies \:n(B) = 30$

Now,

To find the number of students who speak English only,

$\red{\rm :\longmapsto\:n(B - A)}$

$\rm \:  =  \:  \: n(B) - n(A\cap B)$

$\rm \:  =  \:  \: 30 - 15$

$\rm \:  =  \:  \: 15$

So, it implies

The number of students who speak English only = 15

Additional Information :-

$\boxed{ \rm{ n(A - B) = n(A) - n(A\cap B)}}$

$\boxed{ \rm{ A'\cup B' = A'\cap B'}}$

$\boxed{ \rm{ A'\cap B' = A'\cup B'}}$

$\boxed{ \rm{ A\cap A' = \phi \: }}$

$\boxed{ \rm{ A\cup A' = U \: }}$

$\boxed{ \rm{ U' \: = \: \phi \: }}$

$\boxed{ \rm{ \phi \: ' \: = \: U}}$

$\boxed{ \rm{ n(A\cup B) = n(A - B) + n(A\cap B) + n(B - A)}}$

Answer 2

Answer:

Let students who speak Hindi as H and who speak English as E.

$n(HUE)=n(H)+n(E)-n(H \cap E)$

where

• n(HUE)=40
• n(H)=25
• $n(H \cap \: E) = 15$
• n(E)=?

$n(HUE)=n(H)+n(E)-n(H \cap E) \\ 40 = 25 + n(E) - 15 \\ 40 = 10 + n(E) \\ n(E) = 40 - 10 \\ n(E) = 30$

n(E-H)=n(E)-$n(H \cap \: E)$

n(E-H)=30-15

n(E-H)=15

Number of students who speak only English = 15