Question: In a group of 950 persons, 750 can speak Hindi, and 460 can speak English. Find how many can speak H...

Question

In a group of 950 persons, 750 can speak Hindi, and 460 can speak English. Find how many can speak Hindi only.

Answer 1

Solution

Basic knowledge of unions and sets would be required to solve questions of these kinds. The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as $A\cup B$ or “A or B”. The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as $A\cap B$ or “A and B”.
The shaded region:

The number of elements in a set is called the cardinal number, or cardinality, of the set. This is denoted as N(A)
$N(A\cup B)= N(A) + N(B) – N(A\cap B)$

Complete step-by-step solution:
Let A be the set of people that speak Hindi and B be the set of people that speak English.
Then, we have the number of people that speak Hindi which is 750
$N(A) = 750$
The number of people that speak in English is 460
$N(B)=460$
Now in our question, it is mentioned that in a group there are a total of 950 persons
N( $A\cup B$ ) = 950
Thus, we can interpret that N( $A\cap B$ ) will give the number of people that speak both Hindi and English
Now let us substitute the values in the formula $N(A\cup B)= N(A) + N(B) – N(A\cap B)$
$950 = 750 + 460 - N(A\cap B)$
$\Rightarrow N(A\cap B) = 750 + 460 -950$
$\Rightarrow N(A\cap B) = 260$
But, in our question, we have been asked to find the number of persons that speak Hindi only. It can be calculated by subtracting the number of people who speak both Hindi and English from the number of people that speak Hindi.
Let us assume N(H) to represent the number of people that speak in Hindi only
$N(H) = N(A) – N (A\cap B)$
$\Rightarrow N(H) = 750 - 260$
$\Rightarrow N(H) = 490$
Therefore, the number of people that speak only Hindi is 490 out of 950.

Note: In questions like these students often get confused with the data given, for example, here it is mentioned that 750 is the number of people that speak Hindi so it should be understood that it isn’t the number of people that speak ‘only’ Hindi. The same goes for 460, 460 isn’t the number of people that speak only English. 750 also has people that can speak both Hindi as well as in English. To find the number of people that speak only in Hindi we have to subtract the number of people that speak both in Hindi and in English.