Understanding the Nursery Class Language Problem

This problem involves a common scenario using set theory concepts, often visualized with Venn diagrams. We need to determine the number of students speaking Hindi, only Hindi, and only English based on the total number of students and information about those speaking English and both languages.

Given Information:

• Total students admitted to the nursery class: 50
• Number of students who can speak both English and Hindi: 10
• Number of students who can speak English: 21

Calculations:

Let $E$ represent the set of students who can speak English, and $H$ represent the set of students who can speak Hindi.

We are given:

• Total students = 50
• Number of students speaking both English and Hindi, $|E \cap H| = 10$
• Number of students speaking English, $|E| = 21$

1. Finding the number of students who speak only English:

The number of students who speak only English is the total number of English speakers minus those who speak both languages.

Number of students speaking only English = $|E| - |E \cap H|$

Number of students speaking only English = $21 - 10 = 11$

2. Finding the total number of students speaking Hindi:

We know that the total number of students (50) is the sum of students who speak only English, only Hindi, and both English and Hindi, assuming every student speaks at least one language. The formula for the union of two sets is $|E \cup H| = |E| + |H| - |E \cap H|$. If we assume all 50 students speak at least one language, then $|E \cup H| = 50$.

Using the Principle of Inclusion-Exclusion:

Total Students = (Only English) + (Only Hindi) + (Both English and Hindi)

Let's use the total union formula: $|E \cup H| = |E| + |H| - |E \cap H|$

Assuming $|E \cup H| = 50$ (total students):

$50 = 21 + |H| - 10$

$50 = 11 + |H|$

$|H| = 50 - 11$

$|H| = 39$

So, the total number of students who can speak Hindi is 39.

3. Finding the number of students who speak only Hindi:

The number of students who speak only Hindi is the total number of Hindi speakers minus those who speak both languages.

Number of students speaking only Hindi = $|H| - |E \cap H|$

Number of students speaking only Hindi = $39 - 10 = 29$

Summary of Findings:

We have calculated the following:

• Number of students who can speak Hindi ($|H|$): 39
• Number of students who can speak only Hindi: 29
• Number of students who can speak only English: 11

Therefore, the numbers are 39, 29, and 11 respectively.

Verification:

Let's check if these numbers add up correctly:

Students speaking only English + Students speaking only Hindi + Students speaking both

$11 + 29 + 10 = 50$

This matches the total number of students admitted, confirming our calculations.