(i) In a group of 50 people , 35 speak Hindi , 25 speak both English and Hindi and all people speak atleast one of the two languages . How may people speak only English and not Hindi ? How many people speak English?
(ii) In a group of people , 50 speak both English and Hindi and 30 people speak English but not Hindi All the people speak at least one of the two languages . How many people speak English ?

Answer

To solve the given problems step by step, let's break it down.

### Part (i)

1. **Identify the given information:**
- Total number of people (n) = 50
- Number of people who speak Hindi (n(H)) = 35
- Number of people who speak both English and Hindi (n(E ∩ H)) = 25
- All people speak at least one of the two languages.

2. **Use the formula for the union of two sets:**
\[ n(E \cup H) = n(E) + n(H) - n(E \cap H) \]
Since all people speak at least one language:
\[ n(E \cup H) = 50 \]
Plugging in the values:
\[ 50 = n(E) + 35 - 25 \]
Simplifying:
\[ 50 = n(E) + 10 \]
\[ n(E) = 50 - 10 = 40 \]
So, the number of people who speak English (n(E)) = 40.

3. **Find the number of people who speak only English:**
- To find the number of people who speak only English (n(E) - n(E ∩ H)):
\[ n(\text{Only English}) = n(E) - n(E \cap H) \]
Plugging in the values:
\[ n(\text{Only English}) = 40 - 25 = 15 \]
So, the number of people who speak only English and not Hindi = 15.

### Part (ii)

1. **Identify the given information:**
- Number of people who speak both English and Hindi (n(E ∩ H)) = 50
- Number of people who speak English but not Hindi (n(E) - n(E ∩ H)) = 30
- All people speak at least one of the two languages.

2. **Use the information to find the total number of people who speak English:**
- Let n(E) be the total number of people who speak English.
- The total number of people who speak English can be calculated as:
\[ n(E) = n(E \cap H) + n(\text{Only English}) \]
Plugging in the values:
\[ n(E) = 50 + 30 = 80 \]
So, the total number of people who speak English = 80.

### Summary of Answers
- (i) Number of people who speak only English and not Hindi = 15; Total number of people who speak English = 40.
- (ii) Total number of people who speak English = 80.