In a group of 1000 people 700 can speak English and 500 can speak Hindi. If all the people speak atleast one of the two languages find:
(a) how many can speak both the languages?
(b) how many can speak exactly one language ?

Answer

To solve the problem step by step, we will use the principles of set theory. Let's denote:

- \( A \): the set of people who can speak English.
- \( B \): the set of people who can speak Hindi.

Given:
- \( |A| = 700 \) (the number of people who can speak English)
- \( |B| = 500 \) (the number of people who can speak Hindi)
- \( |A \cup B| = 1000 \) (the total number of people who can speak at least one of the two languages)

### Step 1: Find the number of people who can speak both languages.

We can use the formula for the union of two sets:

\[ |A \cup B| = |A| + |B| - |A \cap B| \]

Where \( |A \cap B| \) is the number of people who can speak both languages. Rearranging the formula gives us:

\[ |A \cap B| = |A| + |B| - |A \cup B| \]

Now, substituting the known values:

\[ |A \cap B| = 700 + 500 - 1000 \]

Calculating this:

\[ |A \cap B| = 1200 - 1000 = 200 \]

So, **200 people can speak both languages**.

### Step 2: Find the number of people who can speak exactly one language.

To find the number of people who speak only one language, we will calculate the number of people who speak only English and only Hindi.

- **People who speak only Hindi**:

\[ |B \text{ only}| = |B| - |A \cap B| = 500 - 200 = 300 \]

- **People who speak only English**:

\[ |A \text{ only}| = |A| - |A \cap B| = 700 - 200 = 500 \]

Now, to find the total number of people who speak exactly one language, we add the two results:

\[ \text{Total speaking exactly one language} = |A \text{ only}| + |B \text{ only}| = 500 + 300 = 800 \]

Thus, **800 people can speak exactly one language**.

### Final Answers:
(a) The number of people who can speak both languages is **200**.
(b) The number of people who can speak exactly one language is **800**.

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