There is a group of 50 persons who can speak English or Tamil or both. Out of these persons , 37 can speak English and 30 cm speak Tamil.
How many can speak English only ?

Answer

To solve the problem step by step, we can use the principle of set theory.

### Step-by-Step Solution:

1. **Identify the total number of people and the number of speakers:**
- Let \( N(A) \) be the number of people who can speak English.
- Let \( N(B) \) be the number of people who can speak Tamil.
- We know that:
- \( N(A) = 37 \) (people who can speak English)
- \( N(B) = 30 \) (people who can speak Tamil)
- \( N(A \cup B) = 50 \) (total number of people who can speak either English or Tamil or both)

2. **Use the formula for the union of two sets:**
- The formula for the union of two sets is:
\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]
- Here, \( N(A \cap B) \) represents the number of people who can speak both languages.

3. **Substitute the known values into the formula:**
\[ 50 = 37 + 30 - N(A \cap B) \]

4. **Simplify the equation to find \( N(A \cap B) \):**
- Combine the numbers on the right side:
\[ 50 = 67 - N(A \cap B) \]
- Rearranging gives:
\[ N(A \cap B) = 67 - 50 = 17 \]
- So, 17 people can speak both English and Tamil.

5. **Calculate the number of people who can speak English only:**
- The number of people who can speak English only is given by:
\[ N(A \text{ only}) = N(A) - N(A \cap B) \]
- Substitute the values:
\[ N(A \text{ only}) = 37 - 17 = 20 \]

6. **Final Answer:**
- Therefore, the number of people who can speak English only is **20**.