In a group of 50 persons, 40 speak Hindi and 25 speak English and Hindi both. Find how many persons speak only English ? Also find how many persons speak English ?

Answer

To solve the problem, we can use the principle of set theory. Let's denote:

- Let \( H \) be the set of persons who speak Hindi.
- Let \( E \) be the set of persons who speak English.

From the question, we have the following information:

1. Total number of persons in the group = 50
2. Number of persons who speak Hindi, \( |H| = 40 \)
3. Number of persons who speak both Hindi and English, \( |H \cap E| = 25 \)

We need to find:

1. The number of persons who speak only English.
2. The total number of persons who speak English.

### Step 1: Find the number of persons who speak only Hindi.

To find the number of persons who speak only Hindi, we can use the formula:

\[ |H \text{ only}| = |H| - |H \cap E| \]

Substituting the values:

\[ |H \text{ only}| = 40 - 25 = 15 \]

### Step 2: Find the total number of persons who speak English.

To find the total number of persons who speak English, we can use the formula:

\[ |E| = |H \cap E| + |E \text{ only}| \]

We need to find \( |E \text{ only}| \) first. We know that the total number of persons is 50, so we can express the total as:

\[ |H \text{ only}| + |E \text{ only}| + |H \cap E| = 50 \]

Substituting the known values:

\[ 15 + |E \text{ only}| + 25 = 50 \]

Now, simplifying this gives:

\[ |E \text{ only}| = 50 - 15 - 25 = 10 \]

Now we can find \( |E| \):

\[ |E| = |H \cap E| + |E \text{ only}| = 25 + 10 = 35 \]

### Final Answers:

1. The number of persons who speak only English = 10
2. The total number of persons who speak English = 35

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