If a community of 175 persons , 40 read TOI, 50 read the Samachar Patrika and 100 do not read any How many persons read both the papers

Answer

To solve the problem step-by-step, we can follow these steps:

### Step 1: Understand the given information
We have a community of 175 persons.
- 40 persons read the Times of India (TOI).
- 50 persons read the Samachar Patrika.
- 100 persons do not read any newspaper.

### Step 2: Calculate the number of persons who read at least one newspaper
Since 100 persons do not read any newspaper, we can find the number of persons who read at least one newspaper by subtracting the non-readers from the total population:
\[ \text{Number of persons who read at least one newspaper} = \text{Total persons} - \text{Persons who do not read any} \]
\[ = 175 - 100 = 75 \]

### Step 3: Use the formula for union of two sets
Let:
- \( A \) = number of persons who read TOI = 40
- \( B \) = number of persons who read Samachar Patrika = 50
- \( A \cup B \) = number of persons who read at least one newspaper = 75
- \( A \cap B \) = number of persons who read both newspapers (which we need to find)

According to set theory, the formula for the union of two sets is:
\[ A \cup B = A + B - A \cap B \]

### Step 4: Substitute the known values into the formula
We can rearrange the formula to find \( A \cap B \):
\[ A \cap B = A + B - A \cup B \]
Substituting the values we have:
\[ A \cap B = 40 + 50 - 75 \]

### Step 5: Calculate the number of persons who read both newspapers
Now, we perform the calculation:
\[ A \cap B = 90 - 75 = 15 \]

### Conclusion
Therefore, the number of persons who read both the Times of India and the Samachar Patrika is **15**.

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