Out of 450 students in a school , 193 students read Science Today , 200 students read Science Refresher, while 80 students read neither. How many students read both the magazines ?

Options

A. 137
B. 80
C. 57
D. 23

Answer

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

### Step 1: Define the sets
Let:
- \( A \) = the set of students who read "Science Today"
- \( B \) = the set of students who read "Science Refresher"

From the problem, we know:
- \( n(A) = 193 \) (students reading "Science Today")
- \( n(B) = 200 \) (students reading "Science Refresher")
- Total students = 450
- Students reading neither magazine = 80

### Step 2: Calculate students reading at least one magazine
To find the number of students reading at least one of the magazines, we subtract the number of students reading neither from the total number of students:
\[ n(A \cup B) = \text{Total students} - \text{Students reading neither} \]
\[ n(A \cup B) = 450 - 80 = 370 \]

### Step 3: Use the formula for the union of two sets
The formula for the union of two sets is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Where \( n(A \cap B) \) is the number of students reading both magazines.

### Step 4: Substitute the known values into the formula
Now we can substitute the known values into the formula:
\[ 370 = 193 + 200 - n(A \cap B) \]

### Step 5: Simplify the equation
Combine the numbers on the right side:
\[ 370 = 393 - n(A \cap B) \]

### Step 6: Solve for \( n(A \cap B) \)
Rearranging the equation to solve for \( n(A \cap B) \):
\[ n(A \cap B) = 393 - 370 \]
\[ n(A \cap B) = 23 \]

### Conclusion
Thus, the number of students who read both magazines is **23**.

---