`45%` of the students of a class participated in Physics Olympaid and `65%` of the students of the class participated in Maths Olympiad. 4 students participated in neither of these two and 8 students participated in both. Find how many students
participated only in maths Olympiad ?

Answer

To find how many students participated only in the Maths Olympiad, we can follow these steps:

### Step 1: Define Variables
Let the total number of students in the class be \( x \).

### Step 2: Calculate Total Participants
From the problem, we know:
- 45% of the students participated in the Physics Olympiad:
\[ \text{Physics participants} = \frac{45}{100} \times x = 0.45x \]

- 65% of the students participated in the Maths Olympiad:
\[ \text{Maths participants} = \frac{65}{100} \times x = 0.65x \]

### Step 3: Account for Students Participating in Both Olympiads
We are given that 8 students participated in both Olympiads.

### Step 4: Use the Inclusion-Exclusion Principle
The total number of students who participated in at least one of the Olympiads can be calculated as:
\[ \text{Total participants} = \text{Physics participants} + \text{Maths participants} - \text{Both participants} \]
Substituting the values we have:
\[ \text{Total participants} = 0.45x + 0.65x - 8 \]
This simplifies to:
\[ \text{Total participants} = 1.1x - 8 \]

### Step 5: Account for Students Participating in Neither
We know that 4 students participated in neither Olympiad. Therefore, the equation becomes:
\[ x - (1.1x - 8) = 4 \]

### Step 6: Solve the Equation
Rearranging the equation:
\[ x - 1.1x + 8 = 4 \]
\[ -0.1x + 8 = 4 \]
Subtracting 8 from both sides:
\[ -0.1x = 4 - 8 \]
\[ -0.1x = -4 \]
Dividing both sides by -0.1:
\[ x = \frac{-4}{-0.1} = 40 \]

### Step 7: Calculate Total Maths Participants
Now that we know the total number of students \( x = 40 \), we can find the total number of students who participated in the Maths Olympiad:
\[ \text{Maths participants} = 0.65 \times 40 = 26 \]

### Step 8: Calculate Students Participating Only in Maths
To find the number of students who participated only in the Maths Olympiad, we subtract the number of students who participated in both:
\[ \text{Only Maths participants} = \text{Maths participants} - \text{Both participants} \]
\[ \text{Only Maths participants} = 26 - 8 = 18 \]

### Final Answer
Thus, the number of students who participated only in the Maths Olympiad is **18**.

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