`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 in at least one Olympiad?

Answer

To solve the problem step by step, we will use the information provided to find the number of students who participated in at least one Olympiad.

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

### Step 2: Determine the number of students participating in each Olympiad
- Students participating in Physics Olympiad = \( 45\% \) of \( x \) = \( \frac{45}{100}x = 0.45x \)
- Students participating in Maths Olympiad = \( 65\% \) of \( x \) = \( \frac{65}{100}x = 0.65x \)

### Step 3: Identify the number of students participating in both Olympiads
We are given that 8 students participated in both Olympiads.

### Step 4: Use the principle of inclusion-exclusion
To find the total number of students participating in at least one Olympiad, we can use the formula:
\[ \text{Total participating} = (\text{Physics}) + (\text{Maths}) - (\text{Both}) \]
Substituting the values we have:
\[ \text{Total participating} = 0.45x + 0.65x - 8 \]
This simplifies to:
\[ \text{Total participating} = (0.45 + 0.65)x - 8 = 1.1x - 8 \]

### Step 5: Account for students participating in neither Olympiad
We know that 4 students participated in neither Olympiad. Therefore, the equation can be set up as:
\[ x - (1.1x - 8) = 4 \]
This simplifies to:
\[ x - 1.1x + 8 = 4 \]
\[ -0.1x + 8 = 4 \]

### Step 6: Solve for \( x \)
Rearranging gives:
\[ -0.1x = 4 - 8 \]
\[ -0.1x = -4 \]
Dividing both sides by -0.1:
\[ x = \frac{-4}{-0.1} = 40 \]

### Step 7: Calculate the number of students participating in at least one Olympiad
Now that we have \( x = 40 \), we can find the number of students who participated in at least one Olympiad:
\[ \text{Total participating} = 1.1(40) - 8 = 44 - 8 = 36 \]

### Final Answer
The number of students who participated in at least one Olympiad is **36**.

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