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The problem you're working on can be solved using algebra. Let's break it down step-by-step.

### Problem Summary:
- A group of people were originally supposed to equally divide a restaurant bill of €200.
- Before settling the bill, 3 people leave.
- The remaining people have to pay an extra €15 each compared to what they would have paid if no one left.

We need to find how many people were originally in the group.

### Step-by-Step Solution:

1. **Let the total number of people in the group be \( x \).**

2. **Each person's original share** would have been:
   \[
   \frac{200}{x}
   \]
   since the bill of €200 is divided equally among \( x \) people.

3. **After 3 people leave**, the number of people remaining is \( x - 3 \). The new share for each person is:
   \[
   \frac{200}{x - 3}
   \]

4. **The difference between the new share and the original share is €15**:
   \[
   \frac{200}{x - 3} - \frac{200}{x} = 15
   \]

5. **Solve the equation**:

   First, we simplify the equation by getting a common denominator:
   \[
   \frac{200(x) - 200(x - 3)}{x(x - 3)} = 15
   \]

   Simplify the numerator:
   \[
   \frac{200x - 200(x - 3)}{x(x - 3)} = 15
   \]
   \[
   \frac{200x - 200x + 600}{x(x - 3)} = 15
   \]
   \[
   \frac{600}{x(x - 3)} = 15
   \]

6. **Multiply both sides by \( x(x - 3) \)** to eliminate the denominator:
   \[
   600 = 15x(x - 3)
   \]

7. **Expand and solve the quadratic equation**:
   \[
   600 = 15(x^2 - 3x)
   \]
   \[
   600 = 15x^2 - 45x
   \]
   Divide everything by 15:
   \[
   40 = x^2 - 3x
   \]
   Rearrange the equation:
   \[
   x^2 - 3x - 40 = 0
   \]

8. **Solve using the quadratic formula**:
   The quadratic formula is given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For the equation \( x^2 - 3x - 40 = 0 \), we have \( a = 1 \), \( b = -3 \), and \( c = -40 \).

   Substituting these values into the formula:
   \[
   x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-40)}}{2(1)}
   \]
   \[
   x = \frac{3 \pm \sqrt{9 + 160}}{2}
   \]
   \[
   x = \frac{3 \pm \sqrt{169}}{2}
   \]
   \[
   x = \frac{3 \pm 13}{2}
   \]

   So, the two possible values for \( x \) are:
   \[
   x = \frac{3 + 13}{2} = 8 \quad \text{or} \quad x = \frac{3 - 13}{2} = -5
   \]

   Since \( x \) must be positive, the original number of people in the group is **8**.

### Final Answer:
The group originally had **8 people**.

---

Would you like more details on any part of the process, or do you have additional questions?

### Related Questions:
1. What would the answer be if two people left instead of three?
2. How would the result change if the bill was €250 instead of €200?
3. Can you generalize a formula to solve similar problems for any number of people leaving?
4. How does the solution change if the remaining people paid €20 extra instead of €15?
5. How would you solve this problem using numerical methods instead of algebra?

### Tip:
When solving problems with variables in the denominator, always consider multiplying through to clear fractions early on. This can simplify solving the equation.

A restaurant bill that comes to €200 is to be divided equally among a group of people. However, three people leave before the bill is settled and the remainder of the group have to pay an extra €15 each. How many people were originally in the group?

Learn how to solve a real-world algebra problem where a group of people must split a €200 bill. After three people leave, the others pay €15 extra. This step-by-step guide uses algebra and the quadratic formula to find how many people were originally in the group. Suitable for Grades 8-10.

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