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The problem involves Enzo and Beatriz playing a game where they each earn a different number of tickets per game. Enzo earns 5 tickets per game, while Beatriz earns 11 tickets per game. The question asks for the minimum number of games Enzo must have played to end up with the same number of tickets as Beatriz.

### Problem Breakdown:
- Enzo earns 5 tickets per game.
- Beatriz earns 11 tickets per game.
- After playing, they both have the same number of tickets.

### Solution Approach:
We need to find the minimum number of games \( x \) Enzo must have played, such that the number of tickets he earns is equal to the number of tickets Beatriz earns after \( y \) games. 

Let:
- \( x \) be the number of games Enzo played.
- \( y \) be the number of games Beatriz played.

The number of tickets Enzo earned is \( 5x \) and the number of tickets Beatriz earned is \( 11y \). The two values must be equal for both to have the same number of tickets:
\[
5x = 11y
\]
This is a Diophantine equation, and the solution can be found by finding the smallest \( x \) and \( y \) such that the equation holds true.

The least common multiple (LCM) of 5 and 11 will help us find the smallest solution. The LCM of 5 and 11 is 55, meaning both will have earned 55 tickets when:
\[
5x = 55 \quad \text{and} \quad 11y = 55
\]

Thus:
\[
x = \frac{55}{5} = 11 \quad \text{and} \quad y = \frac{55}{11} = 5
\]

### Conclusion:
The minimum number of games Enzo must have played is **11**.

Would you like more details on how this was calculated, or do you have any questions?

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Here are five related questions to expand on this topic:

1. What if Enzo earned 6 tickets per game instead of 5?
2. How would the result change if Beatriz earned 10 tickets instead of 11?
3. What is the general approach for solving Diophantine equations like this?
4. Could you explain how to calculate the least common multiple (LCM) for different numbers?
5. How does the greatest common divisor (GCD) help in similar types of problems?

**Tip:** When solving Diophantine equations, always look for common multiples to find solutions easily.

Enzo y Beatriz están jugando en las maquinitas. Increíblemente, Enzo gana 5 boletos por cada juego y Beatriz gana 11 boletos por cada juego. Cuando dejaron de jugar, ambos tenían el mismo número de boletos. ¿Cuál es el número mínimo de juegos que Enzo pudo haber jugado?

Learn how to solve a Diophantine equation involving two people earning different amounts of tickets per game. This problem covers key algebraic concepts and uses the Least Common Multiple (LCM) to find the minimum number of games Enzo must have played. Suitable for students in Grades 6-8.

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