5. In a certain school, a teacher prepared a certain number of thank-you cards for the participants of the Math Night. The number of students who registered to participate was $\frac{2}{13}$ more than the number of thank-you cards he prepared, but on the actual Math Night, $\frac{1}{15}$ of the registered students did not show up. How many students participated in the Math Night if the teacher had to print an additional 20 thank-you cards?

## Problems worth 10 points:
5. Let $x$ be the number of thank-you cards the teacher has prepared.

The number of students who registered to participate is $x+\frac{2}{13} x=\frac{15}{13} x$.

1 POINT

At the Math Night, $\frac{14}{15}$ of that number showed up, i.e., $\frac{14}{15} \cdot \frac{15}{13} x=\frac{14}{13} x$ students.

2 POINTS

This is $\frac{1}{13}$ more than the number of prepared thank-you cards.

From $\frac{1}{13} x=20$, we get $x=260$.

2 POINTS

At the Math Night, 280 students participated.

1 POINT

TOTAL 6 POINTS