Find the Total Students: Solving the 1/5 and 20% Participation Problem

To solve this problem, we will find a common equation to account for all students:

• Activity: Playing catch, Fraction: $\frac{1}{5}x$
• Activity: Playing soccer, Fraction: 20% of $x$ which is $\frac{1}{5}x$
• Activity: Watching a movie, Number: $15$ students

The total number of students involved is $x$. Thus, the setup for the equation is:

$\frac{1}{5}x + \frac{1}{5}x + 15 = x$

Simplify and solve for $x$:

$\frac{2}{5}x + 15 = x$

Subtract $\frac{2}{5}x$ from both sides:

$15 = x - \frac{2}{5}x$

$15 = \frac{3}{5}x$

To isolate $x$, multiply both sides by $\frac{5}{3}$:

$x = 15 \times \frac{5}{3}$

$x = 25$

Therefore, the total number of students is 25 students.