1. The price of one ticket is 50 denars. When the price is reduced, the number of visitors increases by $50\%$, and the revenue increases by $20\%$. By how much is the price of the entry ticket reduced?
Solution. First method. Two people, at the old price, would pay 100 denars, while three people (50% more), at the new price, would pay 120 denars (20% more than 100 denars). Therefore, one person, at the new price, would pay 40 denars, from which it follows that the reduction is 20% (from 50 denars to 40 denars).

Second method. Let the number of visitors be \( p \), the earnings \( z \), and the new price of one ticket \( x \). Then, \( 50p = z \) and \( x \cdot \frac{150}{100} p = \frac{120}{100} z \). If we divide the last two equations, we get \( \frac{50p}{x \cdot 1.5p} = \frac{z}{1.2z} \), from which it follows that \( x = 40 \). Therefore, the reduction is 20%.

Third method. Let the number of visitors be \( x \), then the earnings are \( 50x \). If the new price of the ticket is \( y \), then the number of visitors is \( 1.5x \) (50% more), and the earnings are \( 60x \) (20% more than \( 50x \)). Then, \( 1.5x \cdot y = 60x \), from which it follows that \( y = 40 \). Consequently, the reduction is 20%.