People enter the metro from the street evenly. After passing through the turnstiles, they find themselves in a small hall before the escalators. The doors just opened, and the hall in front of the escalators was initially empty, with only one escalator operating for descent. One escalator could not handle the crowd, so in 6 minutes, the hall was half full. Then a second escalator was turned on for descent, but the crowd continued to grow - after another 15 minutes, the hall was full.

How long will it take for the hall to empty if a third escalator is turned on?
When one escalator is turned on, 1/12 of the hall is filled in a minute. When two escalators are turned on, 1/1 of the hall is filled in a minute. We can reason in different ways.

First method. The difference 1/12 - 1/30 = 1/20 shows what part of the hall one escalator empties in a minute. When a third escalator is turned on, the crowd will begin to decrease at a rate of 1/20 - 1/30 = 1/60 of the hall per minute.

Therefore, the hall will be empty in an hour.

Second method. The speed \( v_2 \) of filling the hall with two escalators turned on is the arithmetic mean of the speeds \( v_1 \) and \( v_3 \) of filling with one and three escalators turned on, respectively. Therefore, \( v_3 = 2v_2 - v_1 = 2 \cdot \frac{1}{30} - \frac{1}{12} = -\frac{1}{60} \), which means that 1/60 of the hall is emptied per minute.

## Answer

In an hour.

Author: Frankin B.R.

In the first year, the population of a certain village increased by \( n \) people, and in the second year, it increased by 300 people. During the first year, the population increased by 300%, and during the second year, it increased by \( n \% \). How many residents are there in the village now?

## Solution

Suppose initially there were \( m \) residents in the village. Then after the first year, the population became \( m + 3m = 4m \) residents, so \( n = 3m \). After another year, the population became \( 4m + 300 = 4m + \frac{3m}{100} \cdot 4m \) residents, so \( 12m^2 = 30000 \).

Thus, \( m = 50 \) and now there are \( 4m + 300 = 500 \) people in the village.

## Answer

500 people.