Question

Обычно пешеход проходит расстояние от станции до дачи за 4,5 часа. однажды он шел на 12 км/ч быстрее обычного и пришел на 1.5 часа раньше. какое расстояние между станцией и дачей. решить двумя способами.

Answer 1

1 способ:
х км/ч - обычная скорость пешехода
(х+12) км/ч - новая скорость пешехода
4,5-1,5=3ч. - затратил пешеход в другой раз
4,5х=3(х+12)
4,5х-3х=36
1,5х=36
х=36:1,5
х=24 км/ч - обычная скорость пешехода
4,5·24=108 км - расстояние между станцией и дачей
2 способ:
х км - расстояние между дачей и станцией
х/3 - х/4,5=12
3х-2х=12·9
х=108 км - расстояние между дачей и станцией

Answer 2

Problem Analysis

We are given that a pedestrian usually takes 4.5 hours to walk from the station to the dacha. However, on one occasion, the pedestrian walked 12 km/h faster than usual and arrived 1.5 hours earlier than usual. We need to find the distance between the station and the dacha using two different methods.

Method 1: Using the Usual Speed

Let's assume the usual speed of the pedestrian is x km/h. We know that the pedestrian takes 4.5 hours to walk from the station to the dacha at this speed. Therefore, the distance between the station and the dacha can be calculated using the formula:

Distance = Speed × Time

Substituting the values, we have:

Distance = x km/h × 4.5 h

Method 2: Using the Faster Speed

On this particular occasion, the pedestrian walked 12 km/h faster than usual. Therefore, the speed can be calculated as (x + 12) km/h. We also know that the pedestrian arrived 1.5 hours earlier than usual, which means the time taken is (4.5 - 1.5) hours. Using the same formula as before, we can calculate the distance:

Distance = (x + 12) km/h × (4.5 - 1.5) h

Let's calculate the distance using both methods.

Calculation

Method 1: Distance = x km/h × 4.5 h

Method 2: Distance = (x + 12) km/h × (4.5 - 1.5) h

Solution

To find the distance between the station and the dacha, we need to solve the two equations obtained from the two methods. Let's solve them.

Method 1: Distance = x km/h × 4.5 h

Method 2: Distance = (x + 12) km/h × (4.5 - 1.5) h

Simplifying the equations, we have:

Method 1: Distance = 4.5x km

Method 2: Distance = 3(x + 12) km

Since both methods represent the same distance, we can equate the two equations:

4.5x = 3(x + 12)

Solving for x, we get:

4.5x = 3x + 36 1.5x = 36 x = 24

Now that we have the value of x, we can substitute it back into either equation to find the distance:

Distance = 4.5x km Distance = 4.5 × 24 km Distance = 108 km

Therefore, the distance between the station and the dacha is 108 km.

Alternative Solution

Alternatively, we can solve the problem using a different approach. Let's calculate the time taken to travel the distance using both methods.

Method 1: Time = 4.5 hours

Method 2: Time = 4.5 - 1.5 = 3 hours

Now, we can calculate the distance using the formula:

Distance = Speed × Time

Method 1: Distance = x km/h × 4.5 h

Method 2: Distance = (x + 12) km/h × 3 h

Equating the two distances, we have:

x × 4.5 = (x + 12) × 3

Simplifying the equation, we get:

4.5x = 3x + 36 1.5x = 36 x = 24

Substituting the value of x back into either equation, we find:

Distance = x km/h × 4.5 h Distance = 24 km/h × 4.5 h Distance = 108 km

Therefore, the distance between the station and the dacha is 108 km.

Conclusion

Using both methods, we have determined that the distance between the station and the dacha is 108 km.