Question

Мотоциклист может проехать Расстояние между пунктами за 2 часа а велосипедист за 6 часов Однажды они одновременно отправились навстречу друг другу из этих пунктов Сколько километров проехал каждый до встречи если расстояние между пунктами 60 км Решите задачу двумя способами

Answer 1

60:2=30км/час скорость мотоцикла
60:6=10км/час скорость велосипеда
30+10=40км/час скорость сближения
60:40=1,5 час до встречи
30*1,5=45км проехал мотоциклист
10*1,5=15км проехал велосипедист

60:2=30км/час скорость мотоцикла
60:6=10км/час скорость велосипеда
х проехал до встречи мотоциклист
60-х проехал велосипедист
х/30=(60-х)/10
10х=1800-30х
40х=1800
х=1800:40=45км проехал мотоциклист
60-45=15км проехал велосипедист

Answer 2

Problem Statement

Мотоциклист может проехать расстояние между пунктами за 2 часа, а велосипедист за 6 часов. Однажды они одновременно отправились навстречу друг другу из этих пунктов. Сколько километров проехал каждый до встречи, если расстояние между пунктами 60 км? Решите задачу двумя способами.

Solution 1: Using Relative Speed

To solve this problem, we can use the concept of relative speed. The relative speed is the sum of the speeds of the two objects moving towards each other.

Let's assume that the distance traveled by the motorcyclist before the meeting is x km. Since the motorcyclist can cover the distance between the points in 2 hours, their speed is x/2 km/h.

Similarly, let's assume that the distance traveled by the cyclist before the meeting is y km. Since the cyclist can cover the distance between the points in 6 hours, their speed is y/6 km/h.

The total distance between the two points is given as 60 km.

To find the distances traveled by each before the meeting, we can set up the following equation:

x + y = 60 (equation 1)

Now, let's calculate the relative speed. The relative speed is the sum of the speeds of the motorcyclist and the cyclist:

Relative speed = (x/2) + (y/6) km/h (equation 2)

The time taken to meet is the same for both the motorcyclist and the cyclist. Let's assume it is t hours.

Using the formula distance = speed × time, we can set up the following equation:

60 = (x/2 + y/6) × t (equation 3)

Now, we have three equations (equations 1, 2, and 3) with three unknowns (x, y, and t). We can solve these equations to find the values of x and y.

Let's solve the equations:

From equation 1, we have x = 60 - y.

Substituting this value of x in equation 2, we get:

(60 - y)/2 + y/6 = Relative speed

Simplifying the equation, we get:

30 - y/2 + y/6 = Relative speed

Multiplying through by 6 to eliminate the fractions, we get:

180 - 3y + y = 6 × Relative speed

Simplifying further, we get:

180 - 2y = 6 × Relative speed

Now, let's substitute the value of Relative speed with the sum of the speeds of the motorcyclist and the cyclist:

180 - 2y = 6 × [(x/2) + (y/6)]

Substituting the value of x as 60 - y, we get:

180 - 2y = 6 × [(60 - y)/2 + (y/6)]

Simplifying the equation, we get:

180 - 2y = 6 × [30 - y/2 + y/6]

Multiplying through by 6 to eliminate the fractions, we get:

1080 - 12y = 180 × [30 - y/2 + y/6]

Simplifying further, we get:

1080 - 12y = 180 × [180 - 3y + y]

Simplifying the equation, we get:

1080 - 12y = 180 × [180 - 2y]

Expanding the equation, we get:

1080 - 12y = 32400 - 360y

Bringing all the terms to one side, we get:

348y = 31320

Dividing both sides by 348, we get:

y = 90

Substituting the value of y in equation 1, we get:

x + 90 = 60

Simplifying the equation, we get:

x = -30

Since distance cannot be negative, we discard the negative value of x.

Therefore, the distance traveled by the motorcyclist before the meeting is 0 km and the distance traveled by the cyclist before the meeting is 90 km.

So, the motorcyclist did not travel any distance before the meeting, while the cyclist traveled a distance of 90 km before the meeting.

Solution 2: Using Time and Speed

Another way to solve this problem is by using the concept of time and speed.

Let's assume that the motorcyclist's speed is m km/h and the cyclist's speed is c km/h.

The time taken by the motorcyclist to cover the distance between the points is given as 2 hours, so we have:

m = distance / time = 60 / 2 = 30 km/h

The time taken by the cyclist to cover the distance between the points is given as 6 hours, so we have:

c = distance / time = 60 / 6 = 10 km/h

Now, let's calculate the distances traveled by each before the meeting.

Let's assume that the motorcyclist traveled x km before the meeting, and the cyclist traveled y km before the meeting.

Since the time taken by both the motorcyclist and the cyclist to meet is the same, we can set up the following equation:

x / m = y / c (equation 4)

Substituting the values of m and c, we get:

x / 30 = y / 10

Simplifying the equation, we get:

3x = y (equation 5)

We also know that the sum of the distances traveled by both the motorcyclist and the cyclist is equal to the total distance between the points, which is 60 km:

x + y = 60 (equation 6)

Now, we have two equations (equations 5 and 6) with two unknowns (x and y). We can solve these equations to find the values of x and y.

From equation 5, we have y = 3x.

Substituting this value of y in equation 6, we get:

x + 3x = 60

Simplifying the equation, we get:

4x = 60

Dividing both sides by 4, we get:

x = 15

Substituting the value of x in equation 6, we get:

15 + 3(15) = 60

Simplifying the equation, we get:

15 + 45 = 60

Therefore, the distance traveled by the motorcyclist before the meeting is 15 km and the distance traveled by the cyclist before the meeting is 45 km.

So, the motorcyclist traveled a distance of 15 km before the meeting, while the cyclist traveled a distance of 45 km before the meeting.

Conclusion

In conclusion, when the distance between the points is 60 km, the motorcyclist did not travel any distance before the meeting, while the cyclist traveled a distance of 90 km before the meeting. Alternatively, when the distance between the points is 60 km, the motorcyclist traveled a distance of 15 km before the meeting, while the cyclist traveled a distance of 45 km before the meeting.