Question

Мотоциклист проезжает за 5 ч такое же расстояние, какое автомобилист проезжает за 6 ч. Они выехали одновременно из двух пунктов, расстояние между которыми 15 км, в одном направлении так, что мотоциклист догоняет автомобилиста. Сколько километров проедет автомобилист, пока его не догонит мотоциклист?"

Answer 1

Х км - расстояние, пройденное автомобилистом до встречи
х+15 = 6х/5
х+15=1,2x
1,2x-x=15
0,2x=15
x=15:0,2
х = 75 км - столько проехал автомобилист, пока его не догнал мотоциклист.

Answer 2

Problem Analysis

We are given that a motorcyclist travels for 5 hours and covers the same distance as an automobile driver who travels for 6 hours. They start at the same time from two points that are 15 km apart. The motorcyclist catches up to the automobile driver. We need to determine the distance the automobile driver travels before being caught by the motorcyclist.

Solution

Let's assume that the speed of the motorcyclist is x km/h and the speed of the automobile driver is y km/h.

We know that the distance covered by the motorcyclist is the same as the distance covered by the automobile driver. Therefore, we can set up the following equation:

Distance covered by motorcyclist = Distance covered by automobile driver

Using the formula distance = speed × time, we can write the equation as:

x km/h × 5 h = y km/h × 6 h

Simplifying the equation, we get:

5x = 6y

We also know that the motorcyclist catches up to the automobile driver. This means that the time taken by the motorcyclist to cover the distance between the two points is less than the time taken by the automobile driver. Therefore, we can set up the following equation:

Time taken by motorcyclist < Time taken by automobile driver

Using the formula time = distance / speed, we can write the equation as:

15 km / x km/h < 15 km / y km/h

Simplifying the equation, we get:

y < x

Now we have a system of equations: 1. 5x = 6y 2. y < x

We can solve this system of equations to find the values of x and y.

Solving the System of Equations

To solve the system of equations, we can use substitution or elimination. Let's use substitution.

From equation 2, we have y < x. We can rewrite this as x > y.

Substituting x with 6y/5 in the inequality x > y, we get:

6y/5 > y

Multiplying both sides of the inequality by 5, we get:

6y > 5y

Subtracting 5y from both sides of the inequality, we get:

y > 0

This means that the speed of the automobile driver, y, must be greater than 0.

Now, substituting x with 6y/5 in equation 1, we get:

5(6y/5) = 6y

Simplifying the equation, we get:

6y = 6y

This equation is true for all values of y.

Therefore, the system of equations has infinitely many solutions.

Since the system of equations has infinitely many solutions, we cannot determine the exact values of x and y. However, we can make some observations based on the given information.

From equation 1, we know that 5x = 6y. This means that the speed of the motorcyclist, x, is greater than the speed of the automobile driver, y.

From equation 2, we know that y < x. This confirms that the speed of the motorcyclist is indeed greater than the speed of the automobile driver.

Based on this information, we can conclude that the motorcyclist will always catch up to the automobile driver, regardless of their specific speeds.

Therefore, the distance the automobile driver will travel before being caught by the motorcyclist is 15 km.

Answer

The automobile driver will travel 15 km before being caught by the motorcyclist.

Please let me know if you need any further clarification.