Question

Два туриста вышли из пункта А по направлению к пункту В, одновременно, причем первый проходит каждый километр пути на 5 минут быстрее второго. Первый турист, пройдя пятую часть пути, вернулся обратно в А и пробыв там 10 минут, снова пошел в В. Однако в В оба туриста пришли одновременно. Каково расстояние от А до В, если второй турист прошел его за 2,5 часа.

Answer 1

Первый проходит 1 км за x км, второй за (x+5) км. Скорость первого 1/x км/мин, второго 1/(x+5) км/мин. Второй был в пути 2,5 часа или 2,5*60 = 150 мин. Вышли из А и пришли в В одновременно. Значит, первый затратил на весь путь тоже 150 мин.

Второй за 150 мин прошёл $150\cdot\frac1{x+5}=\frac{150}{x+5}$ км.

Первый прошёл 1/5 пути или $\frac{150}{x+5}\cdot\frac15=\frac{30}{x+5}$ км за $\frac{30}{x+5}:\frac1x=\frac{30x}{x+5}$ мин.

Затем он вернулся в А за то же время, 10 мин пробыл в А, и прошёл ВЕСЬ путь за $\frac{150}{x+5}:\frac1x=\frac{150x}{x+5}$ мин. Всего же он затратил 150 мин, то есть

$\frac{30x}{x+5}+\frac{30x}{x+5}+10+\frac{150x}{x+5}=150\\ \frac{210x}{x+5}=140\\ 210x=140x+700\\ 70x=700 x=10$

Первый проходит 1 км за 10 мин. Значит, весь путь $\frac{150}{x+5}=\frac{150}{15}=10\;\;km$

Answer 2

Problem Analysis

We have two tourists who start at point A and walk towards point B simultaneously. The first tourist walks each kilometer 5 minutes faster than the second tourist. After walking one-fifth of the distance, the first tourist returns to point A, stays there for 10 minutes, and then continues walking towards point B. Both tourists arrive at point B at the same time. We need to find the distance from point A to point B, given that the second tourist took 2.5 hours to complete the journey.

Solution

Let's assume that the distance from point A to point B is d kilometers. We can use the following steps to find the value of d.

1. Let's denote the speed of the second tourist as v km/h. Since the first tourist walks each kilometer 5 minutes faster than the second tourist, the speed of the first tourist can be expressed as (v + 12) km/h (since there are 60 minutes in an hour and 5 minutes is equal to 1/12th of an hour).

2. The time taken by the second tourist to complete the journey is given as 2.5 hours. We can use the formula time = distance / speed to write the equation 2.5 = d / v.

3. The first tourist walks one-fifth of the distance and then returns to point A. The time taken for this part of the journey can be calculated as (1/5) * (d / (v + 12)) + 10 minutes. We convert the 10 minutes to hours by dividing by 60.

4. The remaining distance from point A to point B is (4/5) * d. The time taken by the first tourist to cover this distance is given by ((4/5) * d) / (v + 12).

5. Since both tourists arrive at point B at the same time, we can equate the times taken by the second tourist and the first tourist. This gives us the equation 2.5 = (1/5) * (d / (v + 12)) + 10/60 + ((4/5) * d) / (v + 12).

6. Solving this equation will give us the value of d.

Let's calculate the value of d using the given information.

Calculation

Using the equation derived in step 5, we can solve for d:

2.5 = (1/5) * (d / (v + 12)) + 10/60 + ((4/5) * d) / (v + 12)

Multiplying both sides of the equation by 5(v + 12) to eliminate the denominators:

12.5(v + 12) = d + (v + 12)(10/60) + 4d

Simplifying the equation:

12.5v + 150 = d + (v + 12)(1/6) + 4d

12.5v + 150 = d + (v + 12)/6 + 4d

Multiplying both sides of the equation by 6 to eliminate the fraction:

75v + 900 = 6d + v + 12 + 24d

Combining like terms:

75v + 900 = 25d + v + 12

Simplifying the equation:

74v + 888 = 25d

Now, we can substitute the value of v from the equation 2.5 = d / v:

2.5v = d

Substituting this value in the equation 74v + 888 = 25d:

74v + 888 = 25 * 2.5v

Simplifying the equation:

74v + 888 = 62.5v

Subtracting 62.5v from both sides:

11.5v + 888 = 0

Subtracting 888 from both sides:

11.5v = -888

Dividing both sides by 11.5:

v = -888 / 11.5

Calculating the value of v:

v ≈ -77.2174

Since the speed cannot be negative, we can conclude that there is no valid solution for this problem.