Question

Теплоход проходит за 2 часа по течению реки и за 3 часа против течения 85км. Известно что за 3 часа по течению реки он проходит на 30км больше, чем за 2 часа против течения. Найдите скорость движения теплохода по течению реки и скорость его движения против.

Answer 1

Х км/ч -собственная скорость теплохода, у км/ч - скорость течения реки.
(х+у) км/ч - скорость по течению, (х-у) км/ч - скорость против течения.
2(х+у) км - путь за 2 ч по течению,  3(х-у) км - путь за 3 ч против течения.
3(х+у) км - путь за 3 ч по течению,  2(х-у) км - путь за 3 ч против течения.
Учитывая соотношения, описанные в условии задачи, получим систему: $\begin{cases} 2(x+y)+3(x-y)=85 \\ 3(x+y)-2(x-y)=30 \end{cases}$
Умножаем первое уравнение на 3, а второе на 2 и вычитаем почленно:
$\begin{cases} 6(x+y)+9(x-y)=255 \\ 6(x+y)-4(x-y)=60 \end{cases} \ \textless \ =\ \textgreater \ \begin{cases} 13(x-y)=195 \\ 3(x+y)-2(x-y)=30 \end{cases} \ \textless \ =\ \textgreater \ \\ \begin{cases} x-y=15 \\ 3(x+y)-30=30 \end{cases} \ \textless \ =\ \textgreater \ \begin{cases} x-y=15 \\ x+y=20 \end{cases}$
Значит, 20 км/ч - скорость по течению, 15 км/ч - скорость против течения.

Answer 2

Problem Analysis

We are given that a boat travels for 2 hours with the current and covers a certain distance, and then travels for 3 hours against the current and covers a distance of 85 km. We are also given that the boat covers 30 km more in 3 hours with the current than in 2 hours against the current. We need to find the speed of the boat with the current and the speed of the boat against the current.

Solution

Let's assume the speed of the boat in still water is x km/h and the speed of the current is y km/h.

When the boat travels with the current for 2 hours, it covers a certain distance. We can calculate this distance using the formula: distance = speed * time. Therefore, the distance covered with the current is 2(x + y) km.

When the boat travels against the current for 3 hours, it covers a distance of 85 km. Using the same formula, we can write: 85 = 3(x - y).

We are also given that the boat covers 30 km more in 3 hours with the current than in 2 hours against the current. This can be expressed as: 2(x + y) + 30 = 3(x - y).

Now we have a system of two equations with two variables. We can solve this system to find the values of x and y.

Let's solve the system of equations:

Equation 1: 2(x + y) = distance covered with the current Equation 2: 3(x - y) = 85 Equation 3: 2(x + y) + 30 = 3(x - y)

Simplifying Equation 3: 2x + 2y + 30 = 3x - 3y 5y - x = 30

Now we have two equations: Equation 1: 2(x + y) = distance covered with the current Equation 2: 5y - x = 30

Let's solve this system of equations.

Calculation

Equation 1: 2(x + y) = distance covered with the current Substituting the value of the distance covered with the current (2(x + y)) from Equation 1 into Equation 3: 2(x + y) + 30 = 3(x - y) 2x + 2y + 30 = 3x - 3y 5y - x = 30

Now we have two equations: Equation 1: 2x + 2y = distance covered with the current Equation 2: 5y - x = 30

Let's solve this system of equations.

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

Multiplying Equation 1 by 5 and Equation 2 by 2 to eliminate the x term: 10x + 10y = 5(distance covered with the current) 10x - 2y = 60

Subtracting Equation 2 from Equation 1: 12y = 5(distance covered with the current) - 60 12y = 5(distance covered with the current) - 60

Now we have one equation with one variable. We can solve this equation to find the value of y.

Let's solve for y:

12y = 5(distance covered with the current) - 60 y = (5(distance covered with the current) - 60) / 12

Now that we have the value of y, we can substitute it back into Equation 2 to find the value of x.

Let's solve for x:

5y - x = 30 x = 5y - 30

Now we have the values of x and y, which represent the speed of the boat in still water and the speed of the current, respectively.

Answer

The speed of the boat in still water is x km/h, and the speed of the current is y km/h.

Let's calculate the values of x and y using the given information.

Using the equation y = (5(distance covered with the current) - 60) / 12, we can substitute the value of the distance covered with the current (2(x + y)) into the equation to find the value of y.

Let's calculate the value of y:

y = (5(2(x + y)) - 60) / 12

Simplifying the equation: 12y = 10(x + y) - 60 12y = 10x + 10y - 60 2y = 10x - 60 y = (10x - 60) / 2 y = 5x - 30

Now that we have the value of y, we can substitute it back into Equation 2 (5y - x = 30) to find the value of x.

Let's calculate the value of x:

5y - x = 30 5(5x - 30) - x = 30 25x - 150 - x = 30 24x - 150 = 30 24x = 180 x = 180 / 24 x = 7.5

Therefore, the speed of the boat in still water is 7.5 km/h, and the speed of the current is 5(7.5) - 30 = 7.5 km/h.

Conclusion

The speed of the boat in still water is 7.5 km/h, and the speed of the current is 7.5 km/h.