Катер прошёл 10 км против течения и 4 км по течению, затратив на весь путь столько же времени, сколько ему потребовалось бы, если бы он шёл 18 км по озеру. Какова собственная скорость катера, если известно, что скорость течения реки равна 2 км/ч?

Let's denote the boat's speed as "V" km/h.

To solve the problem, we can use the formula: distance = speed × time.

When the boat travels against the current, its effective speed decreases by the speed of the current. So, the boat's speed when going against the current is (V - 2) km/h.

When the boat travels with the current, its effective speed increases by the speed of the current. So, the boat's speed when going with the current is (V + 2) km/h.

Now, let's calculate the time it took the boat to travel each distance:

Time spent going against the current = distance ÷ speed = 10 ÷ (V - 2) hours.

Time spent going with the current = distance ÷ speed = 4 ÷ (V + 2) hours.

Time taken to travel 18 km on the lake = distance ÷ speed = 18 ÷ V hours.

According to the problem, the time taken for all three cases is the same:

10 ÷ (V - 2) = 4 ÷ (V + 2) = 18 ÷ V.

Let's solve the equation:

10(V + 2) = 4(V - 2) = 18V.

Expanding the equation, we get:

10V + 20 = 4V - 8 = 18V.

Simplifying further, we have:

10V - 4V + 18V = -20 + 8.

24V = -12.

Dividing both sides by 24, we get:

V = -12 ÷ 24.

Therefore, the value of "V" is -0.5 km/h.

However, since speed cannot be negative in this context, there seems to be no valid solution to this problem.

Please double-check the provided information or rephrase the question if needed.