We want to determine the length of a boat moving parallel to the shore from the shore. If we walk in the same direction as the boat, we find its length to be 200 steps. However, if we walk in the opposite direction, we find that the entire length of the boat passes by us in just 40 steps. How many steps is the length of the boat?
Let the length of the boat be $x$ steps, and its speed be $k$ times our own speed. We need to distinguish between two cases, depending on whether $k>1$ or $k<1$. 

If $k>1$, that is, the boat's speed is greater than ours. In this case, when moving in the direction of the boat, it will catch up to us, and thus we start our calculation from the bow of the boat. While we take 200 steps, the boat travels $200k$ steps. Since it just passed us during this time, the difference in our distances is the length of the boat, that is,

$$
200k - 200 = x
$$

While we take 40 steps in the direction opposite to the boat's course, the boat travels $40k$ steps. Since it just passed us during this time, the sum of our distances is the length of the boat, so

$$
40k + 40 = x
$$

From (1) and (2),

$$
200k - 200 = 40k + 40
$$

from which

$$
k = \frac{3}{2}, \quad \text{and thus} \quad x = 100 \text{ steps.}
$$

b) If $k<1$, that is, we move faster than the boat. In this case, when moving in the direction of the boat, we will catch up to the boat, so we start our calculation from the stern of the boat.

In this case, the equation is as follows

$$
200 - 200k = x
$$

Equation (2) remains unchanged. From (2) and (3),

$$
40k + 40 = 200 - 200k
$$

from which

$$
k = \frac{2}{3}, \quad \text{and thus} \quad x = \frac{200}{3} = 66 \frac{2}{3} \text{ steps.}
$$

Imre Szlanka (Aszód, Petőfi g. II. o. t.)