a motorboat whose speed is 18km/h in still water takes 1 hour more to go upstream and 24km/h to return downstream at the same spot. find the speed of the stream

Let's denote the speed of the stream as \( x \) km/h.

### Given:
- Speed of the motorboat in still water = 18 km/h.
- Speed of the motorboat downstream = 24 km/h.
- Time difference between upstream and downstream journey = 1 hour.

### Step 1: Speed relationships
- When going upstream, the boat's effective speed is reduced by the speed of the stream, i.e., \( 18 - x \) km/h.
- When going downstream, the boat's effective speed is increased by the speed of the stream, i.e., \( 18 + x = 24 \) km/h.

From the second equation:
\[
18 + x = 24
\]
\[
x = 6 \text{ km/h}.
\]

Thus, the speed of the stream is 6 km/h.

### Step 2: Time relationship
Let the distance traveled (to the upstream point) be \( d \) km.

- Time taken to travel upstream = \( \frac{d}{18 - x} = \frac{d}{18 - 6} = \frac{d}{12} \) hours.
- Time taken to travel downstream = \( \frac{d}{18 + x} = \frac{d}{24} \) hours.

Given that the time difference between upstream and downstream is 1 hour, we set up the equation:
\[
\frac{d}{12} - \frac{d}{24} = 1.
\]

### Step 3: Solve for \( d \)
\[
\frac{2d - d}{24} = 1
\]
\[
\frac{d}{24} = 1
\]
\[
d = 24 \text{ km}.
\]

Thus, the distance to the upstream point is 24 km.

### Final Answer:
The speed of the stream is \( 6 \) km/h.

Would you like further clarification or more details on the solution?

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Here are 5 related questions for practice:
1. If the boat took 2 hours longer to go upstream, what would be the speed of the stream?
2. How much time would the boat take to travel downstream if the speed of the stream was 8 km/h?
3. What is the total travel time (upstream + downstream) for the boat at this stream speed?
4. If the distance traveled upstream was doubled, how would the time taken change?
5. How does the speed of the stream affect the boat's downstream speed?

**Tip:** In motion problems, it's essential to set up speed and time relationships carefully based on the direction of travel and the effect of external factors like currents or winds.

A motorboat whose speed is 18 km/h in still water takes 1 hour more to go upstream and 24 km/h to return downstream at the same spot. Find the speed of the stream.

Solve the problem of finding the speed of a stream when a motorboat's speed in still water is 18 km/h, and its downstream speed is 24 km/h. This solution uses algebra and the time-speed-distance relationship, ideal for students in Grades 8-10.

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