Let's denote the speed of the in-line skater as $x$ kilometers per hour.

Since the cyclist traveled 10 kilometers per hour faster than the skater, the cyclist's speed is $x + 10$ kilometers per hour.

Now, we can set up an equation using the relationship between distance, speed, and time:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

The time taken by the cyclist to travel 40 kilometers is:

$\frac{40}{x + 10}$

The time taken by the skater to travel 20 kilometers is:

$\frac{20}{x}$

Since both the cyclist and the skater took the same amount of time, we can equate the two expressions:

$\frac{40}{x + 10} = \frac{20}{x}$

Solving the equation:

1. Cross-multiply to eliminate the denominators:

$40x = 20(x + 10)$

2. Distribute the 20 on the right-hand side:

$40x = 20x + 200$

3. Subtract $20x$ from both sides:

$20x = 200$

4. Divide both sides by 20:

$x = 10$

Thus, the speed of the skater is 10 kilometers per hour.

Summary:

• The skater's speed is 10 km/h.
• The cyclist's speed is 10 km/h + 10 km/h = 20 km/h.

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. If the cyclist traveled 60 kilometers instead of 40, how would the skater's speed change?
2. How long did it take both the skater and cyclist to complete their respective distances?
3. What if the skater's speed was increased by 5 km/h, how would the new cyclist's speed compare?
4. How would the equation change if the cyclist's speed was 15 km/h faster than the skater's?
5. What would happen if the skater traveled 30 kilometers instead of 20 at the same speed?

Tip: When two objects travel different distances in the same time, equating the times using the formula $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$ is a powerful method to solve for unknown speeds.