To solve this problem, let's break it down step by step.

Key Information:

• The car travels at 60 km/h for every 90 km it covers at 80 km/h.
• The total distance traveled is 600 km.

Let:

• $x$ be the distance traveled at 60 km/h.
• $y$ be the distance traveled at 80 km/h.

From the problem, we know that:

1. The total distance is 600 km, so: $x + y = 600$
2. For every 90 km traveled at 80 km/h, the car travels 60 km at 60 km/h. Therefore, the ratio of distances at each speed is: $\frac{x}{60} = \frac{y}{90}$ Simplifying this gives: $9x = 6y \quad \text{or} \quad 3x = 2y$ which we can rearrange to: $x = \frac{2}{3}y$

Step 1: Solve the System of Equations

We now have the system:

1. $x + y = 600$
2. $x = \frac{2}{3}y$

Substitute $x = \frac{2}{3}y$ into the first equation: $\frac{2}{3}y + y = 600$ Combine the terms: $\frac{5}{3}y = 600$ Multiply both sides by 3 to eliminate the fraction: $5y = 1800$ Solve for $y$: $y = \frac{1800}{5} = 360$

Step 2: Find $x$

Now, substitute $y = 360$ into the equation $x = \frac{2}{3}y$: $x = \frac{2}{3} \times 360 = 240$

Final Answer:

• The car traveled 240 km at 60 km/h.
• The car traveled 360 km at 80 km/h.

Would you like a detailed breakdown of the steps, or is there another question related to this problem you'd like to explore?

Follow-up Questions:

1. How can we verify that the solution is correct by checking the total distance?
2. How do we calculate the total time spent at each speed?
3. If the car travels a total of 800 km, how would the distances at each speed change?
4. What other real-life problems can be modeled using the ratio of distances and speeds like this?
5. How would the time taken at each speed affect the total travel time?

Tip: When dealing with problems involving ratios or proportionality, it helps to set up the relationship first and express all variables in terms of one unknown.