To solve for the value of $x$, we can use the concept of average speed, which is defined as the total distance traveled divided by the total time taken.

Step 1: Calculate the total distance traveled

1. Distance traveled in the first part of the journey: $\text{Distance}_1 = \text{Speed}_1 \times \text{Time}_1 = 60 \, \text{km/h} \times 2 \, \text{h} = 120 \, \text{km}$

2. Distance traveled in the second part of the journey: $\text{Distance}_2 = \text{Speed}_2 \times \text{Time}_2 = x \, \text{km/h} \times 3 \, \text{h} = 3x \, \text{km}$

3. Total distance: $\text{Total Distance} = \text{Distance}_1 + \text{Distance}_2 = 120 \, \text{km} + 3x \, \text{km}$

Step 2: Calculate the total time taken

The total time is: $\text{Total Time} = 2 \, \text{h} + 3 \, \text{h} = 5 \, \text{h}$

Step 3: Use the average speed formula

The average speed for the whole journey is given by: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$ Substituting the values we have: $64.2 = \frac{120 + 3x}{5}$

Step 4: Solve for $x$

Multiply both sides of the equation by 5: $321 = 120 + 3x$ Subtract 120 from both sides: $201 = 3x$ Finally, divide by 3: $x = \frac{201}{3} = 67 \, \text{km/h}$

So, the value of $x$ is 67 km/h.

Would you like a more detailed explanation or have any questions?

Here are some related problems you might find interesting:

1. What would be the average speed if the second part of the journey was 4 hours instead of 3?
2. How would the value of $x$ change if the car traveled 70 km/h in the first part?
3. If the total time was reduced to 4 hours, what would be the new average speed?
4. How would the value of $x$ change if the total average speed was 66 km/h?
5. What is the total distance traveled if $x$ is 75 km/h?

Tip: Always double-check your units when solving speed, distance, and time problems to ensure accuracy.