Let's denote:

• The average speed on the winding road as $x$ miles per hour.
• The average speed on the straight road as $x + 20$ miles per hour.

Given Information:

• Time on the winding road = 3 hours.
• Time on the straight road = 2 hours.
• Total distance of the trip = 210 miles.

Step 1: Express the distances traveled.

• Distance on the winding road = $x \times 3$ miles.
• Distance on the straight road = $(x + 20) \times 2$ miles.

Step 2: Set up the equation for the total distance.

The sum of the distances traveled on both roads is 210 miles: $3x + 2(x + 20) = 210$

Step 3: Simplify the equation.

First, expand the equation: $3x + 2x + 40 = 210$ Combine like terms: $5x + 40 = 210$ Now, subtract 40 from both sides: $5x = 170$ Finally, divide by 5: $x = 34$

Step 4: Calculate the speeds.

• Average speed on the winding road = $x = 34$ mph.
• Average speed on the straight road = $x + 20 = 54$ mph.

Step 5: Verify the total distance.

• Distance on the winding road: $34 \times 3 = 102$ miles.
• Distance on the straight road: $54 \times 2 = 108$ miles.
• Total distance = $102 + 108 = 210$ miles (which matches the total trip distance).

Thus, the average speed on the winding road is 34 mph, and on the straight road is 54 mph.

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

Here are 5 follow-up questions:

1. How would the solution change if the time on the winding road was 4 hours?
2. What would the total trip distance be if the average speed on the winding road was 40 mph?
3. How can we solve this problem if the total time spent on the trip was 5 hours?
4. What if the average speed on the winding road was 10 mph slower than the straight road, how would that affect the answer?
5. How can we set up an equation if the trip had 4 different sections with different speeds and times?

Tip: When solving distance, speed, and time problems, always make sure to define your variables clearly and set up an equation based on the total distance or total time.