We can solve this problem by first converting the time to hours and then setting up distance equations for both the uphill and downhill rides.

Step 1: Convert time to hours

• You ride up the mountain for 45 minutes, which is: $\frac{45}{60} = 0.75 \, \text{hours}$
• You ride down the mountain for 20 minutes, which is: $\frac{20}{60} = 0.3333 \, \text{hours}$

Step 2: Set up distance equations

Let $x$ be your speed in miles per hour (mph) going uphill. The speed going downhill is 4 mph faster, or $x + 4$ mph.

• The distance going up the mountain is the speed times time: $\text{Distance up} = x \times 0.75$
• The distance going down the mountain is: $\text{Distance down} = (x + 4) \times 0.3333$

Step 3: Equate the distances

Since the distance going up and the distance going down are the same, we set the two expressions equal to each other: $x \times 0.75 = (x + 4) \times 0.3333$

Step 4: Solve for $x$

Expand both sides: $0.75x = 0.3333(x + 4)$ Distribute the $0.3333$ on the right-hand side: $0.75x = 0.3333x + 1.3332$ Subtract $0.3333x$ from both sides: $0.4167x = 1.3332$ Solve for $x$: $x = \frac{1.3332}{0.4167} \approx 3.2 \, \text{mph}$

Step 5: Find the total distance

Now that we know $x = 3.2$ mph, we can calculate the distance up the mountain (which is the same as the distance down): $\text{Distance up} = 3.2 \times 0.75 = 2.4 \, \text{miles}$ $\text{Distance down} = (3.2 + 4) \times 0.3333 = 7.2 \times 0.3333 \approx 2.4 \, \text{miles}$

So, the total distance is: $\text{Total distance} = 2.4 + 2.4 = 4.8 \, \text{miles}$

Final Answer:

You rode a total of 4.8 miles.

Would you like more details, or do you have any questions?

Here are some related questions:

1. How would the total distance change if your downhill speed was 5 mph faster instead?
2. How long would it take if you increased your uphill speed by 1 mph?
3. What if the mountain was 6 miles total—how much faster would you need to ride downhill?
4. How would a 15-minute rest at the top affect your total time spent riding?
5. What is the average speed for the entire trip?

Tip: To calculate average speed for a round trip, use total distance divided by total time.