you ride your bike x miles per hour up a mountain for 45 mintues. On the way down the mountain, you ride you bike 4 miles per hour faster. It takes 20 minutes to ride your bike down the mountain. How far did you rider your bike in total?

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.

You ride your bike x miles per hour up a mountain for 45 minutes. On the way down the mountain, you ride your bike 4 miles per hour faster. It takes 20 minutes to ride your bike down the mountain. How far did you ride your bike in total?

Solve the bike riding problem involving speed and distance calculation. Learn how to set up equations for uphill and downhill rides and solve for the total distance traveled. Suitable for Grades 8-10, this problem focuses on the distance-speed-time relationship and algebraic equations.

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