Brian hiked up the hill in 3 hours and returned down hill another 2 hours. His speed downhill was 1mph faster than his speed up hill. How far up hill did he hike?

Let's define:

- \( d \) as the distance Brian hiked uphill (in miles),
- \( v \) as his speed uphill (in mph).

From the problem, we know the following:

1. **Uphill**: He took 3 hours to hike uphill. The distance-speed-time relation is:
   \[
   \text{Distance} = \text{Speed} \times \text{Time}, \quad d = v \times 3
   \]
   So, \( d = 3v \).

2. **Downhill**: His speed downhill is 1 mph faster than his speed uphill, i.e., \( v + 1 \). He took 2 hours to return downhill, so:
   \[
   d = (v + 1) \times 2
   \]

Now, since both equations represent the same distance \( d \), we can set them equal:
\[
3v = 2(v + 1)
\]

### Solving for \( v \):

1. Expand the right side:
   \[
   3v = 2v + 2
   \]
2. Subtract \( 2v \) from both sides:
   \[
   v = 2
   \]

So, his speed uphill was \( v = 2 \) mph.

### Finding the distance \( d \):

Now that we know his uphill speed, we can find the distance using \( d = 3v \):
\[
d = 3 \times 2 = 6 \text{ miles}.
\]

Therefore, Brian hiked **6 miles** uphill.

Would you like further details or have any questions?

Here are some related questions for you to consider:

1. What would the distance be if Brian's downhill speed was 2 mph faster instead of 1 mph?
2. How long would it take if Brian hiked both uphill and downhill at the same speed of 2 mph?
3. What is the average speed for the entire trip (uphill and downhill)?
4. How would the solution change if Brian took 4 hours uphill and 3 hours downhill?
5. If the hill was 10 miles, how long would it take Brian to hike uphill if he maintained the same speed?

**Tip**: In distance-speed-time problems, equations involving both time and speed are often key to solving for unknowns.

Brian hiked up the hill in 3 hours and returned downhill another 2 hours. His speed downhill was 1 mph faster than his speed uphill. How far uphill did he hike?

Discover how to calculate the distance Brian hiked uphill based on his hiking time and varying speeds. This problem focuses on algebraic equations related to distance, speed, and time. Suitable for students in Grades 6-8.

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