Lauren rode her bike from her house to a friend's house `3 1/2` miles away. On the first leg of her trip , she rode uphill at 3 miles per hour. The second part of the trip covered a larger distance but was downhill, and Lauren rode at 5 miles per hour. If the downhill part of the ride took half an hour, how many minutes did the uphill part take ?

Answer

To solve the problem step by step, we will break down the information provided and use it to find the time Lauren took for the uphill part of her trip.

### Step 1: Define Variables
Let \( x \) be the distance Lauren rode uphill in miles. Since the total distance to her friend's house is \( 3.5 \) miles, the distance she rode downhill would be \( 3.5 - x \) miles.

**Hint:** Define variables for unknowns to simplify calculations.

### Step 2: Use Given Information
Lauren rode uphill at a speed of \( 3 \) miles per hour and downhill at \( 5 \) miles per hour. We also know that the downhill part of the ride took half an hour.

**Hint:** Write down the speeds and times for both parts of the journey.

### Step 3: Set Up the Equation for Downhill Distance
The time taken for the downhill part can be expressed using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For the downhill part:
\[ \text{Time} = \frac{3.5 - x}{5} \]
Since the downhill part took half an hour, we can set up the equation:
\[ \frac{3.5 - x}{5} = \frac{1}{2} \]

**Hint:** Use the formula for time to relate distance and speed.

### Step 4: Solve for \( x \)
Multiply both sides of the equation by \( 5 \) to eliminate the fraction:
\[ 3.5 - x = \frac{5}{2} \]
Now convert \( 3.5 \) to a fraction:
\[ 3.5 = \frac{7}{2} \]
So the equation becomes:
\[ \frac{7}{2} - x = \frac{5}{2} \]
Now, isolate \( x \):
\[ x = \frac{7}{2} - \frac{5}{2} = \frac{2}{2} = 1 \]
Thus, the distance Lauren rode uphill is \( 1 \) mile.

**Hint:** Rearranging equations can help isolate the variable you need.

### Step 5: Calculate Time for Uphill Part
Now that we know the uphill distance is \( 1 \) mile, we can find the time taken for the uphill part using the speed:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1}{3} \text{ hours} \]

**Hint:** Use the same formula for time to find how long it took for the uphill ride.

### Step 6: Convert Time to Minutes
Since the time we calculated is in hours, we need to convert it to minutes:
\[ \text{Time in minutes} = \frac{1}{3} \times 60 = 20 \text{ minutes} \]

**Hint:** Remember that there are \( 60 \) minutes in an hour when converting.

### Conclusion
The time Lauren took for the uphill part of her journey is \( 20 \) minutes.

**Final Answer:** 20 minutes