Driving to your friend's house, you travel at an average rate of 35 miles per hour. On your way home, you travel at an average rate of 40 miles per hour. If the round trip took you 45 minutes how far is it from your house to your friend’s house?

Let the distance between your house and your friend’s house be \( d \) miles.

### Step 1: Set up the time formulas
- On your way to your friend's house, you travel at an average rate of 35 miles per hour, so the time it takes to get there is:
  \[
  t_1 = \frac{d}{35}
  \]
- On your way back, you travel at an average rate of 40 miles per hour, so the time it takes to return is:
  \[
  t_2 = \frac{d}{40}
  \]

### Step 2: Total time of the round trip
The total time of the round trip is 45 minutes, which is equivalent to:
\[
45 \, \text{minutes} = \frac{45}{60} \, \text{hours} = 0.75 \, \text{hours}
\]

Thus, the sum of the time to go to your friend's house and the time to return is:
\[
t_1 + t_2 = 0.75
\]

### Step 3: Write the equation
Substitute the expressions for \( t_1 \) and \( t_2 \) into the equation:
\[
\frac{d}{35} + \frac{d}{40} = 0.75
\]

### Step 4: Solve the equation
First, find a common denominator for the fractions. The least common denominator of 35 and 40 is 280, so rewrite the fractions:
\[
\frac{d}{35} = \frac{8d}{280}, \quad \frac{d}{40} = \frac{7d}{280}
\]
Now the equation becomes:
\[
\frac{8d}{280} + \frac{7d}{280} = 0.75
\]
Combine the fractions:
\[
\frac{15d}{280} = 0.75
\]

Multiply both sides by 280:
\[
15d = 0.75 \times 280
\]
\[
15d = 210
\]

### Step 5: Solve for \( d \)
Now, divide both sides by 15:
\[
d = \frac{210}{15} = 14
\]

Thus, the distance between your house and your friend's house is **14 miles**.

Would you like more details on any step?

Here are 5 questions to deepen your understanding:
1. How would the solution change if the speeds were different?
2. What if the total round-trip time was 1 hour instead of 45 minutes?
3. How can you solve this problem if only the average speed for the entire round trip is given?
4. How can we interpret the use of common denominators when dealing with fractions in this context?
5. What are other methods to solve a system of equations involving rates, times, and distances?

**Tip:** Always convert time units consistently (e.g., minutes to hours) to avoid calculation errors.

Learn how to calculate the distance between two locations based on different average speeds for each leg of a round trip. This example uses a rate of 35 mph on the way there and 40 mph on the way back, with a total round-trip time of 45 minutes. Suitable for students in Grades 8-10.

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