Elena bikes 20 minutes each day for exercise.

Write an equation to describe the relationship between her distance in miles, D, and her biking speed, in miles per hour, when she bikes at a constant speed of M miles per hour for the first 5 minutes, then at N miles per hour for the last 15 minutes

We can break down the problem into two parts based on Elena's biking speeds:

1. **First 5 minutes**: Elena bikes at a speed of \( M \) miles per hour.
2. **Last 15 minutes**: Elena bikes at a speed of \( N \) miles per hour.

### Step 1: Convert time to hours
- **5 minutes** = \( \frac{5}{60} = \frac{1}{12} \) hours.
- **15 minutes** = \( \frac{15}{60} = \frac{1}{4} \) hours.

### Step 2: Write the distance formula
For each time period, the distance Elena covers is given by the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}.
\]

#### Distance in the first 5 minutes:
\[
D_1 = M \times \frac{1}{12}.
\]

#### Distance in the last 15 minutes:
\[
D_2 = N \times \frac{1}{4}.
\]

### Step 3: Total distance
The total distance \( D \) is the sum of the two distances:
\[
D = D_1 + D_2.
\]
Substituting the expressions for \( D_1 \) and \( D_2 \):
\[
D = M \times \frac{1}{12} + N \times \frac{1}{4}.
\]
Simplifying further:
\[
D = \frac{M}{12} + \frac{N}{4}.
\]

This is the equation describing the relationship between Elena's distance \( D \) and her speeds \( M \) and \( N \).

Would you like more details on the solution or have any questions?

Here are 5 related questions:
1. How would the equation change if she biked for 30 minutes instead of 20 minutes?
2. What if her speed for the first 5 minutes was variable over time?
3. Can we express the equation in terms of total time if it's not always 20 minutes?
4. How would you solve for \( M \) or \( N \) given the total distance?
5. What happens to the total distance if \( M = N \)?

**Tip:** Always convert time to hours when working with speed in miles per hour!

Elena bikes 20 minutes each day for exercise. Write an equation to describe the relationship between her distance in miles, D, and her biking speed, in miles per hour, when she bikes at a constant speed of M miles per hour for the first 5 minutes, then at N miles per hour for the last 15 minutes.

Learn how to write an equation to describe the relationship between Elena's distance in miles and her biking speeds over a 20-minute period, with a detailed step-by-step breakdown of speed, time, and distance calculations. Ideal for middle school students.

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