Suppose that a family has two card: an SUV and a hybrid. The SUV gets 20 miles per gallon, and the hybrid gets 48 miles per gallon. Each gets driven 10,000 miles/year. What is the family’s overall average fuel efficiency?  How does this compare to the simple average of values 20 and 48? Explain why simple average is not correct in this case.

To calculate the family’s overall average fuel efficiency, we need to use the concept of the **harmonic mean** rather than the simple arithmetic mean because the problem involves averaging rates (miles per gallon).

### Step 1: Calculate the total fuel consumption for each vehicle.
- **SUV:**  
  \[
  \text{Fuel consumption by SUV} = \frac{10,000 \text{ miles}}{20 \text{ miles per gallon}} = 500 \text{ gallons}
  \]

- **Hybrid:**  
  \[
  \text{Fuel consumption by Hybrid} = \frac{10,000 \text{ miles}}{48 \text{ miles per gallon}} = \approx 208.33 \text{ gallons}
  \]

### Step 2: Calculate the total fuel consumption for both vehicles combined.
\[
\text{Total fuel consumption} = 500 \text{ gallons} + 208.33 \text{ gallons} = 708.33 \text{ gallons}
\]

### Step 3: Calculate the total distance driven by both vehicles.
\[
\text{Total distance} = 10,000 \text{ miles} + 10,000 \text{ miles} = 20,000 \text{ miles}
\]

### Step 4: Calculate the overall average fuel efficiency.
\[
\text{Overall average fuel efficiency} = \frac{\text{Total distance driven}}{\text{Total fuel consumption}} = \frac{20,000 \text{ miles}}{708.33 \text{ gallons}} \approx 28.22 \text{ miles per gallon}
\]

### Comparison with the Simple Average
The simple average of the fuel efficiencies (20 mpg and 48 mpg) is:
\[
\text{Simple average} = \frac{20 + 48}{2} = 34 \text{ miles per gallon}
\]

### Explanation: Why is the Simple Average Not Correct?
The simple average is not appropriate in this context because it does not account for the fact that fuel efficiency is a rate, and the total fuel consumption depends on both the distance driven and the efficiency of the vehicle. When combining rates like fuel efficiency, the harmonic mean gives a more accurate measure because it appropriately weights the contributions of each rate based on the distance driven. The simple average assumes equal weighting without considering the actual fuel used, which can lead to an overestimation of the efficiency.

### Summary:
- **Overall average fuel efficiency:** Approximately 28.22 miles per gallon.
- **Simple average:** 34 miles per gallon.
- **Why simple average is incorrect:** It does not consider the actual fuel consumed, leading to an inaccurate representation of the average efficiency.

Would you like more details on any part of this explanation, or do you have any other questions?

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**Related Questions:**
1. How do you calculate the harmonic mean for different rates?
2. What is the difference between arithmetic and harmonic means?
3. Can you provide more examples where the harmonic mean is used?
4. How would the overall average efficiency change if the vehicles drove different distances?
5. How does this concept apply to averaging speeds?

**Tip:** Always use the harmonic mean when dealing with rates and ratios, such as speed, fuel efficiency, or any other scenario where the quantities are inversely related.

Learn how to calculate the family’s overall average fuel efficiency when comparing an SUV (20 miles per gallon) and a hybrid (48 miles per gallon) driven 10,000 miles/year each. Understand why the simple average is not appropriate in this case and the use of harmonic mean in averaging rates. Approximate values: Overall average fuel efficiency: 28.22 miles per gallon; Simple average: 34 miles per gallon.

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