A man motors from A to B. In motoring a distance uphill, he gets a mileage of only 10 miles per gallon of gasoline. On the return trip, he markes 15 miles per gallon. Find the harmonic mean of his mileage. Verify that this is the proper average to be used here, assuming that the distance from A to B is 60 miles.

Answer

To find the harmonic mean of the man's mileage from A to B and back, we can follow these steps:

### Step 1: Identify the mileages
The man has two different mileages:
- Uphill (from A to B): 10 miles per gallon
- Downhill (from B to A): 15 miles per gallon

### Step 2: Use the formula for harmonic mean
The formula for the harmonic mean (HM) of two values \(x_1\) and \(x_2\) is given by:

\[ HM = \frac{2 \cdot x_1 \cdot x_2}{x_1 + x_2} \]

### Step 3: Substitute the values into the formula
Substituting \(x_1 = 10\) and \(x_2 = 15\):

\[ HM = \frac{2 \cdot 10 \cdot 15}{10 + 15} \]

### Step 4: Calculate the numerator and denominator
Calculating the numerator:

\[ 2 \cdot 10 \cdot 15 = 300 \]

Calculating the denominator:

\[ 10 + 15 = 25 \]

### Step 5: Divide the numerator by the denominator
Now, we can calculate the harmonic mean:

\[ HM = \frac{300}{25} = 12 \]

### Conclusion
The harmonic mean of the man's mileage is 12 miles per gallon.

### Verification
To verify that the harmonic mean is the proper average to use in this scenario, we can consider the total distance traveled and the total gasoline consumed.

1. **Distance from A to B**: 60 miles (uphill)
2. **Distance from B to A**: 60 miles (downhill)
3. **Total distance**: \(60 + 60 = 120\) miles

4. **Gasoline consumed uphill**:
- Mileage = 10 miles per gallon
- Gasoline used = \(\frac{60 \text{ miles}}{10 \text{ miles/gallon}} = 6\) gallons

5. **Gasoline consumed downhill**:
- Mileage = 15 miles per gallon
- Gasoline used = \(\frac{60 \text{ miles}}{15 \text{ miles/gallon}} = 4\) gallons

6. **Total gasoline consumed**: \(6 + 4 = 10\) gallons

7. **Average mileage**:
- Total distance / Total gasoline = \(\frac{120 \text{ miles}}{10 \text{ gallons}} = 12\) miles per gallon

This confirms that the harmonic mean of 12 miles per gallon is indeed the correct average to use in this context.