A car travels from Twon A to Twon B at an average speed of 40 miles per hour, and returns immediately along the same route at an average speed of 50 miles per hour. What is the average speed in miles per hour for the round - trip?

Answer

To find the average speed for the round trip from Town A to Town B and back, we can follow these steps:

### Step 1: Define the Distance
Let the distance between Town A and Town B be \( x \) miles.

**Hint:** Define the distance as a variable to simplify calculations.

### Step 2: Calculate Time for the First Leg of the Trip
The car travels from Town A to Town B at an average speed of 40 miles per hour. The time taken for this leg of the trip can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{x}{40} \text{ hours} \]

**Hint:** Remember that time is calculated as distance divided by speed.

### Step 3: Calculate Time for the Return Trip
On the return trip, the car travels back from Town B to Town A at an average speed of 50 miles per hour. The time taken for this leg of the trip is:
\[ \text{Time} = \frac{x}{50} \text{ hours} \]

**Hint:** Use the same formula for time, but substitute the new speed.

### Step 4: Calculate Total Distance and Total Time
The total distance for the round trip is:
\[ \text{Total Distance} = x + x = 2x \text{ miles} \]
The total time taken for the round trip is:
\[ \text{Total Time} = \frac{x}{40} + \frac{x}{50} \]

**Hint:** Add the two time values to get the total time.

### Step 5: Simplify the Total Time
To add the fractions, we need a common denominator. The least common multiple (LCM) of 40 and 50 is 200. We can rewrite the times as:
\[ \frac{x}{40} = \frac{5x}{200} \quad \text{and} \quad \frac{x}{50} = \frac{4x}{200} \]
Thus, the total time becomes:
\[ \text{Total Time} = \frac{5x}{200} + \frac{4x}{200} = \frac{9x}{200} \text{ hours} \]

**Hint:** Finding a common denominator is essential for adding fractions.

### Step 6: Calculate the Average Speed
The average speed for the round trip is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{\frac{9x}{200}} \]
This simplifies to:
\[ \text{Average Speed} = 2x \cdot \frac{200}{9x} = \frac{400}{9} \text{ miles per hour} \]

**Hint:** When dividing by a fraction, multiply by its reciprocal.

### Step 7: Convert to Mixed Number
To express \( \frac{400}{9} \) as a mixed number, divide 400 by 9:
\[ 400 \div 9 = 44 \quad \text{(quotient)} \quad \text{with a remainder of } 4. \]
Thus, we can write:
\[ \frac{400}{9} = 44 \frac{4}{9} \text{ miles per hour} \]

**Hint:** Converting an improper fraction to a mixed number involves division to find the whole number and the remainder.

### Final Answer
The average speed for the round trip is:
\[ \boxed{44 \frac{4}{9}} \text{ miles per hour} \]