A car travels up a hill at a constant speed of 35 km/h and returns down the hill at a constant speed of 60 km/h. Calculate the average speed for the round trip.

Answer

To solve the problem of calculating the average speed for a car that travels up a hill at a speed of 35 km/h and returns down the hill at a speed of 60 km/h, we can follow these steps:

### Step 1: Understand the Concept of Average Speed
The average speed for a round trip can be defined as the total distance traveled divided by the total time taken.

### Step 2: Define the Distances and Speeds
Let’s assume the distance to the top of the hill is \( x \) km. The car travels this distance twice (once up and once down).

- **Distance Up the Hill**: \( x \) km
- **Distance Down the Hill**: \( x \) km
- **Total Distance**: \( 2x \) km

### Step 3: Calculate the Time Taken for Each Leg of the Trip
The time taken to travel up the hill (time \( t_1 \)) and down the hill (time \( t_2 \)) can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

- **Time Up the Hill**:
\[ t_1 = \frac{x}{35} \text{ hours} \]
- **Time Down the Hill**:
\[ t_2 = \frac{x}{60} \text{ hours} \]

### Step 4: Calculate Total Time Taken
The total time taken for the round trip is:
\[ \text{Total Time} = t_1 + t_2 = \frac{x}{35} + \frac{x}{60} \]

To add these fractions, we need a common denominator. The least common multiple of 35 and 60 is 420.

- Convert \( t_1 \) and \( t_2 \):
\[ t_1 = \frac{x}{35} = \frac{12x}{420} \]
\[ t_2 = \frac{x}{60} = \frac{7x}{420} \]

Now, add them:
\[ \text{Total Time} = \frac{12x}{420} + \frac{7x}{420} = \frac{19x}{420} \text{ hours} \]

### Step 5: Calculate Average Speed
Now, we can calculate the average speed using the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{\frac{19x}{420}} \]

This simplifies to:
\[ \text{Average Speed} = \frac{2x \cdot 420}{19x} = \frac{840}{19} \text{ km/h} \]

### Step 6: Calculate the Numerical Value
Now, we can compute the numerical value:
\[ \text{Average Speed} \approx 44.21 \text{ km/h} \]

### Final Answer
The average speed for the round trip is approximately **44.21 km/h**.

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