A car travels from point A to point B at a speed of 60 mph and returns to point A at a speed of 80 mph. What is the average speed for the round trip?

To find the average speed for the round trip, we need to consider the total distance traveled and the total time taken for the trip. Here are the steps to solve the problem:

1. Define the variables:

   • Let $d$ be the distance from point A to point B.
   • The speed from A to B is 60 mph.
   • The speed from B to A is 80 mph.

2. Calculate the time taken for each leg of the trip:

   • Time taken to travel from A to B: $t_1 = \frac{d}{60}$ hours.
   • Time taken to travel from B to A: $t_2 = \frac{d}{80}$ hours.

3. Calculate the total distance traveled:

   • The total distance for the round trip is $2d$.

4. Calculate the total time taken for the round trip:

   • The total time is $t_1 + t_2 = \frac{d}{60} + \frac{d}{80}$.

5. Find a common denominator to add the times:

   • The common denominator of 60 and 80 is 240.
   • Convert the times to have the same denominator: $\frac{d}{60} = \frac{4d}{240}$ $\frac{d}{80} = \frac{3d}{240}$
   • Add the times: $t_1 + t_2 = \frac{4d}{240} + \frac{3d}{240} = \frac{7d}{240}$

6. Calculate the average speed:

   • The average speed is the total distance divided by the total time: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{2d}{\frac{7d}{240}}$
   • Simplify the expression: $\text{Average speed} = 2d \times \frac{240}{7d} = \frac{480d}{7d} = \frac{480}{7} \approx 68.57 \text{ mph}$

So, the average speed for the round trip is approximately 68.57 mph.