Its 80 kilometers from point A to point B. Someone travels at an average speed of 40 kilometers per day and takes a negligible amount of time at point B to turn around to head back to point A. At what average speed must the person travel back to point A so that for the entire round trip journey, their average speed is 80 kilometers per day.

To determine the average speed for the round trip journey, we can use the formula:

Average Speed = Total Distance / Total Time

We know that the total distance is 80 kilometers (from point A to point B and back to point A), and the average speed for the first leg of the journey is 40 kilometers per day. Let's denote the average speed for the return leg as V.

First, we need to find the total time for the journey. Since the person travels at an average speed of 40 kilometers per day for the first leg (point A to point B), the time taken for this leg is given by:

Time for first leg = Distance / Speed = 80 kilometers / 40 kilometers per day = 2 days

Since the person spends a negligible amount of time at point B to turn around, we can assume it takes zero days. Therefore, the total time for the journey is:

Total Time = Time for first leg + Time at point B + Time for return leg
= 2 days + 0 days + Time for return leg
= 2 days + Time for return leg

Now, let's calculate the total time:

Total Time = 2 days + Time for return leg

We want the average speed for the entire round trip journey to be 80 kilometers per day. Therefore, the total time should be:

Total Time = Total Distance / Average Speed = 80 kilometers / 80 kilometers per day = 1 day

Now, we can equate the expressions for the total time:

2 days + Time for return leg = 1 day

By rearranging the equation, we can solve for the time for the return leg:

Time for return leg = 1 day - 2 days = -1 day

It is not possible to have a negative time, so we need to consider the absolute value of the time. Therefore, the time for the return leg should be 1 day.

Now that we know the time for the return leg is 1 day, we can calculate the average speed for the return leg:

Average Speed = Distance / Time
= 80 kilometers / 1 day
= 80 kilometers per day

Therefore, the person must travel at an average speed of 80 kilometers per day to maintain an overall average speed of 80 kilometers per day for the entire round trip journey.