A man cycles to his destination with the wind at 8m/s and returns home against the wind at 4m/s what is his average speed

To find the average speed of the man, we need to consider the total distance traveled and the total time taken.

Let's assume the distance to the destination is D.

When cycling with the wind, the effective speed of the man is the sum of his cycling speed and the wind speed. So, his effective speed with the wind is 8m/s + 4m/s = 12m/s.

The time taken to reach the destination with the wind can be calculated by dividing the distance by the effective speed:
Time = D / 12m/s

When cycling against the wind, the effective speed of the man is the difference between his cycling speed and the wind speed. So, his effective speed against the wind is 8m/s - 4m/s = 4m/s.

The time taken to return home against the wind can be calculated by dividing the distance by the effective speed:
Time = D / 4m/s

The total time taken for the entire journey (both to the destination and the return) can be found by summing the two individual times:
Total Time = (D / 12m/s) + (D / 4m/s)

To find the average speed, we divide the total distance (2D, as he covers the distance to the destination and back) by the total time:
Average Speed = 2D / [(D / 12m/s) + (D / 4m/s)]

Simplifying this expression, we get:
Average Speed = 2 / [(1 / 12m/s) + (1 / 4m/s)]

Now, let's calculate the numerical value of the average speed:

Average Speed = 2 / [(1 / 12) + (1 / 4)]
= 2 / [(1/12) + (3/12)]
= 2 / (4/12)
= 2 / (1/3)
= 2 * 3
= 6 m/s

So, the average speed of the man for the entire journey is 6 m/s.

+ 8 going with wind

- 4 against the wind

-4+8/2 = 2 is the average speed