A cyclist covers his outward journey downhill at 24mph. His return journey, over exactly the same distance uphill, is covered at only 12mph. What is the cyclist's average speed over the entire journey?

If the distance each way is x, then since speed = distance/time,

2x/(x/24 + x/12) = 2x/(3x/24) = 16

To find the cyclist's average speed over the entire journey, we need to calculate the total distance traveled and divide it by the total time taken.

Let's assume the distance of the outward and return journeys is 'd' miles.

The time taken for the outward journey can be calculated using the formula:

Time = Distance / Speed

For the outward journey, the speed is 24 mph, so the time taken is:

Time(outward) = d / 24

Similarly, for the return journey, the time taken can be calculated as:

Time(return) = d / 12

Now, let's find the total time for the entire journey by adding the time for the outward and return journeys:

Total Time = Time(outward) + Time(return)
= d / 24 + d / 12

To calculate the total distance, we add the distances of the outward and return journeys:

Total Distance = Distance(outward) + Distance(return)
= d + d
= 2d

Now, to find the average speed, we divide the total distance by the total time:

Average Speed = Total Distance / Total Time
= 2d / (d / 24 + d / 12)

Simplifying this expression:

Average Speed = 2 / ((1 / 24) + (1 / 12))

To evaluate this expression, we can find the common denominator, then combine the fractions:

Average Speed = 2 / ((1/24) + (2/24))
= 2 / (3/24)
= 2 / (1/8)
= 16 mph

Therefore, the cyclist's average speed over the entire journey is 16 mph.