A man flies between two cities; on a windless day, he averages 120 mph round trip; on a windy day he averaged 140 mph with the wind; he averaged 100 mph against the wind; on the windy day, it took 15 extra minutes to make the round trip; what is the distance between the two cities?

let the distance be d miles

time against the wind = d/100
time with the wind = d/140

d/100 - d/140 = 1/4
times 700
7d - 5d = 175
d = 87.5

the two cities are 87.5 miles apart

check
time at slower speed = 87.5/100 = .875 hrs
times at faster speed = 87.5/140 = .625 hrs
difference = .875 - .625 = .25 hrs or 15 minutes

To find the distance between the two cities, we can use the equation:

Distance = Speed × Time

Let's assume the distance between the two cities is 'D' miles.

On a windless day, he averages 120 mph round trip. This means that his average speed going from one city to the other is 120 mph, and his average speed coming back from that city is also 120 mph.

Therefore, the time taken to travel from one city to the other is:
Time = Distance / Speed = D / 120

Similarly, the time taken to travel back from the other city is also D / 120.

On a windy day, he averaged 140 mph with the wind. This means that his speed going from one city to the other is 140 mph, whereas his speed coming back is reduced by the wind resistance.

On the windy day, it took him 15 extra minutes to make the round trip compared to the windless day. This extra time is due to the slower speed when traveling against the wind.

The time taken to travel from one city to the other with the wind is:
Time = Distance / Speed = D / 140

The time taken to travel back from the other city against the wind is:
Time = Distance / Speed = D / 100

Since the total time for the round trip is 15 minutes longer on the windy day compared to the windless day, we can set up the equation:
(D / 140) + (D / 100) = (D / 120) + (D / 120) + 15/60

First, let's consolidate and simplify the equation:
(7D + 5D) / 700 = (6D + 6D)/720 + 15/60

12D / 700 = 12D / 720 + 1/4

Now, let's solve for D by cross-multiplying:
4(12D) = 700(12D / 720 + 1/4)

48D = 700(12D / 720 + 1/4)

48D = 700(12D + 180) / 720

48D = 700(12D + 180) / 720

48D = 700(12D + 180) / 720

48D = 8400D + 1260000 / 720

48D - 8400D = 1260000 / 720

-8352D = 1260000 / 720

D = (1260000 / 720) / -8352

D ≈ 22.47 miles

Therefore, the distance between the two cities is approximately 22.47 miles.