A plane has a cruising speed of 100 miles per hour when there is no wind. At this speed, the plane flew 600 miles with the wind in the same amount of time it flew 400 miles against the wind. Find the speed of the wind.

since time = distance/speed, if the wind has speed w,

600/(100+w) = 400/(100-w)

Now just find w.

To find the speed of the wind, let's assume the speed of the wind is represented by 'w' (in miles per hour).

We know that the plane has a cruising speed of 100 miles per hour in still air.

When flying with the wind, the effective speed of the plane is increased by the speed of the wind. So the actual speed of the plane is (100 + w) miles per hour.

When flying against the wind, the effective speed of the plane is decreased by the speed of the wind. So the actual speed of the plane is (100 - w) miles per hour.

We are given that the plane flew 600 miles with the wind in the same amount of time it flew 400 miles against the wind. Let's use the formula:

Time = Distance / Speed

For the first scenario (with the wind), we have:
Time with wind = 600 miles / (100 + w) miles per hour

For the second scenario (against the wind), we have:
Time against wind = 400 miles / (100 - w) miles per hour

Since the times are the same, we can set up an equation:
Time with wind = Time against wind

600 / (100 + w) = 400 / (100 - w)

To get rid of the fractions, we can cross-multiply:
600(100 - w) = 400(100 + w)

Now, let's simplify the equation:
60000 - 600w = 40000 + 400w

Combine like terms:
-600w - 400w = 40000 - 60000
-1000w = -20000

Divide both sides of the equation by -1000:
w = -20000 / -1000
w = 20

Therefore, the speed of the wind is 20 miles per hour.

To find the speed of the wind, let's assume that the speed of the wind is "w" miles per hour.

When the plane flies with the wind, its effective speed is the sum of its cruising speed and the speed of the wind. Therefore, the effective speed is 100 mph + w mph.

When the plane flies against the wind, its effective speed is the difference between its cruising speed and the speed of the wind. Therefore, the effective speed is 100 mph - w mph.

Now, we have the following information:

1) When flying with the wind, the plane traveled 600 miles in the same amount of time it took to fly 400 miles against the wind.
2) Time = Distance / Speed

Using equation (2), we can set up the following equation for when the plane is flying with the wind:

Time with wind = 600 miles / (100 mph + w mph)

Using equation (2) again, we can set up the following equation for when the plane is flying against the wind:

Time against wind = 400 miles / (100 mph - w mph)

Since the plane takes the same amount of time for both scenarios:

Time with wind = Time against wind

Therefore, we can set up the equation:

600 miles / (100 mph + w mph) = 400 miles / (100 mph - w mph)

Multiplying both sides of the equation by (100 mph + w mph) and (100 mph - w mph), we get:

600 miles * (100 mph - w mph) = 400 miles * (100 mph + w mph)

Now, let's solve for w.

60000 mph - 600w miles = 40000 mph + 400w miles

Combining like terms:

1000w miles = 20000 mph

Dividing both sides by 1000:

w = 20 mph

Therefore, the speed of the wind is 20 miles per hour.

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