Flying with the wind, a plane traveled 390 miles in 3 hours. Flying against the wind, the plane traveled the same distance in 5 hours. Find the rate of the plane in calm air and the rate of the wind. rate of plane ? mph

rate of wind ? mph

If the speed of the plane and the wind are p and w, then since distance = speed * time,

3(s+w) = 390
5(s-w) = 390

s = 104
w = 26

Oh boy, it looks like the plane has some serious wind resistance going on! But don't worry, I've got your back. Let's solve this puzzle.

Let's call the rate of the plane in calm air "p" and the rate of the wind "w". When the plane is flying with the wind, it's like a superhero with a tailwind pushing it forward, making it go faster. So we can set up an equation:

Speed with wind = plane's rate + wind's rate

390 miles in 3 hours = p + w

But when the plane is flying against the wind, it's like facing a villainous headwind that slows it down. So we have:

Speed against wind = plane's rate - wind's rate

390 miles in 5 hours = p - w

Now we have a system of equations, and we can solve for p and w. Let's do this!

From the first equation, we can express p in terms of w:

p = 390 miles in 3 hours - w

Now let's substitute this expression for p in the second equation:

390 miles in 5 hours = (390 miles in 3 hours - w) - w

Simplify it:

390 * 5 = 390 * 3 - 2w

Divide both sides by 2:

2w = 390 * 3 - 390 * 5

2w = 1170 - 1950

2w = -780

Finally, divide both sides by 2 to solve for w:

w = -780 / 2

So the rate of the wind is -390 mph! Wait a minute, negative wind speed? That doesn't sound right. Let's take a step back and question the problem itself because wind doesn't typically have negative speeds. It seems we might have made a mistake somewhere.

I apologize for the confusion, but it seems like there might be an error in the given information or calculations. I'm here to bring a smile to your face, not to cause confusion.

To solve this problem, let's use the formula for distance:

Distance = Rate x Time

We have two equations:

1. Flying with the wind: Distance = 390 miles, Time = 3 hours
2. Flying against the wind: Distance = 390 miles, Time = 5 hours

Let's assign variables:
Let the rate of the plane in calm air be P mph.
Let the rate of the wind be W mph.

1. Flying with the wind equation: 390 = (P + W) x 3
2. Flying against the wind equation: 390 = (P - W) x 5

Simplifying these equations:

1. 390 = 3P + 3W
2. 390 = 5P - 5W

Now we have a system of two equations and two variables. We can solve this system using various methods, such as substitution or elimination. Let's use the elimination method:

Multiply the first equation by 5 and the second equation by 3 to eliminate the W term:

1. 1950 = 15P + 15W
2. 1170 = 15P - 15W

Now, add the two equations together:

1950 + 1170 = 15P + 15W + 15P - 15W

3120 = 30P

Divide both sides by 30:

P = 3120 / 30

P = 104

So, the rate of the plane in calm air is 104 mph.

Now, substitute this value of P in either of the original equations to solve for the rate of the wind.

Let's use the first equation:

390 = (104 + W) x 3

Simplify:

390 = 312 + 3W

Subtract 312 from both sides:

390 - 312 = 3W

78 = 3W

Divide both sides by 3:

W = 78 / 3

W = 26

So, the rate of the wind is 26 mph.

To summarize:
- The rate of the plane in calm air is 104 mph.
- The rate of the wind is 26 mph.

To find the rate of the plane in calm air and the rate of the wind, we can set up a system of equations using the given information.

Let's assume the rate of the plane in calm air is represented by "p" (in mph) and the rate of the wind is represented by "w" (in mph).

When the plane is flying with the wind, the effective speed of the plane is increased by the speed of the wind. So, the speed of the plane flying with the wind can be represented as "p + w" (in mph).

Similarly, when the plane is flying against the wind, the effective speed of the plane is decreased by the speed of the wind. So, the speed of the plane flying against the wind can be represented as "p - w" (in mph).

Using the formula Distance = Speed × Time, we can write the following equations:

For flying with the wind:
(p + w) * 3 = 390

For flying against the wind:
(p - w) * 5 = 390

Let's solve this system of equations to find the values of "p" and "w".

Starting with the first equation, let's simplify it:

3p + 3w = 390 (Dividing both sides by 3)

Next, let's simplify the second equation:

5p - 5w = 390 (Dividing both sides by 5)

Now, we have a system of equations:

3p + 3w = 390
5p - 5w = 390

To solve this system of equations, we can use the method of substitution or elimination. Let's use substitution.

From the first equation, solve for "p":
p = (390 - 3w) / 3

Substitute this value of "p" in the second equation:

5[(390 - 3w) / 3] - 5w = 390

Now, simplify this equation:

(5 * 390 - 15w) / 3 - 5w = 390

Multiply through by 3 to eliminate the fraction:

5 * 390 - 15w - 15w = 3 * 390

Simplify further:

1950 - 15w - 15w = 1170

Combine like terms:

1950 - 30w = 1170

Subtract 1950 from both sides:

-30w = 1170 - 1950

Simplify:

-30w = -780

Divide by -30 to solve for "w":

w = -780 / -30
w = 26

Now that we have the value of "w" which represents the rate of the wind, we can substitute it back into either of the original equations to find the value of "p".

Let's choose the first equation:

3p + 3(26) = 390

Simplify:

3p + 78 = 390

Subtract 78 from both sides:

3p = 390 - 78

Simplify:

3p = 312

Divide by 3 to solve for "p":

p = 312 / 3
p = 104

Therefore, the rate of the plane in calm air is 104 mph and the rate of the wind is 26 mph.