a plane can travel 192 mi with the wind and then 180 mi back against the wind in 6hr. if the wind blows at 15 mph, what is the speed of the plane

Question

substitute and solve 

Answer 1

I think the exact intention of this problem is that the full trip is 192 miles + 180 miles (for a total of 192 + 180, or 372 miles,) and that this whole trip takes 6 hours.

The key to this is to realize that you don't know how long the trip "with wind" takes, and we don't know either how long the trip "against wind" takes, but just that the trip altogether takes 6 hours.

This being said, our variables will be:
v = speed of the plane
t = time spent traveling with the wind

Now, we use the formula speed = distance / time (which can be expressed as speed * time = distance) for each one of the two legs of the trip.

When traveling with the wind, the ground speed of the plane will be v + 15, and when traveling against the wind, the ground speed of the plane will be v - 15, so:

a. With the wind:
(v+15) * t = 192
v*t + 15t = 192
v*t = 192 - 15t
v = (192 - 15t)/t [1]

b. Against the wind:
(v-15) * (6-t) = 180
6v -v*t - 90 + 15t = 180 [2]

Now, we can substitute equation [1] in [2], to obtain:

6[(192 - 15t)/t] - [(192 - 15t)/t] * t - 90 + 15t = 180 [3]

Here are the next steps you need to perform (I won't include them explicitly here, but you can solve the problem following these instructions):

i. Multiply both sides of equation [3] by t.
ii. Then, use basic algebra to transform into a quadratic equation in t of the form a*t^2 + b*t + c = 0
iii. Solve this quadratic equation for t. (Notice one of the solutions is extraneous, meaning that it is not really a solution. Basically, only keep the solution of t which is less than 6 hours, since we know the full trip took 6 hours, so t must be less than 6 hours.)
iv. Now, the only thing you need to do is substitute back this value of t into equation [1] above (v = (192 - 15t)/t), in order to get your solution for v.

I solved this completely, and checked my answer by plugging in the values I obtained for t and v into the problem statement.

I hope this is useful to you, and let me know if you have any other questions.