Question 1033225
rate * time = distance.


rate is composed of the speed of the plane plus or minus the speed of the wind.


if we let p = the speed of the plane and w = the speed of the wind and t = the time and d = the distance, you get:


when the plane is flying against the wind, you get (p-w)*t = d.
when the plane is flying with the wind, you get (p+w)*t = d.


when the plane is flying against the wind, you are given that d = 125.
when the plane is flying with the wind, you are given that d = 155.


you get:


(p-w)*t = 125
(p+w)*t = 255


you are given that w = 30, so these formulas become:


(p-30)*t = 125
(p+30)*t = 155


these are two equations that need to be solved simultaneously because the same value of p and the same value of t applies to both of them.


solve for t in both equation to get:


t = 125/(p-30)
t = 155/(p+30)


since both expressions on the right side of these equations are equal to t, you can set those expressions equal to each other to get:


125/(p-30) = 155/(p+30)


multiply both sides of this equation by (p-30)*(p+30) to get:


125 * (p+30) = 155 * (p-30)


simplify to get 125 * p + 125*30 = 155 * p - 155 * 30


simplify to get 125 * p + 3750 = 155 * p - 4650


subtract 125 * p from both sides of the equation and add 4650 to both sides of the equation to get 3750 + 4650 = 155 * p - 125 * p


simplify to get 8400 = 30 * p


divide both sides of this equation by 30 to get 280 = p


that's the speed of the plane in still air (without any wind).


this one is a little more difficult than the usual rate * time problem because you can get into trouble when eliminating variables.


the way above was one way.


another way would be:


start with:


(p-30)*t = 125
(p+30)*t = 155


simplify to get:


pt - 30t = 125
pt + 30t = 155


add the second equation to the first equation to get:


2pt = 280


divide both equations by 2 to get pt = 140.


solve for t to get t = 140/p.


you still don't have a solution, but you do have a relationship that you can use.


go back to one of your equations that you started with.


the one i chose is (p-30)*t = 125


replace t with 140/p to get:


(p-30)*140/p = 125


multiply both sides of this equation by p to get (p-30)*140 = 125*p


simplify to get 140*p - 30*140 = 125*p


simplify to get 140*p - 4200 = 125*p


subtract 125*p from both sides of the equation and add 4200 to both sides of the equation to get:


140*p - 125*p = 4200


simplify to get 15*p - 4200


solve for p to get p = 4200 / 15 = 280.


you get the plane's speed in still air is 280.


that's the same as we got above using a different method.