how to solve this problem

Question

An airplane travels 1228 kilometers against the wind in 2 hours and 1568 kilometers with the wind in the same amount of time. What is the rate of the plane in still air and what is the rate of the wind?

Answer 1

Since distance equals rate times time (D = RT), solving for T means T = D/R.

Let p = rate of plane in still air and w = rate of the wind.

Travelling against the wind reduces the still air rate, so the plane is moving at a rate represented by p - w.

Travelling with the wind the still air speed is helped by the wind to its rate with the wind is p + w.

We can now write two equations.

1) Against the wind the time is 2, so 2 = 1228/(p-w). ; T = D/R

2p - 2w = 1228

p - w = 614

2) With the wind:

2 = 1568 / (p + w)

2p + 2w = 1568

p + w = 784

We have two equations with two unknowns, easily solved by using the addition/elimination method.

p - w = 614

p + w = 784

2p = 1398

p = 699 (rate of plane in still air).

w = 784 - 699 = 85 (rate of the wind).

Check:

R_against= 699 - 85 = 614

T = 1228 / 614 = 2 hours

R_with= 699 + 85 = 784

T = 1568 / 784 = 2 hours