Plz help me solve this equation

Question

In a certain county there are two radio stations: the Maisha FM and the Jambo FM. A researcher interested in the loyalty to stations of this county found the following:

Of those who listen to the Maisha FM on a given day 50% will do so the following day while rest change to the Jambo FM. Of those who listen to Jambo FM on a given day 40% change to Maisha FM the following day.

Yesterday the loyalty levels were 30% Maisha FM and 70% Jambo FM. Assume all conditions hold.

Required:

Determine levels of both stations

       I.           Today                                                 

    II.           Tomorrow                                                      

 III.           If the process persists long enough, what will be the eventual loyalty         

Answer 1

Let n be total number of listeners.

Yesterday .7n are M and .3n are J

Today: M = ..3n(.5) + .7n(.4) = ..43n

J = .3n(.5) + .7n(.6) = .57n

Tomorrow

M = .43(.5)n + .57(.4)n = .443n

J = .43(.5)n + .57(.6)n = .557n

Equilibrium:

Looking at fraction of M by dividing through by n and knowing that f_M + f_J = 1:

f = f(.5) + (1-f).4 which leads to f_m= 4/9 or .444...

Please consider a tutor. Take care.