WXXX, WYYY, AND WZZZ are three radio stations that compete for listeners. Each m

Question

WXXX, WYYY, AND WZZZ are three radio stations that compete for listeners. Each month some of the listeners switch from one station to another while some remain loyal to their current station.

It has been found that each month:

20% of the listeners of WXXX switch to WYYY
10% of the listeners of WXXX switch to WZZZ
30% of the listeners of WYYY switch to WXXX
10% of the listeners of WYYY switch to WZZZ
20% of the listeners of WZZZ switch to WXXX
30% of the listeners of WZZZ switch to WYYY

Currently, the listening audience is divided as follows:

40% listen to WXXX
40% listen to WYYY
20% listen to WZZZ

Use this information to answer the following questions. (Give your answers correct to three decimal places. Note that you are asked to give decimal values and not percentages. For example, you would need to enter 0.628 as opposed to 62.800%.)

(a) What fraction of the listeners will WZZZ have after 2 months?


(b) What fraction of the listeners will WXXX have after 3 months?


(c) What fraction of the listeners will WYYY have after 5 months?

Explanation / Answer

Dear Student Thank you for using Chegg !! Given 3Rdio Stations that Compete for listeners Let the total Number of listeners be X Number of present Listeners for WXXX = 0.40X (Assume X) Number of present Listeners for WYYY = 0.40X (Assume Y) Number of present Listeners for WZZZ = 0.20X (Assume Z) Now as per Information given in question WXXX WYYY WZZZ Initial X Y Z 20% of the listeners of WXXX switch to WYYY 0.8X Y + 0.2X Z 10% of the listeners of WXXX switch to WZZZ 0.7X Y + 0.2X 0.1X+Z 30% of the listeners of WYYY switch to WXXX 0.7X + 0.3Y 0.7Y + 0.2X 0.1X+Z 10% of the listeners of WYYY switch to WZZZ 0.7X + 0.3Y 0.6Y + 0.2X 0.1X+0.1Y+Z 20% of the listeners of WZZZ switch to WXXX 0.7X + 0.3Y+ 0.2Z 0.6Y + 0.2X 0.1X+0.1Y+0.8Z 30% of the listeners of WZZZ switch to WYYY 0.7X + 0.3Y+ 0.2Z 0.6Y + 0.2X + 0.3Z 0.1X+0.1Y+0.5Z X=0.4, Y = 0.4, Z = 0.2 Hence After a Month 0.44 0.38 0.18 Using X = 0.44 Y = 0.38 Z = 0.18 WXXX WYYY WZZZ After 2 Months 0.458 0.37 0.172 Using X = 0.458 Y = 0.37 Z = 0.172 WXXX WYYY WZZZ After 3 Months 0.466 0.3652 0.1688 Using X = 0.466 Y = 0.3652 Z = 0.1688 WXXX WYYY WZZZ After 4 Months 0.46952 0.36296 0.16752 Using X = 0.46952 Y = 0.36296 Z = 0.16752 WXXX WYYY WZZZ After 5 Months 0.471056 0.361936 0.167008 Now coming to questions as asked a) What fraction of the listeners will WZZZ have after 2 months? From the above explanation, WZZZ shall have 0.172 of total listeners b) What fraction of the listeners will WXXX have after 3 months? From the above explanation, WXXX shall have 0.466 of total listeners c) What fraction of the listeners will WYYY have after 5 months? From the above explanation, WYYY shall have 0.362 of total listeners Solution

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