Question

Through an aggressive advertising campaign, each month station A is capturing 12% of station B's customers and 5% of station C's customers. (Hint: The phrase "station A is capturing 12% of station B's customers" means that 12% of the listeners are moving from B to A, not the other way around.) At the same time, station A is losing only 2% of its customers to B and 2% of its customers to C. Additionally, each month 1% of B's customers switch to C, and 1% of C's customers switch to B. Initially, A and B each have 30% of the listeners and C has 40%.

Answer 1

At the End of 3 months

Station A has 0.415 of all the listeners

Station B has 0.226 of all the listeners

Station C has 0.359 of all listeners

Explanation:

The loss and gain in listeners for radio station A is listed below.

- A is gaining 12% of station B's listeners monthly.

- A is gaining 5% of station C's listeners monthly.

- A is losing 2% of its listeners to station B monthly.

- A is losing 2% of its listeners to station C monthly.

For station B

- B is gaining 2% of station A's listeners monthly.

- B is gaining 1% of station C's listeners monthly.

- B is losing 12% of its listeners to station A monthly.

- B is losing 1% of its listeners to station C monthly.

For station C

- C is gaining 2% of station A's listeners monthly.

- C is gaining 1% of station B's listeners monthly.

- C is losing 5% of its listeners to station A monthly.

- C is losing 1% of its listeners to station B monthly.

A starts with 30% of the listeners

B starts with 30% of the listeners

C starts with 40% of the listeners

First month

Station A

A gains 12% of B's 30% = +12% of 30% = +3.6% of all listeners.

A gains 5% of C's 40% = +5% of 40% = +2% of all listeners.

A loses 2% of its 30% to B = -2% of 30% = -0.6% of all listeners.

A loses 2% of its 30% to C = -2% of 30% = -0.6% of all listeners.

Total number of station A listeners at the end of the first month = 30% + 3.6% + 2% - 0.6% - 0.6% = 34.4%

For station B

B gains 2% of A's 30% = +2% of 30% = +0.6% of all listeners.

B gains 1% of C's 40% = +1% of 40% = +0.4% of all listeners.

B loses 12% of its 30% to A = -12% of 30% = -3.6% of all listeners.

B loses 1% of its 30% to C = -1% of 30% = -0.3% of all listeners.

Total number of station B listeners at the end of the first month = 30% + 0.6% + 0.4% - 3.6% - 0.3% = 27.1%

Station C

C gains 2% of A's 30% = +2% of 30% = +0.6% of all listeners.

C gains 1% of B's 30% = +1% of 30% = +0.3% of all listeners.

C loses 5% of its 40% to B = -5% of 40% = -2% of all listeners.

C loses 1% of its 40% to C = -1% of 40% = -0.4% of all listeners.

Total number of station C listeners at the end of the first month = 40% + 0.6% + 0.3% - 2% - 0.4% = 38.5%

Second month

Station A

A gains 12% of B's 27.1% = +12% of 27.1% = +3.252% of all listeners.

A gains 5% of C's 38.5% = +5% of 38.5% = +1.925% of all listeners.

A loses 2% of its 34.4% to B = -2% of 34.4% = -0.688% of all listeners.

A loses 2% of its 34.4% to C = -2% of 34.4% = -0.688% of all listeners.

Total number of station A listeners at the end of the second month = 34.4 + 3.252 + 1.925 - 0.688 - 0.688 = 38.201%

For station B

B gains 2% of A's 34.4% = +2% of 34.4% = +0.688% of all listeners.

B gains 1% of C's 38.5% = +1% of 38.5% = +0.385% of all listeners.

B loses 12% of its 27.1% to A = -12% of 27.1% = -3.252% of all listeners.

B loses 1% of its 27.1% to C = -1% of 27.1% = -0.271% of all listeners.

Total number of station B listeners at the end of the second month = 27.1 + 0.688 + 0.385 - 3.252 - 0.271 = 24.65%

Station C

C gains 2% of A's 34.4% = +2% of 34.4% = +0.688% of all listeners.

C gains 1% of B's 27.1% = +1% of 27.1% = +0.271% of all listeners.

C loses 5% of its 38.5% to B = -5% of 38.5% = -1.925% of all listeners.

C loses 1% of its 38.5% to C = -1% of 38.5% = -0.385% of all listeners.

Total number of station C listeners at the end of the second month = 38.5 + 0.688 + 0.271 - 1.925 - 0.385 = 37.149%

Third month

Station A

A gains 12% of B's 24.65% = +12% of 27.1% = +2.958% of all listeners.

A gains 5% of C's 37.149% = +5% of 37.149% = +1.85745% of all listeners.

A loses 2% of its 38.201% to B = -2% of 38.201% = -0.76402% of all listeners.

A loses 2% of its 38.201% to C = -2% of 34.201% = -0.76402% of all listeners.

Total number of station A listeners at the end of the third month = 38.201 + 2.958 + 1.85745 - 0.76402 - 0.76402 = 41.48841%

For station B

B gains 2% of A's 38.201% = +2% of 38.201% = +0.76402% of all listeners.

B gains 1% of C's 37.149% = +1% of 37.149% = +0.37149% of all listeners.

B loses 12% of its 24.65% to A = -12% of 24.65% = -2.958% of all listeners.

B loses 1% of its 24.65% to C = -1% of 24.65% = -0.2465% of all listeners.

Total number of station B listeners at the end of the third month = 24.65 + 0.76402 + 0.37149 - 2.958 - 0.2465 = 22.58101%

Station C

C gains 2% of A's 38.201% = +2% of 38.201% = +0.76402% of all listeners.

C gains 1% of B's 24.65% = +1% of 24.65% = +0.2465% of all listeners.

C loses 5% of its 37.149% to B = -5% of 37.149% = -1.85745% of all listeners.

C loses 1% of its 37.149% to C = -1% of 37.149% = -0.37149% of all listeners.

Total number of station C listeners at the end of the third month = 37.149 + 0.76402 + 0.2465 - 1.85745 - 0.37149 = 35.93058%

Hope this Helps!!!