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
Step-by-step 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!!!
