Suppose that in any given year, the number of cases of a disease is reduced by 20%...?

If there are 10,000 cases today, how many year will it take to A) reduce the number of cases to 1000? B) to eliminate the disease; that is, to reduce the number of cases to less than 1?Suppose that in any given year, the number of cases of a disease is reduced by 20%...?
It'll take between 10 and 11 years to reduce the number of cases to 1000.





It'll take 42 years to reduce the number of cases to less than 1.





Since you didn't specify a method, I used a spread sheet and let it do the work:





year reduction Total this year


1 . . . . . . . . . 10000


2 . . . 2000 . . . 8000


3 . . . 1600 . . . 6400


4 . . . 1280 . . . 5120


5 . . . 1024 . . . 4096


6 . . . 819.20 . . . 3276.8


7 . . . 655.36 . . . 2621.44


8 . . . 524.29 . . . 2097.15


9 . . . 419.43 . . . 1677.72


10 . . . 335.54 . . . 1342.18


11 . . . 268.44 . . . 1073.74


12 . . . 214.75 . . . 858.99


13 . . . 171.80 . . . 687.19


14 . . . 137.44 . . . 549.76


15 . . . 109.95 . . . 439.8


16 . . . 87.96 . . . 351.84


17 . . . 70.37 . . . 281.47


18 . . . 56.29 . . . 225.18


19 . . . 45.04 . . . 180.14


20 . . . 36.03 . . . 144.12


21 . . . 28.82 . . . 115.29


22 . . . 23.06 . . . 92.23


23 . . . 18.45 . . . 73.79


24 . . . 14.76 . . . 59.03


25 . . . 11.81 . . . 47.22


26 . . . 9.44 . . . 37.78


27 . . . 7.56 . . . 30.22


28 . . . 6.04 . . . 24.18


29 . . . 4.84 . . . 19.34


30 . . . 3.87 . . . 15.47


31 . . . 3.09 . . . 2.38


32 . . . 2.48 . . . 9.9


33 . . . 1.98 . . . 7.92


34 . . . 1.58 . . . 6.34


35 . . . 1.27 . . . 5.07


36 . . . 1.01 . . . 4.06


37 . . . 0.81 . . . 3.25


38 . . . 0.65 . . . 2.6


39 . . . 0.52 . . . 2.08


40 . . . 0.42 . . . 1.66


41 . . . 0.33 . . . 1.33


42 . . . 0.27 . . . 1.06


43 . . . 0.21 . . . 0.85Suppose that in any given year, the number of cases of a disease is reduced by 20%...?
Logs base 10





A. 1000/10000 = 10%


10,000*.8^x=1000


.8^x = .1


log.8^x = log.1


xlog.8 = log.1


x = log.1/log.8


x = 10.3188512 years





B. 10,000*.8^x = .999999


.8^x = .000099


xlog .8 = log .000099


x = log .000099/log .8


x = 41.3204444 years , roughly