Help with equations to figure a word problem

Question

According to a recent survey, in 2000, 75% of households had one or more pets. Throughout the next decade, this decreased by appoximately 25% per year. If this trend continues, in which year will 35% of households have pets?

Answer 1

at 2000, t = 0  

 

Let H = total number of households

      n = H%

 

n = (0.75*H)*(1 - 0.25)^t x 100%           Solution

 

Therefore, set n = 35%H and solve for t (yrs):

 

divide through by 100% and solve for t

 

(0.35 H)/(0.75*H) = (0.75)^t

 

(0.35/0.75) = (0.75)^t       (take ln of both sides and solve for t)

 

ln[(0.35)/(0.75)] = t*ln(0.75)

 

t = ln[(0.35)/(0.75)]/ln(0.75) = 2.649 yr      0.649 yr x 12 months/yr = 7.8 months

 

Therefore, in year (2000 + 2.65) or year 2002 in July/August