A survey found that among people who planned to buy a car this year, 65% owned a

Question

A survey found that among people who planned to buy a car this year, 65% owned an american car. a different survey found that amoung people who own an american built car 20% will switch to a foreign make on their next purchase and amoug thos who own a foreign make 30% will switch to an american built car on their next purchase.

C. FIND THE LONG TERM PERCENTAGE OF THE PEOPLE WHO WILL OWN AN AMERICAN BUILT CAR IF THE TREND HOLDS.

Explanation / Answer

Let total number of people be 100 Initially, % of people owning american car = x => % of people owning foreign built = 100-x Next year (as pe trend) % of people owning american car = x-(0.2x)+ 0.3(100-x) = (0.95x)+30 => % of people owning foreign built = (100-x)- 0.3(100-x)+(0.2x) = 70 - 0.95x Again in next year % of people owning american car =(0.95x)+30-(0.2(0.95x)+30)+ 0.3(100-(0.95x)+30) = (0.95(0.95x)+30)+30 = (0.95^2)x + (30*1.95) => % of people owning foreign built = (100-(0.95^2)x + (30*1.95)) Further next year, % of people owning american car = (0.95^2)x + (30*1.95)-(0.2[(0.95^2)x + (30*1.95)])+ 0.3(100-[(0.95^2)x + (30*1.95)]) = (0.95[(0.95^2)x + (30*1.95))+30 = (0.95^3)x + 30*(1+1.95*0.95) Similarly the trend continues after infinite number of years: (0.95^(infinite) approaches zero. So, the final answer will be the second qty. only.

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