A certain car depreciates such that its value at the end of each year is p % less than its value at the end of the previous year. If that car was worth a dollars on December 31, 2010 and was worth b dollars on December 31, 2011, what was the car worth on December 31, 2013 in terms of a and b?

b (b/a) ^2

Step-by-step explanation:

Given that the value of the car depreciates such that its value at the end of each year is p % less than its value at the end of the previous year and that car was worth a dollars on December 31, 2010 and was worth b dollars on December 31, 2011, then

b = a - (p% * a) = a (1-p%)

b/a = 1 - p%

p% = 1 - b/a = (a-b) / a

Let the worth of the car on December 31, 2012 be c

then

c = b - (b * p%) = b (1-p%)

Let the worth of the car on December 31, 2013 be d

then

d = c - (c * p%)

d = c (1-p%)

d = b (1-p%) (1-p%)

d = b (1-p%) ^2

d = b (1 - (a-b) / a) ^2

d = b ((a-a+b) / a) ^2

d = b (b/a) ^2 = b^3/a^2

The car's worth on December 31, 2013 = b (b/a) ^2 = b^3/a^2