Question

An antique automobile is depreciating, i.e. losing value i.e. going down in price at a rate of 7% per annum compounded annually. Betty and Bob have an account, which earns interest at a rate of 12% per annum continuously compounded. They originally had $9,000 in the account and withdrew $2000 after the second year and $2,500 after the third year. They had enough money (exactly) to buy the automobile after 6 years.
Algebraically find the original value of the automobile.

(I need a step-by-step explanation of the formulas used. Thanks)

Answer 1

Given automobile is losing value at a rate of 7% per annum compounded annually

Let the initial value of automobile be x

At the end of first year the value of automobile will be = x * (1-0.07) = 0.93x

Similarly the value of the automobile at the end of second year will be x * (1-0.07) ^2

Continuing in a similar manner the value of the automobile at the end of 6th tear will be x *(1-0.07)^6   --->(1)

Betty and Bob had originally 9000 in the account

Account value at the end of first year = 9000*(1+12%) = $10080 (where 12% is the annual interest rate earned)

Similarly account value at the end of 2nd year will be = 10080*(1 +12%) - 2000 = $9289.6 ($2000 withdrawn from account at the end of 2nd year)

Similarly account value at the end of 3rd year will be =9289.6 *(1+12%) - 2500 = $7904.352

Similarly account value at the end of 6th year will be = 7904.352*(1+12%)^3 = $11105.0454 ----->(2)

Given they buy the automobile after 6 years implies from equation 1 and 2

=> x*(1-0,07)^6 = 11105.0454

On solving x = 17164.1637