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(Related to The Business of​ Life: Saving for Your First​ House) ​ (Future value) You are hoping to buy a house in the future and recently received an inheritance of ​$18 comma 000. You intend to use your inheritance as a down payment on your house. a. If you put your inheritance in an account that earns 9 percent interest compounded​ annually, how many years will it be before your inheritance grows to ​$32 comma 000​? b. If you let your money grow for 9.75 years at 9 percent​, how much will you​ have? c. How long will it take your money to grow to ​$32 comma 000 if you move it into an account that pays 4 percent compounded​ annually? How long will it take your money to grow to ​$32 comma 000 if you move it into an account that pays 12 percent​? d. What does all this tell you about the relationship among interest​ rates, time, and future​ sums?

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Answer:

A) We need to determine the n in the following equation: FV = PV (1 + r)ⁿ

r = 9% ; FV = 32,000 ; PV = 18,000

32,000 = 18,000 (1.09ⁿ)

32,000 / 18,000 = 1.09ⁿ

1.7778 = 1.09ⁿ

n = (log 1.7778) / (log 1.09) = 6.68 years

B) FV = 18,000 (1.09⁹°⁷⁵) = 18,000 x 2.32 = $41,704

C) if r = 4% ; FV = 32,000 ; PV = 18,000

n = (log 1.7778) / (log 1.04) = 14.67 years

if r = 12% ; FV = 32,000 ; PV = 18,000

n = (log 1.7778) / (log 1.12) = 5.08 years

D) The higher the interest rate, the shorter it takes for money to grow to a certain amount. The higher the interest rate, the more money you will have after a certain amount of years.  

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