Question

Problem#1. At the end of three years, how much is an initial deposit of $100 worth, assuming a compound interest rate of (i) 100 percent? (ii) 10 percent? (iii) zero percent?

Problem#2. At the end of five years, how much is an initial $500 deposit followed by five year-end, annual $100 payment worth, assuming a compound annual interest rate of (i) 10 percent? (ii) 5 percent? (iii) zero percent?

Answer 1

To calculate the total amount at the end of three years with compound interest, use the formula A = P(1+r)^n. For Problem #1, the total amounts are $200 (100% interest), $133.10 (10% interest), and $100 (0% interest). For Problem #2, the total amounts are $689.07 (10% interest), $677.63 (5% interest), and $600 (0% interest).

Step-by-step explanation:

To calculate the total amount at the end of three years with compound interest, we use the formula:A = P(1+r)nWhere A is the total amount, P is the principal (initial deposit), r is the interest rate, and n is the number of compounding periods.(i) For a 100% interest rate, the total amount is $200.(ii) For a 10% interest rate, the total amount is $133.10.(iii) For a 0% interest rate, the total amount remains $100.For problem #2, we need to calculate the future value of the deposit and the future value of the annual payments with compound interest. Then, we add them together to get the total amount at the end of five years.(i) For a 10% interest rate, the total amount is $689.07.(ii) For a 5% interest rate, the total amount is $677.63.(iii) For a 0% interest rate, the total amount is $600.

Answer 2

To calculate the future value of a deposit with compound interest, we use a particular formula depending on whether we have a single deposit or an initial deposit with additional payments. Over three years, $100 grows to $800 at 100% interest, $133.10 at 10% interest, and remains $100 at 0% interest. For a $500 deposit with five annual payments of $100 each, at a 10% interest rate, the final amount is $1415.76.

Step-by-step explanation:

Calculating Future Value with Compound Interest

When calculating the future value of a deposit with compound interest, we use the formula A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.

Problem #1: Future Value of a Single Deposit

1. 100% interest rate: A = $100(1 + 1)^3 = $100(2)^3 = $800
2. 10% interest rate: A = $100(1 + 0.10)^3 = $100(1.1)^3 = $133.10
3. Zero percent interest: A = $100(1 + 0)^3 = $100(1)^3 = $100

Problem #2: Future Value of Initial Deposit and Annual Payments

1. 10% interest rate: The future value of an annuity formula is used here, FV = P[(1 + r)^n - 1] / r. The initial deposit of $500 will accumulate separately from the annual $100 payments. Using compound interest for the deposit:
2. A = $500(1 + 0.10)^5 = $500(1.1)^5 = $805.25
3. For the annuity part: $100 payment at the end of each year for 5 years at 10% interest rate will accumulate to a total of: FV = $100[((1.1)^5 - 1) / 0.10] = $610.51
4. The combined future value of the initial deposit and the annuity is: $805.25 + $610.51 = $1415.76
5. The calculations for 5% interest and zero percent follow the same principles but with the respective interest rates.