Continuous Compounding Interest Worksheet (Age 15, Math) — 15 A = Pe^(rt) Problems

Instructions

Use the formula for continuously compounding interest, A = Pe^rt, to solve the following problems. Remember:

• A is the final amount of money.
• P is the principal, or the initial amount of money.
• e is Euler's number (approximately 2.71828). Use the 'e' button on your calculator.
• r is the annual interest rate. Remember to convert percentages to decimals (e.g., 5% = 0.05).
• t is the time in years.

Round all monetary answers to the nearest cent. Round all time answers to two decimal places. Round all rate answers to two decimal places when expressed as a percentage.

1. You invest $5,000 in an account that earns 3% annual interest, compounded continuously. How much will be in the account after 10 years?


2. An initial investment of $1,200 is made into a savings account with a 5.5% annual interest rate, compounded continuously. What is the value of the account after 8 years?


3. A college fund is started with $25,000. If the interest rate is 4.2% compounded continuously, what will the balance be after 20 years?


4. If you deposit $800 into an account paying 7% annual interest compounded continuously, how much money will you have in 5 years?


5. You want to have $10,000 in your account in 7 years. If the account pays 6% interest compounded continuously, how much principal should you invest today?


6. How much principal must be deposited in an account that pays 3.5% interest compounded continuously to have a balance of $50,000 in 18 years?


7. A very ambitious goal is to become a millionaire in 40 years. How much would you need to invest right now in an account that provides 8% annual interest compounded continuously to achieve this goal?


8. You invest $10,000 in an account with an interest rate of 5% compounded continuously. How long will it take for your investment to double to $20,000?


9. How many years will it take for an investment of $6,000 to grow to $15,000 if it is invested at 4% annual interest compounded continuously?


10. An account earns 6.5% interest compounded continuously. How long will it take for the money in the account to triple?


11. You have $4,200 to invest. The account you've chosen offers a 2.8% interest rate compounded continuously. How long will it take for your account to reach a balance of $7,500?


12. An investment of $5,000 grew to $9,000 in 10 years. Assuming the interest was compounded continuously, what was the annual interest rate?


13. What annual interest rate, compounded continuously, is required for an $8,000 investment to double in 12 years?


14. Maria invests $3,000 in an account with a 4.5% interest rate compounded continuously. At the same time, Carlos invests $2,800 in an account with a 5% interest rate compounded continuously. After 15 years, who will have more money in their account?


15. An investment of $7,000 is made in an account that compounds continuously. For the first 5 years, the interest rate is 3%. After 5 years, the rate increases to 4% for the next 10 years. What is the final value of the investment after the full 15 years?

Answer Key

1. A = 5000 * e^{(0.03 * 10)} = $6,749.29


2. A = 1200 * e^{(0.055 * 8)} = $1,862.33


3. A = 25000 * e^{(0.042 * 20)} = $57,833.05


4. A = 800 * e^{(0.07 * 5)} = $1,135.53


5. P = 10000 / e^{(0.06 * 7)} = $6,570.47


6. P = 50000 / e^{(0.035 * 18)} = $26,510.98


7. P = 1000000 / e^{(0.08 * 40)} = $40,656.97


8. t = ln(20000/10000) / 0.05 = ln(2) / 0.05 = 13.86 years


9. t = ln(15000/6000) / 0.04 = ln(2.5) / 0.04 = 22.91 years


10. t = ln(3) / 0.065 = 16.90 years


11. t = ln(7500/4200) / 0.028 = 20.70 years


12. r = ln(9000/5000) / 10 ≈ 0.0588 = 5.88%


13. r = ln(16000/8000) / 12 = ln(2) / 12 ≈ 0.0578 = 5.78%


14. Maria: A = 3000 * e^{(0.045 * 15)} ≈ $5,886.23
    Carlos: A = 2800 * e^{(0.05 * 15)} ≈ $5,928.31
    Carlos will have more money.


15. Step 1 (First 5 years): A = 7000 * e^{(0.03 * 5)} ≈ $8,129.41
    Step 2 (Next 10 years): A = 8129.41 * e^{(0.04 * 10)} ≈ $12,126.70