A = Pe^rt Worksheet for 15-Year-Olds — Solve for A or P (Math)

Instructions

This worksheet focuses on the formula for continuously compounded interest: A = Pe^rt.

Where:

• A is the final amount of money after the investment period.
• P is the principal, or the initial amount of money.
• e is the natural number (approximately 2.71828). Use the 'e' button on your calculator for accuracy.
• r is the annual interest rate, written as a decimal (e.g., 5% = 0.05).
• t is the time in years.

For each problem, carefully identify the given values and determine whether you need to solve for the final amount (A) or the principal amount (P). Remember, to solve for P, you will need to rearrange the formula to P = A / e^rt.

Please round all your final monetary answers to two decimal places.

Practice Problems

1. You invest $2,000 into a savings account with an annual interest rate of 4% compounded continuously. How much money will be in the account after 10 years?


2. To have $25,000 in an account in 8 years, how much money must be invested now? The account pays 6.5% annual interest compounded continuously.


3. An initial investment of $15,000 is made into an account with a 7.2% interest rate compounded continuously. What is the value of the account after 15 years?


4. You want to save up $5,000 for a car in 3 years. If you find an investment that offers 3.8% interest compounded continuously, what is the principal amount you need to invest today?


5. A college fund is started with an initial deposit of $10,000. If the interest rate is 5.5% compounded continuously, what will the balance be after 18 years?


6. What was the initial deposit on an account that now holds $8,534.51 after 12 years of being compounded continuously at a rate of 4.2%?


7. Your grandparents put $500 into a savings bond for you when you were born. The bond earns 6% interest compounded continuously. How much will the bond be worth on your 18th birthday?


8. An entrepreneur wants to have $100,000 in a business account in 5 years to expand their company. How much must they deposit now into an account that earns 8% interest compounded continuously?


9. An art collector purchases a painting for $50,000. If the value of the painting is expected to appreciate at a rate of 9% per year, compounded continuously, what will its estimated value be in 20 years?


10. A winning lottery ticket is worth $1,000,000 in 30 years. If the prize is invested at a continuously compounded rate of 5%, what is the present-day cash value (the principal) of the ticket?

Answer Key

1. Solve for A
   P = $2,000, r = 0.04, t = 10 years
   A = 2000 * e^{(0.04 * 10)}
   A = 2000 * e^0.4
   A ≈ 2000 * 1.49182
   A ≈ $2,983.65


2. Solve for P
   A = $25,000, r = 0.065, t = 8 years
   P = 25000 / e^{(0.065 * 8)}
   P = 25000 / e^0.52
   P ≈ 25000 / 1.68203
   P ≈ $14,862.95


3. Solve for A
   P = $15,000, r = 0.072, t = 15 years
   A = 15000 * e^{(0.072 * 15)}
   A = 15000 * e^1.08
   A ≈ 15000 * 2.94468
   A ≈ $44,170.20


4. Solve for P
   A = $5,000, r = 0.038, t = 3 years
   P = 5000 / e^{(0.038 * 3)}
   P = 5000 / e^0.114
   P ≈ 5000 / 1.12075
   P ≈ $4,461.50


5. Solve for A
   P = $10,000, r = 0.055, t = 18 years
   A = 10000 * e^{(0.055 * 18)}
   A = 10000 * e^0.99
   A ≈ 10000 * 2.69123
   A ≈ $26,912.35


6. Solve for P
   A = $8,534.51, r = 0.042, t = 12 years
   P = 8534.51 / e^{(0.042 * 12)}
   P = 8534.51 / e^0.504
   P ≈ 8534.51 / 1.6553
   P ≈ $5,155.80


7. Solve for A
   P = $500, r = 0.06, t = 18 years
   A = 500 * e^{(0.06 * 18)}
   A = 500 * e^1.08
   A ≈ 500 * 2.94468
   A ≈ $1,472.34


8. Solve for P
   A = $100,000, r = 0.08, t = 5 years
   P = 100000 / e^{(0.08 * 5)}
   P = 100000 / e^0.4
   P ≈ 100000 / 1.49182
   P ≈ $67,032.00


9. Solve for A
   P = $50,000, r = 0.09, t = 20 years
   A = 50000 * e^{(0.09 * 20)}
   A = 50000 * e^1.8
   A ≈ 50000 * 6.04965
   A ≈ $302,482.35


10. Solve for P
    A = $1,000,000, r = 0.05, t = 30 years
    P = 1000000 / e^{(0.05 * 30)}
    P = 1000000 / e^1.5
    P ≈ 1000000 / 4.48169
    P ≈ $223,130.16