Continuous Compound Interest Worksheet (A = Pe^(rt)) — Math for 15-Year-Olds

Instructions

This worksheet will help you practice solving problems involving money and continuously compounded interest. You will use the formula:

A = Pe^rt

Where:

• A = the final amount of money after the time has passed.
• P = the principal, or the initial amount of money.
• e = Euler's number, a special mathematical constant approximately equal to 2.71828. (Your calculator should have an e^x button).
• r = the annual interest rate, written as a decimal.
• t = the time the money is invested or borrowed for, in years.

Important: Always convert the interest rate percentage to a decimal before using it in the formula. For example, 5% becomes 0.05. Round all final monetary answers to the nearest cent (two decimal places).

Practice Problems

1. You invest $2,000 in a savings account with an annual interest rate of 3.5% compounded continuously. How much money will be in the account after 8 years?


2. A recent high school graduate receives $500 as a gift and deposits it into an account that earns 4.2% annual interest compounded continuously. If they don't touch the money, how much will they have after 15 years?


3. Your parents invest $10,000 for your college fund on the day you are born. The account has an interest rate of 6% compounded continuously. How much money will be in the account on your 18th birthday?


4. How long would it take for an investment of $5,000 to grow to $7,500 if the account earns 5% interest compounded continuously? (Hint: You'll need to solve for 't'. Use the natural logarithm, ln).


5. An investor wants to have $50,000 in an account in 10 years. If the account pays 4.8% interest compounded continuously, what is the initial principal (P) they need to invest today? (Hint: Rearrange the formula to solve for P).


6. Challenge: Maria invests $4,000 in an account with a 3% interest rate compounded continuously. David invests $3,500 in an account with a 4% interest rate compounded continuously. Who will have more money after 20 years, and by how much more?

Answer Key

1. You invest $2,000 at 3.5% for 8 years.

• A = 2000 * e^{(0.035 * 8)}
• A = 2000 * e^0.28
• A = 2000 * 1.3231298...
• A = $2,646.26

2. You invest $500 at 4.2% for 15 years.

• A = 500 * e^{(0.042 * 15)}
• A = 500 * e^0.63
• A = 500 * 1.87761...
• A = $938.81

3. You invest $10,000 at 6% for 18 years.

• A = 10000 * e^{(0.06 * 18)}
• A = 10000 * e^1.08
• A = 10000 * 2.944679...
• A = $29,446.80

4. How long for $5,000 to become $7,500 at 5%?

• 7500 = 5000 * e^{(0.05 * t)}
• 1.5 = e^0.05t
• ln(1.5) = 0.05t
• 0.405465 = 0.05t
• t = 0.405465 / 0.05
• t ≈ 8.11 years

5. What principal is needed for $50,000 in 10 years at 4.8%?

• 50000 = P * e^{(0.048 * 10)}
• 50000 = P * e^0.48
• 50000 = P * 1.616074...
• P = 50000 / 1.616074...
• P = $30,939.19

6. Challenge: Maria vs. David after 20 years.

• Maria: A = 4000 * e^{(0.03 * 20)} = 4000 * e^0.6 = $7,288.48
• David: A = 3500 * e^{(0.04 * 20)} = 3500 * e^0.8 = $7,789.04
• Comparison: David will have more money.
• Difference: $7,789.04 - $7,288.48 = $500.56 more.