$500 are deposited in an account with $7 \%$ interest rate, compounded continuously. What is the balance after 10 years?

$F=\$[?]

Identify the principal amount $P = 500 , interest rate r = 0.07 , and time t = 10 years.
Apply the continuous compounding formula: F = P e r t .
Substitute the given values: F = 500 × e 0.07 × 10 .
Calculate the final balance: F ≈ 1006.88 ​ .

Explanation

Problem Setup We are given a principal amount of $500 deposited in an account with a 7% interest rate, compounded continuously. We want to find the balance after 10 years.

Formula for Continuous Compounding The formula for continuous compounding is given by:


F = P e r t
where:

F is the future value (balance)
P is the principal amount
r is the interest rate (as a decimal)
t is the time in years


Substituting the Values We are given:


P = $500
r = 7% = 0.07
t = 10 years

Substituting these values into the formula, we get:
F = 500 × e 0.07 × 10

Calculating the Future Value Now, we calculate the value of e 0.07 × 10 :

e 0.07 × 10 = e 0.7 ≈ 2.01375
So, the future value is:
F = 500 × 2.01375 = 1006.8763537352384

Final Balance Therefore, the balance after 10 years is approximately $1006.88.

Examples
Continuous compounding is a concept widely used in finance. For example, when calculating the future value of an investment or loan, banks and financial institutions often use continuous compounding to model the growth of money over time. This is particularly useful for long-term investments, where the effect of compounding can significantly increase the final value. Understanding continuous compounding helps investors make informed decisions about their investments and plan for their financial future.

Answered by GinnyAnswer | 2025-07-05