Question

Jasmine invests a sum of money in a savings account with a fixed annual interest rate of 8% compounded continuously. After 9 years, the balance reaches $18,091.34. What was the amount of the initial investment?

Answer 1

To calculate the accrued amount of an account that compounds continuously you have to apply the following formula:

`A=Pe^(rt)`

Where

A is the accrued or final amount

P is the principal or initial amount

r is the interest rate expressed as a decimal value

t is the time period in years

You have to determine the principal amount given that after 9 years the final balance of an account that has an annual interest of 8% is $18,091.34

- The first step is to write the formula for the principal amount P, to do so, divide both sides of the expression by e^(rt)

`\begin{gathered} (A)/(e^(rt))=(Pe^(rt))/(e^(rt)) \\ P=(A)/(e^(rt)) \end{gathered}`

- Next, divide the interest rate by 100 to express it as a decimal value

`\begin{gathered} r=(8)/(100) \\ r=0.08 \end{gathered}`

- Now you can calculate the principal amount, replace the expression using A=18,091.34, r=0.08, and t=9

`\begin{gathered} P=(18091.34)/(e^(0.08\cdot9)) \\ P=(18091.34)/(e^(0.72)) \\ P=8806.000558\cong8806.00 \end{gathered}`

The initial amount of the investment was $8806