How much would you have to deposit in an account with a 9% interest rate, compounded continuously, to have $1500 in your account 5 years later?
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Answer (2)
Identify the formula for continuous compounding: A = P e r t .
Substitute the given values: A = 1500 , r = 0.09 , and t = 5 .
Solve for the principal P : P = e 0.09 × 5 1500 .
Calculate the value of P : P ≈ 956.44 .
Explanation
Understanding the Problem We are given the future value of an investment, the interest rate, and the time period. We need to find the present value (principal) that would result in the given future value with continuous compounding.
Recalling the Formula The formula for continuous compounding is: A = P e r t where:
A is the future value of the investment
P is the principal (initial deposit)
r is the interest rate (as a decimal)
t is the time in years
Identifying Given Values We are given:
A = $1500
r = 0.09
t = 5 years We need to find P .
Solving for the Principal Substitute the given values into the formula: 1500 = P e 0.09 × 5 Now, solve for P :
P = e 0.09 × 5 1500 P = e 0.45 1500 Using a calculator, we find that: P ≈ 956.44
Final Answer Therefore, you would need to deposit approximately $956.44 to have $1500 in your account after 5 years with a 9% interest rate compounded continuously.
Examples
Continuous compounding is a concept widely used in finance. For example, if you want to know how much to invest today to reach a specific savings goal in the future, you can use the continuous compounding formula. This is also applicable in calculating loan payments or understanding the growth of investments over time. Understanding this concept helps in making informed financial decisions, such as planning for retirement or saving for a down payment on a house. By knowing the initial investment, interest rate, and time, you can accurately predict the future value of your investment.
You need to deposit approximately $956.44 in an account with a 9% interest rate, compounded continuously, to have $1500 in 5 years. This is calculated using the formula for continuous compounding. By rearranging the formula, we substituted the values and found the principal amount needed.
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