How much would you have to deposit in an account with a 7% interest rate, compounded monthly, to have $1100 in your account 10 years later?
[tex]
\begin{array}{c}
P=\$[?]
F=P\left(1+\frac{r}{n}\right)^{n t}
\end{array}
[/tex]
Round to the nearest cent.
Answer (1)
Use the future value formula: F = P ( 1 + n r ) n t .
Plug in the given values: F = 1100 , r = 0.07 , n = 12 , and t = 10 .
Solve for the principal amount P : P = ( 1 + 12 0.07 ) 120 1100 .
Calculate P and round to the nearest cent: P ≈ 547.36 .
Explanation
Understanding the Problem We are given the future value of an account, the interest rate, the compounding period, and the time. We need to find the principal amount that needs to be deposited to reach the given future value.
Stating the Formula We will use the formula for future value with compound interest: F = P ( 1 + n r ) n t where:
F is the future value
P is the principal amount (initial deposit)
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the number of years
Identifying the Given Values We are given:
F = $1100
r = 7% = 0.07
n = 12 (compounded monthly)
t = 10 years We need to find P .
Plugging in the Values Plug in the given values into the formula: 1100 = P ( 1 + 12 0.07 ) 12 ⋅ 10 Now, solve for P :
P = ( 1 + 12 0.07 ) 120 1100
Calculating the Principal Calculate the value of P :
P = ( 1 + 12 0.07 ) 120 1100 ≈ 547.36
Final Answer Therefore, you would have to deposit $547.36 in the account to have $1100 in 10 years.
Examples
Imagine you want to save up for a new gaming PC that costs $1100. You find a savings account with a 7% interest rate, compounded monthly. This problem helps you determine how much money you need to deposit initially to reach your goal of $1100 in 10 years. Understanding compound interest and calculating the present value can help you plan your savings and investments to achieve your financial goals.
