To determine the principal amount that needs to be deposited in an account with a 7% annual interest rate, compounded monthly, to have [tex]$1100 in the account 10 years later, follow these steps:
1. Identify the given values:
- Future value (F) = $[/tex]1100
- Annual interest rate (r) = 0.07 (7%)
- Number of times the interest is compounded per year (n) = 12 (monthly compounding)
- Number of years (t) = 10
2. Understand the formula:
The formula to calculate the future value based on the principal amount (P) with compound interest is:
[tex]\[
F = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
We need to solve for P:
[tex]\[
P = \frac{F}{\left(1 + \frac{r}{n}\right)^{nt}}
\][/tex]
3. Plug in the given values:
[tex]\[
P = \frac{1100}{\left(1 + \frac{0.07}{12}\right)^{12 \times 10}}
\][/tex]
4. Calculate the term inside the parentheses:
[tex]\[
1 + \frac{0.07}{12} = 1 + 0.0058333 \approx 1.0058333
\][/tex]
5. Raise this term to the power of `nt` (12 * 10 = 120):
[tex]\[
\left(1.0058333\right)^{120}
\][/tex]
6. Divide the future value (F) by this result to find the principal amount (P):
[tex]\[
P = \frac{1100}{1.507563}
\][/tex]
7. Calculate the principal amount (P):
[tex]\[
P \approx 547.3558942594921
\][/tex]
8. Round the principal amount to the nearest cent:
[tex]\[
P \approx 547.36
\][/tex]
Hence, you would need to deposit approximately [tex]$547.36 in the account to have $[/tex]1100 after 10 years with a 7% interest rate compounded monthly.
