Certainly! Let's break down the solution step-by-step for the given problem. You deposit [tex]$400 each month into an account that earns an 8% annual interest rate, compounded monthly, for 15 years.
### a) How much will you have in the account in 15 years?
To find the future value of an annuity where you make regular monthly deposits, we use the future value formula:
\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]
Where:
- \( P \) is the monthly deposit amount, which is $[/tex]400.
- [tex]\( r \)[/tex] is the monthly interest rate. Since the annual interest rate is 8%, the monthly interest rate is [tex]\( \frac{8\%}{12} \)[/tex] or 0.08/12.
- [tex]\( n \)[/tex] is the total number of deposits. Over 15 years, making monthly deposits, [tex]\( n = 15 \times 12 \)[/tex].
Substituting these values into the formula, we compute the future value (FV).
From the provided details, the future value turns out to be approximately:
[tex]\[ FV \approx \$138,415.29 \][/tex]
### b) How much total money will you put into the account?
Total money deposited is very straightforward. It is simply the monthly deposit amount multiplied by the total number of months:
[tex]\[ \text{Total Deposited} = P \times n \][/tex]
[tex]\[ \text{Total Deposited} = 400 \times 180 \][/tex]
[tex]\[ \text{Total Deposited} = \$72,000 \][/tex]
### c) How much total interest will you earn?
To find the total interest earned, subtract the total amount deposited from the future value of the account:
[tex]\[ \text{Total Interest} = FV - \text{Total Deposited} \][/tex]
[tex]\[ \text{Total Interest} \approx \$138,415.29 - \$72,000 \][/tex]
[tex]\[ \text{Total Interest} \approx \$66,415.29 \][/tex]
### Summary
- Future Value after 15 years (a): [tex]\( \approx \$138,415.29 \)[/tex]
- Total Amount Deposited (b): [tex]\( \$72,000 \)[/tex]
- Total Interest Earned (c): [tex]\( \approx \$66,415.29 \)[/tex]
You end up with approximately [tex]$138,415.29 in the account, having deposited $[/tex]72,000 over the years, earning approximately $66,415.29 in interest.
