You deposit $320 every six months into an account earning 7.6% interest compounded semiannually.
a) How much will you have in the account in 35 years?
$
b) How much total money will you put into the account?
$
c) How much total interest will you earn?
$

This problem involves calculating the future value of an annuity, where regular deposits are made, and interest is compounded semiannually. We will also determine the total amount deposited and the total interest earned.

### Given dataThis problem involves calculating the future value of an annuity, where regular deposits are made, and interest is compounded semiannually. We will also determine the total amount deposited and the total interest earned.

### Given 
- Deposit per period ($ PMT $) = $320
- Interest rate per period ($ i $) = $ \frac{7.6\%}{2} = 3.8\% = 0.038 $
- Number of periods per year = 2 (because it's compounded semiannually)
- Total time = 35 years
- Total number of periods ($ n $) = $ 35 \times 2 = 70 $

#### a) How much will you have in the account in 35 years?
To find the future value $ FV $ of an annuity compounded periodically, we use the formula:
\[
FV = PMT \times \frac{(1 + i)^n - 1}{i}
\]
Substituting the values:
\[
FV = 320 \times \frac{(1 + 0.038)^{70} - 1}{0.038}
\]
Let's compute this.

#### b) How much total money will you put into the account?
This is simply the total number of deposits multiplied by the amount of each deposit:
\[
\text{Total deposits} = PMT \times n = 320 \times 70 = 22,400
\]

#### c) How much total interest will you earn?
The total interest earned is the future value minus the total deposits:
\[
\text{Total interest} = FV - \text{Total deposits}
\]

Now, I'll calculate these values.### a) The amount you will have in the account in 35 years is approximately:
\[
\boxed{106,178.03} \, \text{dollars}
\]

### b) The total amount of money you will put into the account is:
\[
\boxed{22,400} \, \text{dollars}
\]

### c) The total interest you will earn is:
\[
\boxed{83,778.03} \, \text{dollars}
\]

Would you like further details on any part of this, or do you have any other questions?

### Related Questions:
1. What if the interest rate was compounded quarterly instead of semiannually?
2. How would the future value change if the deposits were made monthly instead of semiannually?
3. How does increasing the interest rate to 8% affect the future value of the account?
4. What happens to the future value if you deposit $500 every six months instead?
5. How would reducing the time to 25 years impact the total interest earned?

### Tip: When calculating future values, always ensure the compounding frequency matches the deposit frequency to use the correct interest rate per period.

You deposit $320 every six months into an account earning 7.6% interest compounded semiannually.
a) How much will you have in the account in 35 years?
b) How much total money will you put into the account?
c) How much total interest will you earn?

Learn how to calculate the future value of an annuity with $320 deposited every six months into an account earning 7.6% interest compounded semiannually. This solution covers compound interest, annuities, and total interest earned over 35 years. Suitable for Grades 10-12.

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