Suppose you want to have $400,000 for retirement in 30 years. Your account earns 6% interest.
a) How much would you need to deposit in the account each month?
$
b) How much interest will you earn?

To solve this problem, we will use the **future value of an ordinary annuity formula**. The formula for the future value $ FV $ of regular monthly deposits $ PMT $ is:

\[
FV = PMT \times \frac{(1 + r)^n - 1}{r}
\]

where:
- $ FV $ is the future value (the amount you want to have for retirement), which is $400,000.
- $ PMT $ is the monthly deposit (the unknown value we need to find in part a).
- $ r $ is the monthly interest rate (6% annual interest, so $ r = \frac{6\%}{12} = 0.005 $ per month).
- $ n $ is the total number of deposits (30 years, so $ n = 30 \times 12 = 360 $ months).

### Part a) Finding the monthly deposit:

We need to rearrange the future value formula to solve for $ PMT $:

\[
PMT = \frac{FV \times r}{(1 + r)^n - 1}
\]

Substituting the given values:

\[
PMT = \frac{400,000 \times 0.005}{(1 + 0.005)^{360} - 1}
\]

Now, let's calculate it:

\[
PMT = \frac{400,000 \times 0.005}{(1.005)^{360} - 1}
\]

### Part b) Finding the total interest earned:

To find the total interest earned, we first calculate the total amount deposited, which is:

\[
\text{Total deposits} = PMT \times n
\]

The interest earned is the difference between the future value and the total deposits:

\[
\text{Interest} = FV - \text{Total deposits}
\]

I'll calculate the specific values for you next.### Part a) Monthly deposit:

You would need to deposit approximately **$398.20** each month to reach $400,000 in 30 years.

### Part b) Interest earned:

The total interest earned over 30 years would be approximately **$256,647.24**.

Let me know if you need any further clarification, and here are some related questions for deeper understanding:

1. How would the required monthly deposit change if the interest rate were higher or lower?
2. What if you wanted to save $500,000 instead of $400,000?
3. How would the calculation change if you contributed quarterly instead of monthly?
4. What would happen if you made a lump sum deposit at the beginning instead of monthly contributions?
5. How does compound interest impact the total amount saved over time?

### Tip:
Always start saving early—the power of compound interest grows significantly with time!

Suppose you want to have $400,000 for retirement in 30 years. Your account earns 6% interest. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

Find out how to save $400,000 for retirement by making monthly deposits in an account with 6% interest. Learn how much you need to deposit each month and how much interest you will earn over 30 years. This problem covers compound interest and annuity calculations, ideal for students in Grades 11-12.

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