You would like to have $900,000 when you retire in 30 years. How much should you invest each quarter if you can earn a rate of 3.3% compounded quarterly?

a) How much should you deposit each quarter? Round your answer to the nearest cent.

$

b) How much total money will you put into the account? Round your answer to the nearest cent.

$

c) How much total interest will you earn? Round your answer to the nearest cent.

$

To solve this problem, we need to use the **future value of an ordinary annuity formula**, since you're making regular investments (deposits) over time. The formula is:

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

Where:
- $ FV $ is the future value (the amount you want when you retire, which is $900,000),
- $ P $ is the deposit made each period (the amount we are solving for in part a),
- $ r $ is the interest rate per period (quarterly),
- $ n $ is the total number of periods.

Let's break it down.

### Step 1: Identify the variables
- **Future Value (FV):** $900,000
- **Annual interest rate:** 3.3%, but we need the quarterly rate, so divide by 4:  
  $ r = \frac{3.3\%}{4} = \frac{0.033}{4} = 0.00825 $
- **Number of years:** 30, but since deposits are made quarterly, convert to quarters:  
  $ n = 30 \times 4 = 120 $ quarters

### a) How much should you deposit each quarter?
We are solving for $ P $, the quarterly deposit, using the formula rearranged:

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

Substitute the known values:

\[
P = \frac{900,000 \times 0.00825}{(1 + 0.00825)^{120} - 1}
\]

Let me calculate that for you.

\[
P = \frac{900,000 \times 0.00825}{(1.00825)^{120} - 1}
\]

Now calculating:

\[
P = \frac{7,425}{1.09096 - 1} = \frac{7,425}{0.09096} \approx 81,612.35
\]

Now, \( P \approx 81, max

Learn how to calculate the quarterly investment needed to accumulate $900,000 in 30 years with a 3.3% interest rate compounded quarterly. This detailed guide covers the future value of an annuity formula, compound interest, and step-by-step calculations. Ideal for college and university students.

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